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weyls law

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Chapter 1 of a Wiley-VCH volume, Mathematical Analysis of Evolution, Information, and Complexity (2009), by Wolfgang Arendt, Robin Nittka, Wolfgang Peter and Frank Steiner. It covers the history of Weyl's law from Rayleigh, Sommerfeld and Lorentz through Weyl's 1911-1915 work, remainder terms, the torus and Riemann surfaces, Robin boundary conditions, a heat-kernel proof, and Kac's question of hearing the shape of a drum. It appears to be reference material filed with Phil's Stakgold notes.

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1 1 Weyl’s Law: Spectral Properties of the Laplacianin Mathematics and Physics Wolfgang Arendt, Robin Nittka, Wolfgang Peter,1)Frank Steiner 1.1 Introduction Weyl’s law is in its simplest version a statement on the asymptotic growth of the eigenvalues of the Laplacian on bounded domains with Dirichlet and Neumann boundary conditions. In the typical applications in physics one deals either with the Helmholtz wave equation describing the vibrations of a string, a membrane (drum), a mass of air in a concert hall, the heat radiation from a body in ther- mal equilibrium, the ﬂuctuations of the gravitational ﬁeld in cosmology, or the Schrödinger equation of a quantum system which may be a simple quantum bil- liard, an atom, a molecule or a compound nucleus. While Weyl’s seminal work was provoked by the famous black body radiation problem, i.e. an electromagneticcavity problem, in particular by a conjecture put forward independently by Som- merfeld and Lorentz in 1910, Weyl’s law has its roots in music and, respectively, acoustics. Already in 1877, Lord Rayleigh had, in his famous book, “The Theory ofSound” treated the overtones of a violin or piano string and the natural notes of an organ pipe or the air contained within a room. For a room of cubical shape he de- rived the correct asymptotic behavior for the overtones. The trick used by Rayleigh to count the vibrational modes was to reduce the problem to a three-dimensional lattice-point problem from which he could derive that the number of overtones withfrequency between νandν+dνgrows at high frequencies, ν→∞ , asymptotically asV·ν 3(Weyl’s law!), where Vis the volume of the room or analogously of an organ pipe. In 1900, Rayleigh realized that the same formula can be applied to a physical-ly completely different, but mathematically equivalent problem: the heat radiation from a body in thermal equilibrium with radiation, the importance of which had been pointed out already in 1859 by Kirchhoff. The amount of energy emitted by a body is determined by the high-frequency spectrum of standing electromagnetic waves and that spectrum should be essent ially the same as for the high overtones of an organ pipe, say. 1)Corresponding author. Mathematical Analysis of Evolution, Information, and Complexity. Edited by Wolfgang Arendt and Wolfgang P. Schleich Copyright © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40830-6 21W e y l ’ s L a w In the crucial black body radiation experiments carried out in the 1890s, which led Planck, in 1900, to the famous radiation law named after him and to the dis- covery of quantum theory, one measures the energy density emitted rather than the energy itself, i.e. the energy divided by the volume V. Thus it follows from Rayleigh’s asymptotic result V·ν3, derived for a cubical geometry, that the volume factor is canceled if one considers the energy density, in accordance with the ex- pectations using physical arguments and, very importantly, in complete agreement with the experimental ﬁndings. It was realized, however, and emphasized by Som- merfeld and Lorentz in 1910 that there arises the mathematical problem to prove that the number of sufﬁciently high overtones which lie between νandν+dνisinde- pendent of the shape of the enclosure and is simply proportional to its volume .I tw a s a great achievement when Weyl proved in 1911 that, by applying the Fredholm–Hilbert theory of integral equations, the Sommerfeld–Lorentz conjecture holds! From then on, Weyl himself and many other mathematicians and physicists have studied and generalized Weyl’s law by deriving corrections or even complete ex- pressions for the remainder term. The Weyl asymptotics as discussed above in the three-dimensional case is par- ticularly striking if considered as an inv erse spectral problem, which became quite popular after Kac’s talk in 1966 entitled “Can one hear the shape of a drum?”. Subsequently, several partially afﬁrmative answers to this question have been given. But, on the other hand, a particular ly striking counterexample by Gordon, Webb and Wolpert from 1992 shows that not all geometric information about the domain is contained in the spectrum. This chapter is organized as follows. In Section 1.2 we give a historical account of Weyl’s law. The following two chapters are devoted to Weyl’s law with remain-der term and the statistical behavior of th e latter using trace formulae. We discuss the Laplacian on the torus in Section 1.3 and the Laplace–Beltrami operator on Riemann surfaces in Section 1.4. Then two generalizations of Weyl’s law to Robinboundary conditions and for unbounded quantum billiards are presented in Sec- tion 1.5. In Section 1.6 we provide a self-contained proof of Weyl’s law for bounded Euclidean domains and Dirichlet boundary conditions; the case Weyl himself treat- ed in his ﬁrst article on this topic. However, we follow a different, very fruitful, ap- proach based on heat kernels. In Section 1.7 we give an account on what is knowntoday about Kac’s question. In particular we show under which precise regularity assumptions one can hear whether a body is a ball. 1.2 A Brief History of Weyl’s Law 1.2.1 Weyl’s Seminal Work in 1911–1915 In February 1911, David Hilbert presented to a meeting of the Royal Academy of Sciences of Göttingen a short note [1] written by Hermann Weyl. This note contains 1.2 A Brief History of Weyl’s Law 3 for the ﬁrst time a rigorous proof of the asymptotic behavior of the eigenvalues λn of the two-dimensional (scalar) Helmholtz wave equation (Δ+λ)u(x)=0 (x∈Ω) (1.1) satisfying the Dirichlet boundary condition u(x)=0 (x∈∂Ω), (1.2) where Ω∈R2is an arbitrary bounded domain with area |Ω|,b o u n d a r y∂Ω,a n d Δ=∂2 ∂x2 1+∂2 ∂x2 2 denotes the Laplacian onΩ. The “membrane problem” (1.1, 1.2) has nontrivial solutions uonly for a discrete set of eigenvalues⎫braceleftbigλn⎫bracerightbig n∈N. The corresponding eigen- functions{un}n∈Nprovide an orthonormal basis of L2(Ω), and we may enumerate the eigenvalues in increasing order 0 < λ1uλ2u... Note that the eigenvalues λncan have different physical interpretations. In the case of a vibrating membrane with clamped edge, where udescribes the transversal vibrations of the membrane, one has λn=k2 n,w h e r e kn=(2π/c)νni st h ew a v en u m - ber which is proportional to the eigenfrequency νn,i . e .t ot h ep u r et o n e sw h i c ht h e membrane is capable of producing. The constant cis the sound velocity depend- ing on the physical properties of the membrane, i.e. on the mass density and the tension under which the membrane is held. In the case of quantum mechanics, where uis the wave function having the meaning of a probability amplitude, Equa- tion (1.1) is the time independent Schrödinger equation of a freely moving particlewith mass m,a n d λ n=⎫parenleftBig 2m//planckover2pi12⎫parenrightBig Enis proportional to the quantal energy levels En. (/planckover2pi1denotes Planck’s constant.) Since explicit analytical expressions for the eigenvalues are known only for a few membranes with simple shape (for example equilateral triangles, rectangles, cir- cles) and their numerical computation for large nis very difﬁcult for general do- mains, it is natural to study thei r asymptotic distribution as n→∞ . Applying the Fredholm–Hilbert theory of linear integral equations, Weyl proved that lim n→∞n λn=|Ω| 4π. (1.3) Deﬁning the counting function N (λ): =#⎫braceleftbigλnuλ⎫bracerightbig, (1.3) is equivalent to the asymp- totic behavior N(λ)=|Ω| 4πλ+o(λ) (λ→∞ ). (1.4) These results are now called Weyl’s law . Shortly afterwards, Weyl submitted three papers [2–4] which contain the details of his proof, a generalization of (1.4) to the three-dimensional scalar wave equation ( Ω⊂R3), N(λ)=|Ω| 6π2λ3/2+o(λ3/2) (λ→∞ ), (1.5) 41W e y l ’ s L a w and the extension to the vector Helmholtz wave equation describing the vibrations of the electric ﬁeld Ein an empty cavity Ωwith perfectly reﬂecting walls ∂Ω.A s we shall discuss in more detail in Sections 1.2.3–1.2.8, it is exactly this electro- dynamic cavity problem, studied extensively in those years by theoretical physi-cists, which was one of the open problems that provoked Weyl to start his seminal work. The electromagnetic cavity problem requires of the electric ﬁeld vector boundary conditions which are more involved than the simple boundary condition (1.2). In his ﬁrst papers [2,4] on this problem, Weyl considered some nonphysical boundaryconditions; following a suggestion of Levi–Civita, in his subsequent paper [5] the correct boundary conditions E~n=0a n d∇E=0o n∂Ωwere taken into account. However, the Gauss law ∇E=0o n Ωi.e. throughout the interior of the cavity, was still discarded. In his paper [5] Weyl went even one step further and conjectured the existence of a second asymptotic term N(λ)=|Ω| 4πλ∓|∂Ω| 4π√ λ+o⎫parenleftBig√ λ⎫parenrightBig (λ→∞ ) (1.6) for the two-dimensional problem (1.1), where |∂Ω|denotes the length of the cir- cumference of the membrane and the (–) sign refers to the Dirichlet boundary condition (1.2) and the (+) sign to the Neumann boundary condition (∂u/∂n=0 , x∈∂Ω), and N(λ)=|Ω| 6π2λ3/2∓|∂Ω| 16πλ+o(λ)( λ→∞ ) (1.7) for the three-dimensional case, where |∂Ω|now denotes the surface area of ∂Ω.T h e formulae (1.6) and (1.7) became known as Weyl’s conjecture . It was justiﬁed (under certain conditions on Ω) by Ivrii [6] and Melrose [7] only in 1980. In 1915, Weyl concluded his work on the asymptotic behavior of eigenvalues with a study [8] of the elastic vibrations uof a homogeneous body with volume |Ω|which are determined by the solutions of the differential equation BΔu+Agrad div u+λu= 0 . (1.8) Here λis related to the frequency νbyλ=( 2πν)2,a n d A,Bare positive constants (related to the Lamé coefﬁcients characteri zing the elastomechanical properties of the body). Imposing the boundary conditions ∇u=0a n d n~u= 0 on the boundary ∂Ωof the body, Weyl proved for arbitrary shapes of the body N(λ)=|Ω| 6π2Fλ3/2+o⎫parenleftBig λ3/2⎫parenrightBig (λ→∞ ), (1.9) where Fis a function of the elastic constants, F=2 / c3 T+1 /c3 Lwith cT=√ Bthe transverse and cL=√ A+Bthe longitudinal sound velocity. The Weyl formulae (1.3)–(1.7) and (1.9) are very striking since they tell us that the coefﬁcient of the leading asymptotic term is determined only by the area, resp., 1.2 A Brief History of Weyl’s Law 5 the volume of the domain and is independent of its shape. That is one can “hear” the area of a drum (a membrane, held ﬁxed along its boundary) or the volume of a cavity following Marc Kac’s [9] acoustic rephrasing of the problem. We refer to Sections 1.2.8, 1.3.6, 1.3.7 and 1.7 for more details on Kac’s problem. In his papers [1, 2], Weyl mentions that his asymptotic formulae “provide in par- ticular the solution to a problem the impor tance of which has recently been empha- sized by Sommerfeld and Lorentz” (here and in the following citations, we employ free translations from the original German). 1.2.2 The Conjecture of Sommerfeld (1910) In September 1910, Arnold Sommerfeld delivered a talk at the “82. Naturforscher- Versammlung” in Königsberg [10]. In this talk he studied the solution of the inho- mogeneous differential equation in one, two and three dimensions (ΔΩ+λ)υ=f (1.10) describing forced vibrations. For this purpose, he introduced the resolvent kernel (called the “ Green function ”) GΩ(x,y;λ): =⎫summationdisplay mum(x)um(y) λ–λm⎫parenleftbigx,y∈Ω⎫parenrightbig, (1.11) where um(x) are the eigenfunctions of (1.1). In addition to the Dirichlet boundary condition (1.2), Sommerfeld also considered Neumann boundary conditions and, “as in the theory of heat conduction”, Robin boundary conditions (h1u+h2(∂u/∂n)= 0o n∂Ω,h1,h2constant or arbitrary functions on ∂Ω). A formal application of the operator (ΔΩ+λ)toGΩ(x,y;λ) (acting on the ﬁrst argument x)g i v e s (ΔΩ+λ)GΩ(x,y;λ)=⎫summationtext mum(x)um(y), and Sommerfeld remarks that this expression is zero for x=/y, but is inﬁnite for x=y. He calls this expression “Zackenfunk- tion” (“spike function”), the physical interpretation of it is a “unit source” (point source). This is, of course, an early attempt to introduce the Dirac delta distribu- tion, since the above expressi on is nothing other than the completeness relation of the orthonormal eigenfunctions um∈L2(Ω) ⎫summationdisplay mum(x)um(y)=δ(x–y) . (1.12) The solution of the inhomogeneous problem (1.10) then reads υ(x)=⎫integraldisplay ΩGΩ(x,y;λ)f(y)dy. (1.13) This result is quite remarkable since it allows one to reduce the problem (1.10) of the forced vibrations on Ωto the problem (1.1) of the free vibrations on the same domain Ω. “As some material is fully characterized by its spectral lines i.e. 61W e y l ’ s L a w by its free vibrational frequencies, so is also the behavior of a domain for arbitrary vibratory motions completely determined by the spectrum of its free vibrational possibilities.” [10] Sommerfeld [10] then discusses the convergence of the series (1.11): “In the one-dimensional case the series (1.11) is absolutely convergent, in the two- and three-dimensional case only conditionally convergent. In the ﬁrst case the growth of the denominator λ–λmi ss u f ﬁ c i e n tf o rc o n v e r g e n c e , s i n c e[ ...]t h ed e n o m i n a - tor becomes inﬁnite, as m2. In the latter cases, will λm,n,r e s p . λm,n,l, equally well always approach inﬁnity quadratically in m,n,r e s p . l, as I do not doubt (1). How- ever, such a growth is not sufﬁcient, as is well known, to render the double sum over m,nresp., the triple sum over m,n,l, convergent. Rather, here the change of sign of the nominator um,n(x)um,n(y)p l a y sa ne s s e n t i a lr o l ea si ti sg u a r a n t e e d in its natural ordering by the oscillatory character of the series.” In the above foot- note (1) Sommerfeld adds: “The general and rigorous proof of this asymptotic be- havior of the eigenvalues seems to me an i mportant and grateful mathematical problem.” Here we have Sommerfeld’s conjecture which was one of the motives for the pio- neering work of Weyl. Sommerfeld considers, as an application of his method, the “problem of acous- tics of rooms” (using Neumann boundary conditions on the walls), and he empha-sizes that his “method is fundamentally di fferent from the classical method intro- duced in mathematical physics by Fourier”, whereby he refers to Fourier’s famous work “Théorie [analytique] de la chaleur” from 1822. Here two remarks are in order. i) In his conjecture, Sommerfeld takes for grant- ed that the eigenvalues depend for example in the three-dimensional case on threeintegers (“quantum numbers”) (m,n,l)i.e.λ m,n,l,“ w h i c he a c hr u nf r o m0t o ∞and have the meaning of the number of divisions of the domain by nodal surfaces with respect to the three dimensions. (One may think of the known cases of the paral-lelepiped or the sphere.)” Consequently, he considers the sum in Equation (1.11) as a triple sum running over m,n,a n d l.I ti sk n o w n ,h o w e v e r ,t h a tt h es i t u a t i o n envisaged by Sommerfeld holds in general only for domains for which the wave equation (1.1) is separable in coordinates⎫parenleftbigq 1,q2,q3⎫parenrightbigi.e. where the particular solu- tions can be written as a product um,n,l=um(q1)υn(q2)wl(q3). In the generic case, however, i.e. for a cavity with arbitrary shape, the eigenvalues depend on a sin- gle positive integer only, which just counts the eigenvalues in increasing order, as assumed in (1.11). ii) Sommerfeld points out that the Green function (1.11) “degen-erates ( G=∞)” at the points λ=λ m“according to the general resonance principle , except at special positions of the point source, if, for example, um(x)=0 ,a n dt h e r e - fore the critical eigenvibration is not excited.” In the physics literature, the Green function (1.11) is considered as a distribution by adding a small positive imaginary part ( ε>0 )t o λ∈R, i.e. one considers the kernel of the regularized resolvent operator (λ+iε+Δ)–1. We refer also to Sections 1.3.4, 1.4 and 1.6 where expres- sions similar to (1.11) are given for the Green’s function, for example for the heat kernel. 1.2 A Brief History of Weyl’s Law 7 1.2.3 The Conjecture of Lorentz (1910) At the end of October 1910, i.e. one month after Sommerfeld’s talk, Hendrik An- toon Lorentz delivered six lectures at Göttingen under the title “Old and new prob- lems of physics” published [11] from notes taken by Max Born and talked over with Lorentz. See also the letter dated October 28, 1929, sent by Max Born to Einstein, together with Born’s comment to this letter from 1969 [12]. Lorentz, who had already received in 1902 the second Nobel prize in physics, was at that time probably the most famous living theoretical physicist. He was invited to Göttingen by the Wolfskehl commission of the Göttingen Academy which was to confer a prize for proving Fermat’s last theorem. As long as the prize was not awarded, the proceeds from the principal should be used to invite eminent scientists to lecture at Göttingen. (Paul Wolfskehl (1856–1906), original- ly a physician, fell ill with multiple sclerosis and then became a mathematician working mainly on number theory; he taught at the Technical University of Darm- stadt. The ﬁrst Wolfskehl lecture was given by Poincaré in 1908 and later lectureswere given, among others, by Einstein and Planck; in 1922 Niels Bohr delivered his legendary Wolfskehl lectures on his theory of the atom which later became known as the “Bohr-Festspiele”. In 1997 the Wolfskehl prize was given to AndrewWiles.) In his last three lectures, Lorentz discussed “the phenomenon of radiating heat”. The end of the fourth lecture reads as follo ws. “In conclusion there is a mathemat- ical problem worth mentioning which perhaps will arouse the interest of math- ematicians who are present. It originates in the radiation theory of Jeans. In anenclosure with a perfectly reﬂecting surface there can form standing electromag- netic waves analogous to tones of an organ p ipe; we shall conﬁne our attention only to the very high overtones. Jeans asks for the energy in the frequency interval d ν. To this end he ﬁrst of all calculates the number of overtones which lie between the frequencies νandν+dνand then multiplies this number by the energy which be- longs to the frequency ν, and which according to a theorem of statistical mechanics is the same for all frequencies. In this manner he gets, indeed, the correct law of the radiation at long wavelengths.” “It is here that there arises the mathematical problem to prove that the number of sufﬁciently high overtones which lie between νandν+dνis independent of the shape of the enclosure and is simply proportional to its volume. For several simpleshapes, for which the calculation can be carried out, this theorem will be veriﬁed in a Leiden dissertation. There is no doubt that it holds in general even for multiply connected spaces. Analogous theorems will also hold for other vibrating structures like elastic membranes and air masses etc.” Weyl, who was present at Lorentz’s lectures, writes in a footnote of his second paper [2]: “Lorentz has stated the theorem proven here in Section 1.6 as a plausi- ble conjecture on physical grounds. The simplest cases, for which the proof can be achieved by a direct computation of the eigenvalues, are treated in the Leidendissertation of Fräulein Reudler.” Actually Johanna Reudler veriﬁed [13] that the 81W e y l ’ s L a w asymptotic number of modes depends only on the volume for three special cases, the parallelepiped, the sphere, and the cylinder. There is an apocryphal report that Hilbert predicted that the theorem would not be proved in his lifetime [9]. Well, as we have seen, he was wrong by many, manyyears. Forty years after Lorentz’s lectures, Weyl came back to the “eigenvalue prob- lem” [14]: “H.A. Lorentz had impressed upon the mathematicians the urgency for physics of a settlement of this question. For a pupil of Hilbert around 1910 it was natural to visualize the question as one concerning integral equations.” In the nextsection of this paper [14] Weyl draws atte ntion to a more difﬁcult problem by say- ing: “The physicist will not be satisﬁed wi th a knowledge of the asymptotic behav- ior of the eigenvalues alone; that of the eigenfunctions should also be investigated.”And Weyl mentions in this connection Carleman’s law, see Section 1.4.3. Further on in this paper we read the following sentences: “I feel that these infor- mations about the proper oscillations of a membrane, valuable as they are, are still very incomplete. I have cert ain conjectures on what a complete analysis of their asymptotic behavior should aim at; but since for more than 35 years I have madeno serious attempt to prove them, I think I had better keep them to myself.” 1.2.4 Black Body Radiation: From Kirchhoff to Wien’s Law The study of the heat radiation from a body in thermal equilibrium with radiation has played an eminent role in the history of physics and mathematics for it led Planck in 1900 to the discovery of the quantum theory and Weyl in 1911 to a ﬁrstproof of the eigenvalue asymptotics. (Ther e are several historical studies on this subject. Here we rely on the excellent account given by Pais [15] who, however, does not discuss the aspects concerning Weyl’s law.) The importance of the heatradiation problem was realized already in 1859 by Gustav Kirchhoff [16]. Let the radiation energy which a body absorbs be converted to thermal energy only, not to any other energy form, and denote by E νdνthe amount of energy emitted by the body per unit time per cm2in the frequency interval d ν. (Actually, Kirchhoff uses the wavelength λinstead of the frequency ν.) Furthermore, let Aνbe its absorption coefﬁcient for frequency ν. Kirchhoff showed that the ratio Eν/Aνis a universal function which depends only on νand the equilibrium (absolute) temperature T and is independent of the shape and any other properties of the body i.e. Eν Aν=J(ν,T) . (1.14) A general proof of (1.14) was given much later by Hilbert using the theory of linear integral equations and his “axiomatic method” [17–19]. Kirchhoff called a body perfectly black or just black for short if Aν=1 .T h u s J(ν,T) is the emitted power of a black body which can be measured if we assume (with Kirchhoff) that a perfect black body can be realized by “a space enclosed by bod-ies of equal temperature, through which no radiation can penetrate” [16], i.e. by 1.2 A Brief History of Weyl’s Law 9 an enclosure with perfectly reﬂecting walls. Kirchhoff challenged theorists and ex- perimentalists alike: “It is a highly important task to ﬁnd this function J.G r e a t difﬁculties stand in the way of its experimental determination; nevertheless, there appear grounds for the hope that it can be found by experiments because there isn od o u b tt h a ti th a sas i m p l ef o r m ,a sd oa l lf u n c t i o n sw h i c hd on o td e p e n do nt h e properties of individual bodies and whi ch one has become acquainted with before now.” [16] It is worthwhile to mention that Kirchhoff reports in the same paper about his experiments carried out with sunlight, in terpreted as heat radiation of very high temperature produced in the atmosphere of the sun, and about his discovery of sodium there. He concludes: “Thus a way is found to ascertain the chemical nature of the atmosphere of the sun, and the same way promises also some informationon the chemical nature of the brighter ﬁxed stars.” [16] It will be seen later that Kirchhoff’s statement about the shape-independence of J(ν,T) implicitly implies part of Weyl’s law (1.7) stating that the leading term of the counting function is proportional to the volume V:=|Ω|of the cavity by which the black body is realized. At this point it is convenient to express Jin terms of the spectral energy density ρ(ν,T) which gives the energy per unit volume of the heat radiation in thermal equilibrium at temperature Tfor frequency ν: J(ν,T)=c 8πρ(ν,T) (1.15) (cis the velocity of light in vacuo .) It was conjectured by Josef Stefan on experimen- tal grounds in 1879 and proved theoretically by Ludwig Boltzmann in 1884 [20]that the mean total energy /angbracketleftE/angbracketright(T) radiated by the black body is given by the Stefan– Boltzmann law /angbracketleftE/angbracketright(T)=V∞⎫integraldisplay 0ρ(ν,T)dν=VσT4, (1.16) where σis a universal constant (now called the Stefan–Boltzmann constant ,w h o s e universal value could only be calculated after the discovery of Planck’s law). Boltz-mann’s proof involves thermodynamics and the electromagnetic theory of Maxwell according to which the mean radiation pressure⎫angbracketleftbigp⎫angbracketrightbigobeys the equation of state ⎫angbracketleftbigp⎫angbracketrightbig=1 3/angbracketleftE/angbracketright V. Important progress was made by Wilhelm Wien who proved in 1893 that ρ(ν,T) has to be of the following form ( Wien ’s displacement law ) [21] ρ(ν,T)=ν3f(ν/T) . (1.17) Thus the heat radiation problem was reduced to determining, instead of J(ν,T), the universal function f(x) of the single scaling variable x=ν/T.( O b v i o u s - ly, from (1.17) one immediately derives the Stefan–Boltzmann law (1.16) with σ=⎫integraltext∞ 0x3f(x)dx.) Over the years, many proposals for the correct form of fhave appeared, see for example the four different forms discussed in [22]. Still, 20 years 10 1W e y l ’ s L a w later, Einstein wrote in 1913: “It would be edifying if we could weigh the brain sub- stance which has been sacriﬁced by the theo retical physicists on the altar of this universal function f; and the end of these cruel sacriﬁces is not yet in sight!” [23] In 1896 Wien proposed [24] the exponential form fW(x): = αe–/betatwox(α,/betatwopositive constants), that is ( Wien’s law ) ρW(ν,T)=αν3e–/betatwoν/T. (1.18) At the same time, Friedrich Paschen carried out precise measurements [22, 25] in the near-infrared (for wavelengths λ=c/ν= 1–8 μm,T= 400–1600 K) which were in such a good agreement with Wien’s law (1.18) that he concluded: “It would seem very difﬁcult to ﬁnd another fu nction of the two variables νandT[Equation (1.18)] that represents the observations with as f ew constants.” [25]. Thus it appeared that Wien’s law was the ﬁnal answer to the black-body problem. 1.2.5 Black Body Radiation: Rayleigh’s Law In June 1900, that is several months before Planck’s revolutionary discovery, Lord Rayleigh made another proposal [26] which for the ﬁrst time introduces into the black body radiation problem the density of states D (ν), that is the density of the vibrational modes of a black body cavity. T his step played an important role since Rayleigh’s proposal relies on an assumption which 10 years later led to the conjec- tures of Sommerfeld and Lorentz, and ﬁnally to Weyl’s law (as already discussed inSections 1.2.1–1.2.3). Rayleigh’s starting point is the observation that Wien’s law (1.18) “viewed from t h e t h e o r e t i c a l s i d e [ ...] a p p e a r s t o m e l i t t l em o r e t h a n a c o n j e c t u r e ...” , a n d “ ... the law seems rather difﬁcult of acceptance, especially the implication that as the temperature is raised, the radiation of give n frequency approaches a limit.” [26] In- deed, one obtains from (1.18) lim T→∞ρW(ν,T)=αν3.H ec o n t i n u e s :“ t h eq u e s t i o ni s one to be settled by experiment; but in the meantime I venture to suggest a modi- ﬁcation of (1.18), which appears to me more probable ap r i o r i .” [26] Without further explanation, Rayleigh assumes, ﬁrst of all, that the equilibrium distribution ρis proportional to the density of th e vibrational modes of the cavity per unit volume, that is ρ(ν,T)~D(ν)/V,w h e r e D(ν): =dN dνwithN(ν): =N⎫parenleftBig (2π/cν)2⎫parenrightBig and N(λ) denotes the leading asymptotic term of the counting function expressed in terms of the frequency ν. Secondly, he assumes according to the “Boltzmann– Maxwell doctrine of the partition of energy” (that is the equipartition theorem) that “ e v e r y m o d e o f v i b r a t i o n s h o u l d b e a l i k e f a v o r e d ...” . T h u s h e a s s u m e s t h a t “ t h e energy should be equally divided among all the modes. . . . Since the energy in each mode is proportional to T” (that is proportional to kBTin modern notation, where kBdenotes Boltzmann’s constant, introduced actually only later by Planck!), 1.2 A Brief History of Weyl’s Law 11 Rayleigh’s assumption amounts to ρ(ν,T)~( D(ν)/V)·T. As an “illustration” he ﬁrst considers “the case of a stretched str ing vibrating transversely” and derives the correct Weyl asymptotics N(λ)~√ λ,t h a ti sN(ν)~νand thus D(ν) = constant (“when νis large enough”). Then he continues: “When we pass from one di- mension to three dimensions, and consider for example the vibrations of a cubic mass of air, we have (‘Theory of Sound’, paragraph 267) as the equation for ν2, ν2=p2+q2+r2,w h e r e p,q,rare integers representing the number of subdivisions in the three directions. If we regard p,q,ras the coordinates of points forming a cu- bic array, νis the distance of any point from the origin. Accordingly the number of points for which νlies between νandν+dν, proportional to the volume of the corresponding spherical shell, may be represented by ν2dν, and this expresses the distribution of energy according to the Boltzmann–Maxwell law, so far as regardsthe wavelength or frequency. If we apply this result to radiation, we shall have, since the energy in each mode is proportional to T,Tν 2dν.... ”[ 2 6 ]T h u sR a y l e i g h obtains (apart from the nume rical coefﬁcient) the correct Weyl asymptotics for a three-dimensional cavity, that is N(λ)~Vλ3/2, see (1.5), which leads to N(ν)~Vν3 orD(ν)/V~ν2, and thus to ρREJ(ν,T)=c1ν2T, (1.19) which is commonly known as the Rayleigh–Jeans law but which should rather be referred to as the Rayleigh–Einstein–Jeans law . (We shall discuss below Einstein’s, Rayleigh’s second, and Jeans’ derivation of (1.19) which includes also the explicitexpression for the coefﬁcient c 1.) Here several remarks are in order. i) It is obvious that Rayleigh did not worry about the fact that he used the scalar wave equation in his derivation of the modeasymptotics, by referring to his famous book on acoustics [27], instead of the vector Maxwell equations, which were studied only later by Weyl [2, 4, 5]. ii) In deriving the vibrational mode asymptotics for a cubical box, Rayleigh takes for granted that the result N(ν)~Vν3holds for any shape of the cavity and thus concludes that D(ν)/Vis independent of the shape. In other words, Rayleigh assumes to be true what 10 years later was formulated as a conjecture by Sommerfeld and Lorentz. iii) Although his derivation of D(ν)/V~ν2holds only asymptotically for ν→∞ ,h e derives from this result the law (1.19) stating that it may have the proper form whenν/Tis small! iv) Rayleigh observes that (1.19) is of the general scaling form (1.17) (with f REJ(x)=c1/x), and he regards this “as some conﬁrmation of the suitability of (1.19).” v) Without further comment, Rayleigh writes in his paper [26]: “If we introduce the exponential factor, the complete expression will be ρR=c1ν2Te–c2ν/T.” (1.20) It is this expression which became known as the Rayleigh law .v i )T h e r ei sn o doubt that Rayleigh must have realized that (1.19) is entirely unacceptable since the quadratic dependence of νleads to a physically meaningless divergence (later called “ultraviolet catastrophe” by Ehrenfest) of the total radiation energy (see (1.16)). Of course, by multiplying with the exponenti al “convergence factor” taken over from 12 1W e y l ’ s L a w Wien’s law (1.18), the divergence is avoided and the Stefan–Boltzmann law (1.16) holds with σ=2c1/c3 2. At the end of his paper Rayleigh writes [26]: “Whether (1.20) represents the facts of observation as well as (1.18) I am not in a position to say. It is to be hopedthat the question may soon receive an answer at the hands of the distinguished experimenters who have been occupied with this subject.” We can assume that Rayleigh was well informed about the two teams working in Berlin on black body radiation experiments. The ﬁrst of these, Otto Lummer and Ernst Pringsheim, had already shown in February 1900 that Wien’s law (1.18) failsin the wavelength region λ= 12–18 μm( f o r T= 300–1650 K) [28]. The second team, Heinrich Rubens and Ferdinand Kurlbaum, presented their measurements in the even further infrared ( λ= 30–60 μm,T= –188–1500 ◦C) to the Prussian Academy on October 25, 1900 [29]. In a famous ﬁgure, they plotted ρas a function of Tat the ﬁxed wavelength λ= 51.2 μm and compared their data with some theoretical curves. One of these was the Wien curve, another the Rayleigh curve. Both curves did not work! But then we read in the paper [29] that they had compared their data with a “ﬁfth formula, given by Herr M. Planck after our experiments had alreadyb e e n c o n c l u d e d ...” a n d w h i c h “ r e p r o d u c e s o u r o b s e r v a t i o n w i t h i n t h e l i m i t s o f error.” 1.2.6 Black Body Radiation: Planck’s Law and the Classical Limit According to Pais [15], Planck probably discovered his law in the early evening of Sunday, October 7, 1900, Rubens and his wife had called on the Plancks on theafternoon of that day. In the course of conversation, Rubens mentioned to Planck that he had found ρ(ν,T) to be proportional to Tfor small ν.P l a n c kw e n tt ow o r k after the visitors left and found an interpol ation between his results and Wien’s law, Equation (1.18). He communicated his formula by postcard to Rubens the same evening and stated it publicly [30] in a discussion remark on October 19, following the presentation of a paper by Kurlbaum. Expressed in notations introduced by Planck two months later, Planck’s law reads: ρ P(ν,T)=8πhν3 c31 ehν/kBT–1, (1.21) where hdenotes Planck’s constant and kBis Boltzmann’s constant. Let us consider two limits of Planck’s law. First, in the high-frequency or low- temperature regime, which is now identiﬁed as the quantum limit in which the photon energy hνis much larger than the thermal energy kBT,t h a ti s hν/kBT>> 1, we recover Wien’s law (1.18) with α=( 8 πh)/c3and /betatwo=h/kB.T h i se x p l a i n s why Paschen’s experiments [22, 25], for which hν/kBTW15 holds, were in such a good agreement with Wien’s law, as already mentioned. At the other extreme of low frequency or high temperature, hν/kBT<< 1, which is obtained from Planck’s law in the formal limit when Planck’s constant approaches zero, h→0, and is now identiﬁed as the semiclassical limit , we recover the Rayleigh–Einstein–Jeans 1.2 A Brief History of Weyl’s Law 13 law (1.19) ρP(ν,T)=8πν2 c3(kBT)⎫bracketleftbig1+O(h)⎫bracketrightbig(h→0) . (1.22) A comparison with (1.19) gives the correct value for the constant c1left undeter- mined by Rayleigh, that is c1=8πkB c3=8π c3R NA, (1.23) which does not depend on h,a n dw h e r e Ris the gas constant and NAis Avogadro’s number. Since our main interest here is to understand the role played by Weyl’s law, we are not discussing at this point the argume nts using “resonators” which led Planck to his formula. (Planck’s original deriva tion does not refer to the vibrations of the c a v i t ya n dt h u sd o e sn o ti n v o l v et h ed e n s i t y of states.) Using the fact that the correct formula for the radiating heat, that is Planck’s formula, in the classical limit exactly takes the form of the Rayleigh–Einstein–Jeans law, we can interpret the latter inpurely classical terms by making the general ansatz (valid only for h=0 ! ) ρ class(ν,T): = l i m V→∞⎫parenleftBiggDem(ν) V⎫parenrightBigg kBT, (1.24) where Dem(ν) denotes the density of states of the electromagnetic vibrations in ac a v i t yo fv o l u m e V. Furthermore, we have taken care of the fact that the predic- tions of thermodynamics involve the so-called thermodynamic limit V →∞ .H e r e Dem(ν): =dNem(ν) dνwithNem(ν)=2N(ν)=2 N⎫parenleftBig (2π/cν)2⎫parenrightBig , where N(λ) denotes the two asymptotic terms of the counting function (1.7) for the three-dimensional case, and the factor 2 comes from the two polarizations of the photon. We then obtain Nem(ν)=V8π 3c3ν3+O|∂Ω|(ν2) (1.25) which leads to lim V→∞⎫parenleftBiggDem(ν) V⎫parenrightBigg =8π c3ν2(1.26) since lim V→∞(|∂Ω|/V)=0 ,w h e r e|∂Ω|denotes the surface area of the cavity. In his famous book, originally published in 1928 in German under the title “Gruppentheorie und Quantenmechanik” [31, p. 103–104 and p. 402] Weyl treatsthe black-body radiation and proves that it “is mathematically equivalent to a sys- tem of inﬁnitely many oscillators.” He then states, without proof: “For high fre- quencies νthere are approximately V⎫parenleftBig 8πν 2dν/c3⎫parenrightBig modes of oscillation in the fre- quency interval ν,ν+dν. We are interested above all in the limiting case of an 14 1W e y l ’ s L a w inﬁnitely large cavity; the spectrum then becomes continuous and our formula for the density of frequencies becomes exact.” In a footnote he adds: “This result is easily obtained by elementary methods for a rectangular parallelepiped. For the general proof see H. Weyl [4, 5, 8].” It is clear that the limit V→∞ is an idealiza- t i o nw h i c hc a nn e v e rb er e a l i z e di nap h y s ical experiment. Rather the “assumption must always hold that the linear dimensions of all cavities considered and also the curvature of the radii of all surfaces considered must be large compared with the wavelengths of the radiation. Then we are allowed, without making a noticeable error, to neglect the inﬂuences of the form of the boundaries caused by diffrac-tion.” [32, p. 2] Inserting (1.26) into (1.24), one obtains ρ class(ν,T)=8πkB c3ν2T (1.27) which is precisely the Rayleigh–Einstein –Jeans law (1.19) with the correct power behavior in νand the same coefﬁcient (1.23) as obtained from the exact Planck formula. It is thus seen that heat radiation (in the classical limit) is indeed inde-pendent of the shape of the cavity due to Weyl’s law and the performance of the thermodynamical limit. As shown above, Planck’s radiation law (1.21) from October 1900 can be consid- ered as a simple interpolation formula which smoothly interpolates between the Rayleigh–Einstein–Jeans law (1.27) and Wien’s law (1.18). In fact, it differs from Wien’s law only by the –1 in the denominator. It has rightly been said [15], that even if Planck had stopped after October 19, he would forever be remembered as the discoverer of the radiation law. It is a true measure of his greatness that he went further. He wanted to interpret (1.21). That made him to discover the quantum theory. Already on December 14, 1900, Planck presented a theoretical derivation of his formula to the German Physical Socie ty in Berlin [33] and shortly afterwards (7 January 1901) submitted his famous paper [34]. More and more precise mea- surements carried out during the following years established Planck’s formula as the correct phenomenological law of black body radiation. It is thus quite astonish- ing to learn that several excellent theoreti cal physicists, in particular Lorentz, Lord Rayleigh, and Jeans, worked on alternative theories leading to formulae differentfrom Planck’s. Ironically, since Planck’s derivation does not rely on the density of states, the origin of Weyl’s law lies just in these alternative approaches. Therefore, a history of Weyl’s law without a discussion of these differing theories would beincomplete. 1.2.7 Black Body Radiation: The Rayleigh–Einstein–Jeans Law First of all, one should understand why some theorists were seeking for different theories of black body radiation despite the great empirical success of Planck’s for- mula. The explanation is quite obvious: they realized that Planck’s radiation theorywas not satisfactory from a theoretical point of view; in fact, it was inconsistent! As 1.2 A Brief History of Weyl’s Law 15 the above quotation from Einstein [23] shows, the problem still existed in 1913; the ultimate derivation of Planck’s formula was only provided in 1924 using the correct Bose–Einstein quantum statistics. In 1903, Lorentz [35] derived (1.19) in the low-frequency limit together with the value c1=⎫parenleftBig 16πα/3c3⎫parenrightBig for the coefﬁcient c1where αis a constant such that αT represents the mean kinetic energy of a molecule of a gas. Comparing (1.19) with the low-frequency limit of Planck’s formula (1.21), he obtained α= (3/2) kB(see also (1.23)) and states: “Now the mean kinetic energy of a molecule of a gas would be (3/2) kTa c c o r d i n gt oP l a n c k...t h e r ea p p e a r st h e r e f o r et ob ef u l la g r e e m e n tb e - tween the two theories in the case of long waves, certainly a remarkable conclusion, as the fundamental assumptions are widely different.” The year 1905 is one of the most amazing ones in the history of science: it marks, ﬁrst of all, Einstein’s annus mirabilis with his ﬁve seminal papers, where only the ﬁrst one on the famous light quantum hypothesis [36] concerns us here, since it deals with the radiation problem, and, secondly, the series of papers published by Rayleigh [37, 38] and Jeans [39–42] on the radiation problem using the Weyl asymptotics. From reading these papers it becomes clear that Einstein is the only one who takes Planck’s formula serious since it “agrees with all experiments to date” [36]. But in Section 1.1 of this paper entitled “On a difﬁculty concerning the theory of the« black radiation »” [36] he implicitly expresses his doubts on Planck’s derivation by showing that Planck should have obtained (1.27) instead of his formula (1.21)! The argument is very simple. Planck’s starting point in his derivation is the formula ρ(ν,T)=8πν 2 c3/angbracketleftE/angbracketright(ν,T) , (1.28) where/angbracketleftE/angbracketright(ν,T) is the average energy of a Planck resonator of frequency νat the joint equilibrium of matter and radiation at temperature T.F u r t h e r m o r e ,t h ee q u i - librium energy of a one-dimensional resonator is according to the equipartition theorem given by /angbracketleftE/angbracketright(ν,T)= kBT, and inserting this into (1.28), Einstein ob- tains (1.27). We thus see that the radiation law (1.27), commonly known as the Rayleigh–Jeans law, ought to be called the Rayleigh–Einstein–Jeans law [15]. Many years later Einstein said: “If Planck had drawn this conclusion, he probably wouldnot have made his great discovery, becau se the foundation would have been with- drawn from his deductive reasoning.” [43] Years later Planck himself presented two derivations of (1.27) in his famous book “Theorie der Wärmestrahlung” [32] and concluded: “It is not too much asserted if we say in generalizing: The classical theory leads of necessity to Rayleigh’s radiation law.” Einstein’s paper [36] was submitted on 17 March 1905, and thus is the earliest among the above mentioned papers by Rayleigh and Jeans. (Rayleigh’s ﬁrst pa-per [37] was submitted on 6 May 1905; Jeans’ ﬁrst paper on radiation [39] on 20 May 1905.) As discussed in Section 1.2.5, Rayleigh was the ﬁrst [26] to have already count- ed in 1900 “the number of modes corresponding to any ﬁnite space occupied by 16 1W e y l ’ s L a w radiation” [37] and to obtain the law (1.19), however, without determining the coef- ﬁcient c1. “Elicited by the very clear statement of his view which Mr. Jeans gives in NATURE of April 27 (1900) [44]”, he repeats the arguments of his former paper [26] “with an extension designed to determine the coefﬁcient as well as the law of radia-tion” [37]. By counting the modes within a cube of length l(Weyl’s law), he obtains again (1.19) “as probably representing the truth when νis small.” He remarks that this formula agrees with Planck’s in the limit when νis small apart from the fact that his value for c 1“is eight times as large as that found by Planck.” Rayleigh adds: “A critical comparison of the two processes would be of interest, but not havingsucceeded in following Planck’s reasoning I am unable to undertake it. As apply- ing to all wavelengths, his formula would h ave the greater value if satisfactorily established. On the other hand, the reason ing leading to (1.19) is very simple, and this formula appears to me a necessary consequence of the law of equipartition as laid down by Boltzmann and Maxwell. My difﬁculty is to understand how another process, also based on Boltzmann’s ideas, can lead to a different result.” [37] Two days after Rayleigh’s letter [37] Jeans submitted a short letter [39] in reply to Rayleigh. His main point was “the general question of the applicability of thetheorem of equipartition to the energy of the ether” as opened up by Rayleigh. He takes up “Lord Rayleigh’s example of a stre tched string, say a piano wire” and then discusses the “vibrations of the ether in a ﬁnite enclosure”. He writes: “It is eas-ily seen that the number of slow vibrations is approximately proportional to the volume of the enclosure, so that roughly the energy of ether must be measured per unit volume in order to be independent of the size of the enclosure.” He then arrives at (1.19), but without determining the value for c 1.O nJ u n e7 ,J e a n sa d d s a “postscript” to his paper [40] and calculates again “the number of degrees of free-dom of the æther” by referring to Rayleigh’s book [27](!). From this he obtains the radiation law (1.19) together with the correct value (1.23) for the coefﬁcient c 1.“ T h i s is one-eighth of the amount found by Lord Rayleigh, but agrees exactly with thatgiven by Planck for large values of λ.I ts e e m st om et h a tL o r dR a y l e i g hh a si n t r o - duced an unnecessary factor 8 by counting negative as well as positive values of his integers p,q,r.” (See the discussion before equation (1.19).) A month later, Rayleigh replies to Jeans [38]: “In NATURE, May 18, I gave a calculation of the coefﬁcient of complete radiation at a given absolute temperature for waves of great lengthon principles laid down in 1900, and it appeared that the result was eight times as great as that deduced from Planck’s formula for this case. In connection with similar work of his own, Mr. Jeans has jus t pointed out that I have introduced a re- dundant factor 8 by counting negative as well as positive values of my integers p,q, r– I hasten to admit the justice of this correction. But while the precise agreement of results in the case of very long waves is satisfactory so far as it goes, it does not satisfy the wish expressed in my former letter for a comparison of processes. In the application to waves that are not long , there must be some limitation on the principle of equipartition. Is there any a fﬁnity in this respect between the ideas of Prof. Planck and those of Mr. Jeans?” On July 27, Jeans published another letter [41]: “On two occasions (NATURE, May 18 and July 13) Lord Rayleigh has asked for a critical comparison of two the- 1.2 A Brief History of Weyl’s Law 17 ories of radiation, the one developed by Pr of. Planck and the other by myself, fol- lowing the dynamical principles laid down by Maxwell and Lord Rayleigh. It is with the greatest hesitation that I venture to express my disagreement with some points in the work of so distinguished a physicist as Prof. Planck, but Lord Rayleigh’ssecond demand for a comparison of the two methods leads me to offer the fol- lowing remarks, which would not otherwise have been published, on the theory of Prof. Planck.” Jeans then criticises Planck’s concept of the “entropy of a single res- onator” given by the formula S=k BlogW+constant by saying: “The function W,a s at present deﬁned, seems to me to have no meaning. Planck (in common, I know,with many other physicists) speaks of the ‘probability’ of an event, without spec- ifying the basis according to which the probability is measured. This conception of probability seems to me an inexact conception, and as such to have no place inmathematical analysis.” [41] Jeans’ critique of Planck’s derivation is fully justiﬁed as one can infer from Ein- stein’s “laudatio” for Planck written in 1913: “This [that is Planck’s] calculation which, due to the not sufﬁciently sharp deﬁnition of W, could not be performed w i t h o u ta r b i t r a r i n e s s ,l e dt ot h er a d i a t i o nf o r m u l a( 1 . 2 1 )...”[ 2 3 ] . Jeans then continues [41] by criticising Planck’s introduction of his famous con- stant hvia the fundamental relation ε=hν.“ H e r e εis a small quantity, a sort of indivisible atom of energy, introduced to simplify the calculations. We may legiti- mately remove this artiﬁcial quantity by passing to the limit in which ε= 0...T h e relation ε=hνis assumed by Planck in order that the law ultimately obtained may satisfy Wien’s ‘displacement law’ i.e. may be of the form (1.17). This law is obtained by Wien from thermodynamical c onsiderations on the supposition that the energy of the ether is in statistical equilibrium with that of matter at a uniformtemperature. The method of statistical mechanics, however, enables us to go fur- ther and determine the form of the function f(v/T); it is found to be 8 πk B(T/ν), so that Wien’s law (1.17) reduces to the law given by expression (1.27). In otherwords, Wien’s law directs us to take ε=hν,b u tl e a v e s hindeterminate, whereas statistical mechanics gives us the furt her information that the true value of his h= 0. Indeed, this is sufﬁciently obvious from general principles. The only way of eliminating the arbitrary quantity εis by taking ε= 0, and this is the same as h= 0. – Thus it comes about that in Planck’s ﬁnal law (1.21) the value of his left indeterminate; on putting h= 0, the value assigned to it by statistical mechanics, w ea r r i v ea to n c ea tt h el a w( 1 . 2 7 ) ....Ic a r r yt h em e t h o df u r t h e rt h a nP l a n c k ,s i n c e Planck stops short of the step of putting h= 0. I venture to express the opinion that it is not legitimate to stop short at this point, as the hypotheses upon which Planck h a sw o r k e dl e a dt ot h er e l a t i o n h= 0 as a necessary consequence. Of course, I am aware that Planck’s law is in good agreement with experiment if his given a value different from zero, while my own law, obtained by putting h= 0, cannot possibly agree with experiment. This does not alter my belief that the value h=0i st h eo n l y value which it is possible to take.” [41] Although Jeans’ conclusion [41] that Planck should have arrived at the radiation law (1.27) instead of his formula (1.21) agrees with the conclusions drawn earlierby Einstein [36] and Rayleigh [37, 38]; his belief that the value h=0 i st h e o n l y 18 1W e y l ’ s L a w value which Planck’s constant can possibly take shows that he did not realize the importance of the equation ε=hν(neither did Planck nor Rayleigh!). It was Ein- stein’s revolutionary light-quantum paper [36] (the only contribution he himself called revolutionary) which gave a deep meaning to this equation and thus pavedthe way towards a quantum theory. Einste in put forward the following “heuristic view” [36]. “Monochromatic radiation of low density (within the domain of valid- ity of Wien’s radiation formula) behaves in thermodynamic respect as if it would consist of mutually independent energy quanta of magnitude R/betatwoν/N A[==hνusing /betatwo=h/kB]. – If, in regard to the volume dependence of the entropy, monochromatic radiation (of sufﬁciently low density) behaves as a discontinuous medium, which consists of energy quanta of magnitude [ hν], then this suggests an inquiry as to whether the laws of the generation and cons ervation of light are also constituted as if light were to consist of energy quanta of this kind.” 1.2.8 From Acoustics to Weyl’s Law and Kac’s Question In the previous sections we have discussed how the heat radiation problem was at the origin of Weyl’s famous work. Furt hermore, we have seen that the idea of expressing the spectral energy density ρ(ν,T) of the black body radiation in terms of the density of states D(ν) goes back to Rayleigh [26] who in turn reduced the problem to the “vibrations of a cubical mass of air”. Thus Weyl’s law actually has its roots in acoustics. In view of the fact that Rayleigh was a leading expert in acoustics and the author of the famous book “The Theory of Sound” [27], ﬁrst published in 1877, it is not surprising that he realized that the radiation problemcan be related to the number of vibrational modes of a black body cavity. All the more reason that it is strange to observe that he had difﬁculties in obtaining the correct value for the constant c 1in his radiation law (1.19). The problem was of course a question of the correct boundary co nditions in the electromagnetic case. In his book, Rayleigh writes: “Some of the na tural notes of the air contained within a room may generally be detected on singing the scale. Probably it is somewhat in this way that blind people are able to estimate the size of rooms.” [27] And in a footnote he adds: “A remarkable instance is quoted in Young’s Natural Philoso- phy, II. p. 272, from Darwin’s Zoonomia , II. 487. “The late blind Justice Fielding walked for the ﬁrst time into my room, when he once visited me, and after speak- ing a few words said, ‘This room is about 22 feet long, 18 wide, and 12 high’; allw h i c hh eg u e s s e db yt h ee a rw i t hg r e a ta c c u r a c y . ”A n dt h e nR a y l e i g hc o n t i n u e s : “In long and narrow passages the vibrations parallel to the length are too slow to affect the ear, but notes due to transverse vibrations may often be heard. The relative proportions of the various overtones depend upon the place at which the disturbance is created. In some cases of this kind the pitch of the vibrations, whosedirection is principally transverse, is inﬂuenced by the occurrence of longitudinal m o t i o n ....” These remarks on acoustics lead us directly to Kac’s famous question: “Can one hear the shape of a drum?” [9], which will be discussed in Sections 1.3.6 and 1.3.7, 1.3 Weyl’s Law with Remainder Term. I 19 and the more general question: “Can one hear the periodic orbits of a drum?” to be discussed in Section 1.3.7. 1.3 Weyl’s Law with Remainder Term. I 1.3.1 The Laplacian on the Flat Torus T2 In special cases it is possible to derive exact formulae for the counting function N(λ) which contain in addition to the Weyl t erm (and possible higher order terms) an explicit expression for a remainder function. The most elegant way to derive these formulae is based on trace formulae ; a famous example is the Selberg trace formula [45–47] to be discussed in Section 1.4. To illustrate the method in a sim- ple case, we consider the eigenvalue problem – ΔT2u=λu,w h e r e ΔT2denotes the L a p l a c i a no naﬂ a tt o r u s T2:=S1 L~S1 L=R2/(LZ~LZ) characterized by a length scale L>0 .T2can be represented by the fundamental domain Ω=[0,L]~[0,L]∈R2i.e. by a square with side L, where opposite sides are glued together. Obviously, all of R2is covered by the Γ-translates of Ωwhere Γis the translation group ( LZ)2.T h i s produces a tessellation of R2and leads to the periodic boundary conditions u(x1+μ1L,x2+μ2L)=u(x1,x2),(x1,x2)∈Ω,(μ1,μ2)∈Z2. Note thatT2is a smooth, compact manifold with area |Ω|=L2(but with no boundary). It is easy to see that ( em)m∈Z2=⎫parenleftBig e2πi(m·x)/L⎫parenrightBig m∈Z2is an orthonormal ba- sis of L2(Ω) consisting of eigenvectors of – ΔT2with discrete eigenvalues ( λm)m∈Z2=⎫parenleftBig (4π2)/L2⎫parenleftBig m2 1+m2 2⎫parenrightBig⎫parenrightBig (m1,m2)∈Z2. Letr(n)=#⎫braceleftBig (m1,m2)∈Z2,n=m2 1+m2 2⎫bracerightBig ,n∈N0,w i t h r(0) = 1, i.e. r(n) denotes the number of representations of n∈N0as a sum of two squares of integers. Obviously, the distinct eigenvalues of – ΔT2, (¯λn)n∈N0=⎫parenleftBigg4π2 |Ω|n⎫parenrightBigg n∈N0, occur with multiplicity r(n). Then the counting function on the torus reads N(λ)=⎫summationdisplay ¯λnuλr(n)=⎫summationdisplay 0unu(|Ω|/4π2)λr(n) . (1.29) The very irregular (“valde irregulariter” [48]) number theoretical function r(n)h a d already been studied by Gauss [48] who derived the formula r(n)=4 ( d1(n)–d3(n)), nv1, where d1(n)a n d d3(n) are the number of divisors of nof the form 4 m+1a n d 4m+3 ,m∈N0, respectively. The ﬁrst values are r(0) = 1, r(1) = 4, r(2) = 4, r(3) = 0, r(4) = 4, r(5) = 8. If n==3(mod 4) then r(n)=0 .F o rl a r g e none has r(n)=O(nε)f o r every ε>0 ;r(n)=O⎫parenleftBig (logn)δ⎫parenrightBig is false for every δ. The average order of r(n)i s ¯r:= lim x→∞1 x⎫summationdisplay 0unuxr(n)=π 20 1W e y l ’ s L a w (Gauss resp. the Weyl law, see (1.31)). For further information about r(n), see [49, pp. 241]. 1.3.2 The Classical Circle Problem of Gauss Let ν(x): =⎫summationdisplay 0unuxr(n)=⎫summationdisplay m2 1+m2 2ux (m1,m2)∈Z~Z1 , (1.30) then N(λ)=ν⎫parenleftBigg|Ω| 4π2λ⎫parenrightBigg and the derivation of Weyl’s law is reduced to a lattice point problem ,s i n c e ν(x)h a s a simple geometric interpretation as the n umber of lattice points in the interior and on the boundary of a circle with center (0, 0) and of radius√x.T h ep r o b l e mo fc a l - culating the leading asymptotic behavior of ν(x)f o r x→∞ was already considered by Gauss in 1834 [48] (see also [50, pp. 32–39]). He realized that ν(x) is approxi- mately given by the sum of the areas of all squares of unit side length which are inscribed in the circle of radius√x,a n dt h u s ν(x) is in ﬁrst approximation equal to the area of the circle π⎫parenleftBig√x⎫parenrightBig2=πx. Actually, Gauss proved lim x→∞ν(x) x=π, (1.31) which implies Weyl’s law lim λ→∞N(λ) λ=|Ω| 4π(1.32) for the counting function (1.29). Based on his result (1.31), Gauss considered ν(x)/x as an approximation method to calculate π. To this purpose, he thought about the error one makes at ﬁnite x. Again, by geometrical intuition, one sees that the error should not be larger than the combined area of those squares that are cut by the boundary of the circle i.e. those contained in an annulus of width 2√ 2, and thus is approximately given by 2√ 2 times the perimeter of the circle 2 π√x, and, indeed, Gauss was able to prove ν(x)=πx+O⎫parenleftBig√x⎫parenrightBig (x→∞ ), which implies N(λ)=|Ω| 4πλ+O⎫parenleftBig√ λ⎫parenrightBig (λ→∞ ). Deﬁning a remainder term P(x), ν(x)=πx+P(x), 1.3 Weyl’s Law with Remainder Term. I 21 we are led to the classical circle problem , a famous problem in analytic number theo- ry [51, pp. 181–308]: estimate the remainder function P(x) as accurately as possible. In particular, determine α0=i n f αin the estimate P(x)=O(xα)(x→∞ ). In Figures 1.1 and 1.2 we show plots of ν(x)a n d P(x), respectively, from which it becomes clear that P(x) – due to the erratic behavior of r(n) – is a very irregular function wildly ﬂuctuating about zero. It is therefore no big surprise that to deter- mine the actual size of P(x), and thus the remainder to Weyl’s law, is a difﬁcult problem. Considering the difference ν(n+1 / 2 )– ν(n),n∈N, it is easy to see that P(x)=o(1) is false, and thus 0 uα0u1/2. An important result showing that P(x) is much smaller than the classical result α0u1/2 is due to Sierpi ´nski who proved α0u1/3 in 1906 [52, pp. 73–108]. A famous conjecture by Hardy from 1915 states that α0should be 1/4, i.e. P(x)=O⎫parenleftBig x1/4+ ε⎫parenrightBig for every ε> 0 [53, 54]. Actually, Hardy proved α0v1/4. During the last 100 years, the values for α0decreased only by a tiny amount: α0u37/112 = 0.330...(van der Corput 1923 [55]), α0u12/37 = 0.324...(Wen-Lin Yin 1962 [56]), α0u7/22 = 0.318...(Iwaniec and Mozzochi 1988 [57]). The best bound known today is due to Huxley who proved in 1992 that P(x)=O⎫parenleftBig x23/73⎫parenleftbiglogx⎫parenrightbig315/146⎫parenrightBig ; note that 23/73 = 0.315 ...is still far away from 1/4! (For a review, see [58].) Since P(x) is a wildly ﬂuctuating function, it might be that some very rare spikes exceeding the conjectured x1/4-behavior make it extremely difﬁcult to improve the best existing bound. In order to “tame” these spikes, one can consider moments ofP(x) and hope that the spikes are being washed out. We shall come back to this idea in Section 1.3.9 making use of the trace formula for ν(x)w h i c hw es h a l ln o w derive. Note added in proof: in a recent unpublished paper [59] it is claimed to present a proof of Hardy’s conjecture. 1.3.3 The Formula of Hardy–Landau–Voronoï The counting function ν(x) can be rewritten as ν(x)=⎫summationdisplay m∈Z2θ⎫parenleftBig x–m2⎫parenrightBig , where θ(x) denotes the Heaviside step function. Instead of θ(x–m2), let us consider af u n c t i o n g(m)w i t h –g:R2→C, continuous –g(x)=O⎫parenleftBig 1/(/bardblx/bardbl2+ε)⎫parenrightBig for/bardblx/bardbl2=x2 1+x2 2→∞ ,ε>0 , and let us study the sum⎫summationtext m∈Z2g(m).U s i n gt h e Poisson summation formula ,w eo b t a i n ⎫summationdisplay m∈Z2g(m)=⎫summationdisplay l∈Z2˜g(l), (1.33) 22 1W e y l ’ s L a w where ˜gdenotes the Fourier transform of g: ˜g(l)=⎫integraldisplay R2g(x)e–2πi(l·x)d2x. (1.34) To apply this to the circle problem, we make the further assumption that g(x)i s a radial function which depends only on ρ=/bardblx/bardbl,i . e . g(x)=g(x1,x2)=φ⎫parenleftBig x2 1+x2 2⎫parenrightBig =φ⎫parenleftBig ρ2⎫parenrightBig . Thus ˜g(l)=˜g(l1,l2)=∞⎫integraldisplay –∞∞⎫integraldisplay –∞φ⎫parenleftBig x2 1+x2 2⎫parenrightBig e–2πi(l1x1+l2x2)dx1dx2 =∞⎫integraldisplay 0ρφ⎫parenleftBig ρ2⎫parenrightBig2π⎫integraldisplay 0e–2πi/bardbll/bardblρcosϕdϕdρ=2π∞⎫integraldisplay 0ρφ⎫parenleftBig ρ2⎫parenrightBig J0(2π/bardbll/bardblρ)dρ. Here we have introduced polar coordinates in R2,x1=ρcosϕ,x2=ρsinϕ,0uϕu 2π, and have used the integral representation J0(z)=1 2π2π⎫integraldisplay 0e–izcosϕdϕ for the Bessel function J0(z). Now the Poisson summation formula (1.33) reads ⎫summationdisplay m∈Z2φ⎫parenleftBig m2⎫parenrightBig =2π⎫summationdisplay l∈Z2∞⎫integraldisplay 0ρφ⎫parenleftBig ρ2⎫parenrightBig J0(2π/bardbll/bardblρ)dρ, or, by introducing the multiplicity r(n)a n d ρ=√x,xv0: ∞⎫summationdisplay n=0r(n)φ(n)=π∞⎫summationdisplay n=0r(n)∞⎫integraldisplay 0φ(x)J0⎫parenleftBig 2π√nx⎫parenrightBig dx. (1.35) This is the theorem due to Hardy [54, 60, 61], Landau [51, pp. 189] and Voronoï [62]. 1.3.4 The Trace Formula on the Torus T2and the Leading Weyl Term We recall that the distinct eigenvalues on the torus T2are given by ¯λn=(2π/L)2n= p2 nwith pn:=(2π/L)√n,n∈N0, and multiplicities r(n). Introducing in the theo- rem (1.35) the spectral function h ((2π/L)ρ):=φ⎫parenleftBig ρ2⎫parenrightBig with •h:R→C, continuous •heven i.e. h(–p)=h(p) (1.36) •h(p)=O⎛⎜⎜⎜⎜⎜⎜⎝1 ⎫vextendsingle⎫vextendsingle⎫vextendsinglep⎫vextendsingle⎫vextendsingle⎫vextendsingle2+ε⎞⎟⎟⎟⎟⎟⎟⎠,⎫vextendsingle⎫vextendsingle⎫vextendsinglep⎫vextendsingle⎫vextendsingle⎫vextendsingle→∞ ,ε>0 , 1.3 Weyl’s Law with Remainder Term. I 23 we arrive at the trace formula on the torus T2 ∞⎫summationdisplay n=0r(n)h⎫parenleftbigpn⎫parenrightbig=|Ω| 2π∞⎫integraldisplay 0ph(p)dp+|Ω|∞⎫summationdisplay n=1r(n)ˆh⎫parenleftBig L√n⎫parenrightBig , (1.37) where ˆh(x) denotes the Fourier–Bessel (or Hankel) transform of h(p): ˆh(x): =1 2π∞⎫integraldisplay 0ph(p)J0(px)dp. (In deriving the ﬁrst term on the right-hand side of (1.37), we have used r(0) = 1 = J0(0) and L2=|Ω|.) Note that the left-hand side of (1.37) can be written as the trace of the trace class operator h⎫parenleftBig (–ΔT2)1/2⎫parenrightBig :L2(Ω)→L2(Ω) with h⎫parenleftBig (–ΔT2)1/2⎫parenrightBig f=⎫summationdisplay m∈Z2h⎫parenleftBig⎫radicalbig λm⎫parenrightBig⎫parenleftbigem|f⎫parenrightbigem,for f∈L2(Ω), i.e. ∞⎫summationdisplay n=0r(n)h⎫parenleftbigpn⎫parenrightbig=T r h⎫parenleftBig (–ΔT2)1/2⎫parenrightBig , which explains why (1.37) is called a trace formula. Due to the conditions (1.36) on the spectral function h(p), the operator h⎫parenleftBig (–ΔT2)1/2⎫parenrightBig is actually a Hilbert–Schmidt operator with kernel Gh(x,y)=¯Gh(y,x)∈ L2(Ω~Ω) satisfying, for f∈L2(Ω), ⎫parenleftBig h⎫parenleftBig (–ΔT2)1/2⎫parenrightBig f⎫parenrightBig (x)=⎫integraldisplay ΩGh(x,y)f(y)d2y. (1.38) Furthermore, Gh(x,y) has the uniformly convergent expression in terms of the or- thonormal eigenfunctions em∈L2(Ω) (Mercer’s theorem) Gh(x,y)=⎫summationdisplay m∈Z2h⎫parenleftBig⎫radicalbig λm⎫parenrightBig em(x)¯em(y) , (1.39) which expresses the fact that emis an eigenfunction of the operator h⎫parenleftBig (–ΔT2)1/2⎫parenrightBig with eigenvalue h⎫parenleftBig⎫radicalbig λm⎫parenrightBig . From this one immediately derives the pre-trace formula Trh⎫parenleftBig (–ΔT2)1/2⎫parenrightBig =⎫integraldisplay ΩGh(x,x)d2x. (1.40) 24 1W e y l ’ s L a w Pre-trace formulae of this type are the sta rting point for the derivation of trace formulae in the general case, for example in quantum mechanics, where the right- hand side of (1.40) is expressed by the volume of the classical phase space and the classical actions evaluated along the perio dic orbits of the corresponding classical system [63]. An alternative way to write the left-hand side of (1.37) is ∞⎫summationdisplay n=0r(n)h(pn)=∞⎫integraldisplay 0h⎫parenleftBig√ λ⎫parenrightBig dN(λ), where N(λ) is the counting function, and the integral is understood as a Stieltjes integral. Rewriting in a similar way the ﬁrst term on the right-hand side of (1.37) |Ω| 2π∞⎫integraldisplay 0ph(p)dp=:∞⎫integraldisplay 0h⎫parenleftBig√ λ⎫parenrightBig dN(λ), one obtains d N(λ)=|Ω|/(4π)dλand thus, immediately, the smooth term N(λ)=|Ω| 4πλ, (1.41) w h i c ht u r n so u tt ob ee x a c t l yt h el e a d i n g Weyl term ofN(λ), see (1.4) and Sec- tion 1.3.6. 1.3.5 Spectral Geometry: Interpretation of the Trace Formula on the TorusT2in Terms of Periodic Orbits While the left-hand side of the trace formula (1.37) has a simple spectral interpreta- tion (being just the sum over the “frequencies” pn=⎫radicalbig¯λnof the eigenvibrations on T2, evaluated on a large class of spectral functions h(p), see Equation (1.36)), the inﬁnite series on the right-hand side has a simple geometrical interpretation as can be seen by rewriting (1.37) as follows ∞⎫summationdisplay n=0r(n)h(pn)=|Ω|ˆh(0) +|Ω|∞⎫summationdisplay n=1∞⎫summationdisplay k=1r⎫parenleftBig k2l2 n⎫parenrightBigˆh(kln). (1.42) Here⎫braceleftbigln⎫bracerightbig n∈Ndenotes the primitive length spectrum onT2with ln=L⎫radicalBig m2 1+m2 2=L√n, where nis a square-free integer with r(n)=/0 .lnis the geometrical length of a prim- itive periodic orbit (closed geodesic) of the classical geodesic ﬂow on T2. The non- primitive periodic orbits have lengths kln,kv2, where kcounts the kthtraversal of the corresponding primitive periodic orbit with length ln. The trace formula (1.42) 1.3 Weyl’s Law with Remainder Term. I 25 displays a beautiful relation in spectral geometry relating the spectrum of the Lapla- cian to the length spectrum of the geodesic ﬂow .T h et o r u sT2is a compact Riemann surface of genus 1 and Gaussian curvature K= 0. A generalization to surfaces of higher genus is given by the famous Selberg trace formula [45, 46] which has beenmuch studied in the ﬁeld of quantum chaos (see for example [47,64–66]) and string theory (see for example [67, 68]) and will be discussed in Section 1.4.4. 1.3.6 The Trace of the Heat Kernel on d-Dimensional Tori and Weyl’s Law The trace formula (1.37), respectively (1.42), has the typical structure of a trace for- mula and is in some sense a “meta formula” since it allows one to derive an inﬁnitenumber of relations depending on the spe cial choice of the spectral function h(p) satisfying the conditions (1.36). As a ﬁrst example, let us calculate the trace of the heat kernel , which is obtained for the choice h(p)=e –p2t,t>0 .W i t h ˆh(x)=1 2π∞⎫integraldisplay 0pe–p2tJ0(px)dp=1 4πte–x2/4t we get ( t>0 ) ΘT2(t): =T r etΔT2=∞⎫summationdisplay n=0r(n)e–(4π2/|Ω|)nt=|Ω| 4πt+|Ω| 4πt∞⎫summationdisplay n=1r(n)e–(|Ω|/4)tn =|Ω| 4πt+|Ω| 4πt∞⎫summationdisplay n=1∞⎫summationdisplay k=1r⎫parenleftBig k2l2 n⎫parenrightBig e–k2l2 n/4t.(1.43) Fort→0+o n et h u so b t a i n st h ec o r r e c t |Ω|/(4πt)-term (and no higher order terms of the type∞⎫summationtext n=–1antn/2as occurring in the general case), which yields the correct Weyl term, and an exponentially small remainder term behaving as O⎫parenleftBig t–1e–L2/4t⎫parenrightBig . It is thus seen that the Weyl term corresp onds to the “zero-length contribution” in the periodic orbit sum i.e. to the term obtained for l0:= 0, while the exponen- tial remainder term is determined by the shortest primitive periodic orbit on T2 having the length l1=L. As to physical applications, let us point out that the func- tion ΘT2(t)i sf o r t~1 /T,w h e r e Tdenotes absolute temperature, identical to the partition function in statistical mechanics , and thus the Weyl term determines the high-temperature limit of the corre sponding thermodynamical system. Note that the trace of the heat kernel rewritten as f(q): =∞⎫summationtext n=0r(n)qnwith q=eiπτ, τ=i(4π/|Ω|)t, plays the role of a generating function of the arithmetic func- tion r(n).f(q) was already introduced by Jacobi in 1829 who derived f(q)=⎛⎜⎜⎜⎜⎜⎝⎫summationdisplay m∈Zqm2⎞⎟⎟⎟⎟⎟⎠2 =(θ3(0|τ))2 26 1W e y l ’ s L a w for Im τ> 0 in terms of the elliptic theta function θ3. Using the transformation formula θ3(0|τ)=( – iτ)–1/2θ3(0|–1/τ)derived by Poisson in 1823, one obtains again relation (1.43). It is not difﬁcult to generalize the result (1.43) to d-dimensional ﬂat tori Td:= Rd/Γwith Γ=(LZ)d. The translation group Γhas a natural embedding as a lattice inRd.T oΓthere is associated a uniquely deﬁned dual lattice Γ∗(called a reciprocal lattice in physics): Γ∗=⎫braceleftbigγ∗∈Rn:γ·γ∗∈Zfor all γ∈Γ⎫bracerightbig.W i t h γ=Ln,n∈Zd, γ∗=1 / Lm,m∈Zd,t h ee i g e n v a l u e so f– ΔTdare given by⎫parenleftBig λγ∗⎫parenrightBig γ∗∈Γ∗=4π2⎫vextenddouble⎫vextenddouble⎫vextenddoubleγ∗⎫vextenddouble⎫vextenddouble⎫vextenddouble2 with eigenvectors⎫parenleftBig eγ∗⎫parenrightBig γ∗∈Γ∗=⎫parenleftBig e2πi(γ∗·x)⎫parenrightBig . Furthermore, the length spectrum of the classical periodic orbits on Tdis given by⎫parenleftBig⎫vextenddouble⎫vextenddouble⎫vextenddoubleγ⎫vextenddouble⎫vextenddouble⎫vextenddouble⎫parenrightBig γ∈Γ. Using the Poisson summa- tion formula as in the case d= 2, it is straightforward to derive a trace formu- la onTdfrom which one obtains, for example, for the trace of the heat kernel (t>0 ) ΘTd(t): =T r etΔTd=⎫summationdisplay γ∗∈Γ∗e–4π2/bardblγ∗/bardbl2t=|Ω| (4πt)d/2⎫summationdisplay γ∈Γe–/bardblγ/bardbl2/4t =|Ω| (4πt)d/2+O⎫parenleftBig t–d/2e–L2/4t⎫parenrightBig (t→0+). (1.44) Here the ﬁrst term on the right-hand side corresponding to the identity element I∈Γwith/bardblI/bardbl= 0 yields via the Tauberian theorem of Karamata (see Theorem 1.1 in Section 1.6) Weyl’s law forTd(λ→∞ ) N(λ)=|Ω| (4π)d/2Γ(1 +d/2)λd/2+O⎫parenleftBig λd/2⎫parenrightBig , (1.45) but the trace formula yields, in addition, an exact expression for the remain- der term in the same way as discussed in detail for T2in Section 1.3.9 be- low. The case d= 3 has important applications in several ﬁelds. For example, in solid state physics, chemistry and crystallography, one identiﬁes the lattice Γwith the atomic structure of crystals. Furthermore, the reciprocal lattice Γ∗is very useful in analyzing diffraction phenomena in light and neutron scattering off crystals. Incosmology it has been proposed that the spatial section of our Universe is given by a 3-torus whose fundamental domain is a cube with side length L/similarequal5~10 26m/similarequal 5.6~1010light years [69]. Finally we would like to mention that the case d= 16, i.e. the toriR16/Z16have played an important role in the attempts to answer Kac’s question [9], since it hadalready been noticed by John Milnor in 1964 that the tori T 16give examples of nonisometric compact manifolds with the same spectrum of the Laplacian [70]. The construction of these lattices for d= 16 had already been found by Witt in 1941 [71]. 1.3 Weyl’s Law with Remainder Term. I 27 1.3.7 Going Beyond Weyl’s Law: One can Hear the Periodic Orbits of the Geodesic Flow on the Torus T2 Let us consider another admissible spectral function h(p)i nt h et r a c ef o r m u - la (1.42) which is slightly more general than the one used in the previous section for the heat kernel: h(p): =J0(ps)e–p2t,s∈R,t>0. With ˆh(x)=1 2π∞⎫integraldisplay 0pJ0(ps)e–p2tJ0(px)dp=1 4πte–(s2+x2)/4tI0⎫parenleftbiggsx 2t⎫parenrightbigg (1.46) (I0(z) is the modiﬁed Bessel function) we arrive at the trace formula ( s∈R,t>0 ) G(s,t): =T r⎫parenleftBig J0⎫parenleftBig s(–ΔT2)1/2⎫parenrightBig etΔT2⎫parenrightBig =∞⎫summationdisplay n=0r(n)J0⎫parenleftBigg s⎫radicalBig ¯λn⎫parenrightBigg e–¯λnt =|Ω| 4πte–s2/4t+|Ω| 4πt∞⎫summationdisplay n=1∞⎫summationdisplay k=1r⎫parenleftBig k2l2 n⎫parenrightBig e–(s2+k2l2 n)/4tI0⎫parenleftBiggskln 2t⎫parenrightBigg . (1.47) Since I0(0) = 1, it follows that (1.47) coincides in the limit s→0w i t ht h et r a c e of the heat kernel (1.43), G(0,t)=ΘT2(t). Performing on the other hand for ﬁxed s> 0 the limit t→0+i.e. eliminating the “regulator” t, one obtains the remarkable relation ( s>0 ) G(s,0 )=∞⎫summationdisplay n=0r(n)J0⎫parenleftBigg s⎫radicalBig ¯λn⎫parenrightBigg =|Ω| 2π∞⎫summationdisplay n=1∞⎫summationdisplay k=1r⎫parenleftBig k2l2 n⎫parenrightBig klnδ(s–kln) , (1.48) which is to be understood as an identity in the sense of distributions. Here we have used the asymptotic expansion (valid for z→+∞) I0(z)=1√ 2πzez⎫parenleftBigg 1+O⎫parenleftBigg1 z⎫parenrightBigg⎫parenrightBigg and the delta-sequence 1 2√πte–x2/4t→δ(x)(t→0+). Relation (1.48) tells us that the formal trace G(s,0 )= T r J0⎫parenleftBig s(–ΔT2)1/2⎫parenrightBig yields a well- deﬁned distribution whose singular support is given for s>0b y singsupp G(s,0 )=⎫braceleftbigkln⎫bracerightbig,k∈N, i.e. by the primitive length spectrum⎫braceleftbigln⎫bracerightbigof the geodesic ﬂow on the torus and the nonprimitive length spectrum⎫braceleftbigkln⎫bracerightbig,kv2. Thus the eigenvalues⎫braceleftBig¯λn⎫bracerightBig of the 28 1W e y l ’ s L a w Laplacian onT2together with their multiplicities⎫braceleftbigr(n)⎫bracerightbig“know” the length spectrum of the closed geodesics of the classical motion on T2,i . e .o n ec a nh e a rt h ep e r i o d i c orbits of the torus! Since the torus is uniquely given by its area |Ω|and its length spectrum⎫braceleftbigln⎫bracerightbig, we can conclude that the complete shape of the torus is audible. A slightly different operator has been studied by Chazarain [72], Colin de Verdière [73, 74], and Duistermaat and Guillemin [75, 76], where the Bessel func- tion J0is replaced by cos⎫parenleftBig s(–Δ)1/2⎫parenrightBig respectively exp⎫parenleftBig is(–Δ)1/2⎫parenrightBig . 1.3.8 The Spectral Zeta Function on the Torus T2 Deﬁne for s∈C,R es>1 ,t h e spectral zeta function onT2: /dzetaT2(s): =T r/prime(–ΔT2)–s=∞⎫summationdisplay n=1r(n) ¯λs n=|Ω|s (2π)2s∞⎫summationdisplay n=1r(n) ns, (1.49) where the prime at the trace denotes that the eigenvalue ¯λ0= 0 has been omitted. (Zeta functions of this type for general Laplace–Beltrami operators were introduced in [77, 78] following a suggestion of Weyl. See also [14].) With the help of 1 ns=1 Γ(s)∞⎫integraldisplay 0τs–1e–nτdτ,n>0 ,R e s>0, we obtain for Re s>1 Γ(s)/dzetaT2(s)=1⎫integraldisplay 0ts–1⎫bracketleftbigΘT2(t)–1⎫bracketrightbigdt+∞⎫integraldisplay 1ts–1⎫bracketleftbigΘT2(t)–1⎫bracketrightbigdt. (1.50) Hence /dzetaT2(s) is the Mellin transform of ΘT2(t) with the eigenvalue zero omitted. Since ΘT2(t)=1+ O⎫parenleftBig e–(4π2/|Ω|)t⎫parenrightBig for t→∞ , the second integral has an analytic continuation to the whole complex s-plane as an entire function. Inserting in the ﬁrst integral for ΘT2(t) the expansion (1.43), we obtain for Re s>1 /dzetaT2(s)=|Ω|/(4π) s–1+F(s) , (1.51) where F(s) is an entire function. Thus we can extend the Dirichlet series (1.49) meromorphically to all s∈Chaving only one simple pole at s=1w i t hr e s i d u e |Ω|/(4π), which is given by the area of the torus. This pole is a direct consequence of the leading Weyl term in the expansion (1 .43). It thus follows that the Dirichlet series∞⎫summationtext n=1r(n)/nsdiverges for Re su1, but is convergent for Re s>1 ,w h i c hw i l lb e important in the explicit formula for the remainder term in Weyl’s law. Note that 1.3 Weyl’s Law with Remainder Term. I 29 there exists the following closed expression, which can be considered as another generating function of r(n) (see for example [79, pp. 265]) ∞⎫summationdisplay n=1r(n) ns=4/dzeta(s)L(s) in terms of the Riemann zeta function /dzeta(s) and the Dirichlet L-series L(s): =1 – 1/3s+1 / 5s–...with L(1) = π/4, which has an entire extension. The result (1.51) holds in general for a large class of eigenvalue problems; see for example reference [47] for the Laplace–Beltrami operator on compact Riemannian surfaces of genus gv2. In the case of the Dirichlet Laplacian acting on a smooth bounded open set Ω⊂Rdone can show [80] that /dzetaΩ(s): =T r⎫parenleftBig –ΔD Ω⎫parenrightBig–spossesses a meromorphic analytic continuation into the complex s-place with a leading sim- ple pole at s=d/2 and residue |Ω|/⎫parenleftBig (4π)d/2Γ(d/2)⎫parenrightBig .I np a r t i c u l a r , s= 0 turns out to be a regular point such that the ﬁrst derivative at s=0 ,/dzeta/prime Ω(0) , is well deﬁned. This fact is then used to regularize the functional determinant of –ΔD Ωby [80] det⎫parenleftBig –ΔD Ω⎫parenrightBig := exp⎫parenleftBig –/dzeta/prime Ω(0)⎫parenrightBig . This method was introduced into physics by Stephen Hawking [81] as a convenient way to compute the determinants arising in the Feynman path integral approach to quantum ﬁeld theory and quantum gravity. For applications of this method, see for example [82, pp. 37–43] in the case of quantum mechanics, and [67] in the case of string theory. 1.3.9 An Explicit Formula for the Remainder Term in Weyl’s Law on the TorusT2and for the Circle Problem To derive N(λ) from the trace formula (1.37), we choose the function h(p)= θ⎫parenleftBig λ–p2⎫parenrightBig ,λ>0 .W et h e no b t a i nw i t h 1 2π∞⎫integraldisplay 0ph(p)dp=1 2π√ λ⎫integraldisplay 0pdp=λ 4π and ˆh(x)=1 2π√ λ⎫integraldisplay 0pJ0(px)dp=√ λ 2πxJ1⎫parenleftBig√ λx⎫parenrightBig the relation N(λ)=|Ω| 4πλ+L 2π√ λ∞⎫summationdisplay n=1r(n)√nJ1⎫parenleftBig L√ nλ⎫parenrightBig . (1.52) 30 1W e y l ’ s L a w This equation was found for the ﬁrst ti me in Hardy’s paper [54] who writes in a footnote: “The form of this equation was suggested to me by Mr. S. Ramanujan, ...” . ( A s w e s h a l l s e e b e l o w , t h e s u m i n ( 1 . 5 2 ) i s n o t a b s o l u t e l y c o n v e r g e n t s i n c e the function h(p) used in the derivation is not continuous. Relation (1.52) can be derived, however, by using an appropriate smoothing [65,83].) In order to study the asymptotic behavior of the remainder term, we employ the asymptotic formula J1(x)=⎫radicalbigg 2 πxcos⎫parenleftbigg x–3π 4⎫parenrightbigg +O⎫parenleftBigg1 x3/2⎫parenrightBigg (x→∞ ), and obtain in the limit λ→∞ Nﬂ(λ)=λ1/41 π⎫radicalbigg L 2π∞⎫summationdisplay n=1r(n) n3/4cos⎫parenleftbigg L√ λn–3π 4⎫parenrightbigg +O⎛⎜⎜⎜⎜⎜⎝1 λ1/4∞⎫summationdisplay n=1r(n) n5/4⎞⎟⎟⎟⎟⎟⎠, (1.53) w h e r ew eh a v ed e ﬁ n e dt h e“ ﬂuctuating part ” of the counting function by Nﬂ(λ): = N(λ)–⎫parenleftbig|Ω|/(4π)⎫parenrightbigλ.Nﬂ(λ) describes the ﬂuctuations of N(λ) about the mean be- havior N(λ): =⎫parenleftbig|Ω|/(4π)⎫parenrightbigλgiven by Weyl’s law, see (1.41). In Figure 1.1 we show ap l o to f N(λ)f o r L=2π(which implies N(λ)=ν(λ)a n d P(λ)=Nﬂ(λ)f o rt h er e - mainder term in Gauss’ circle problem) for small values of λ==x(0uxu50). Weyl’s law is indicated as a straight line. One observes that the Weyl term does indeed describe the mean behavior of the staircase function ν(x) very well, even at small values of x.T h eﬂ u c t u a t i n gp a r t P(x) is shown in Figure 1.2 for small values (0uxu50) and for large values (1011uxu1011+1 07)o fxand shows a very erratic behavior ﬂuctuating about zero. I n order to understand this behavior, one has to study the series in (1.53), which is a trigonometric series and therefore more difﬁcult to control than a Fourier series. (Since∞⎫summationtext 1r(n)/n5/4<∞, see Section 1.3.8, the second term in (1.53) is bounded by λ–1/4, and thus can be neglected.) Due to the divergence of the sum∞⎫summationtext 1r(n)/n3/4, the trigonometric sum is only conditionally convergent, explaining the difﬁculty in proving Hardy’s conjecture which amounts to the bound O(λε)for every ε> 0 for this sum. (It is possible, however, to re- place the sharp counting function N(λ) by a smooth counting function depending on a smoothness parameter which leads to better convergence properties, see [65] and [83].) In order to quantify the numerical observation that Nﬂ(λ)o s c i l l a t e sa b o u tz e r o , let us calculate the mean value of P(x)( =Nﬂ(x)f o r L=2π): ¯P(x): =1 xx⎫integraldisplay 0P(y)dy. We then obtain from (1.52) using x⎫integraldisplay 0√yJ1⎫parenleftbig2π√ny⎫parenrightbigdy=x π√nJ2⎫parenleftBig 2π√nx⎫parenrightBig 1.3 Weyl’s Law with Remainder Term. I 31 0 5 10 15 20 25 30 35 40 45 50 x020406080100120140160N Figure 1.1 The counting function ν(x)for the Gaussian circle problem (respectively for a torus with L=2π). The straight line shows the leading term πx(Weyl’s law). and the asymptotics of the Bessel function ( x→∞ ) ¯P(x)=1 π∞⎫summationdisplay n=1r(n) nJ2⎫parenleftBig 2π√nx⎫parenrightBig =x–1/4 π∞⎫summationdisplay n=1r(n) n5/4cos⎫parenleftBigg 2π√nx–5π 4⎫parenrightBigg +O⎫parenleftBig x–3/4⎫parenrightBig , (1.54) which implies, since the sums in (1.54) are now absolutely convergent, lim x→∞⎫vextendsingle⎫vextendsingle⎫vextendsingle¯P(x)⎫vextendsingle⎫vextendsingle⎫vextendsingle= 0 [51, pp. 206]. A method to smooth possible spikes in P(x), which originates in a paper by Cramér in 1922 [84], is to study higher moments of P(x) M k(x): =1 xx⎫integraldisplay 0⎫vextendsingle⎫vextendsingle⎫vextendsingleP(y)⎫vextendsingle⎫vextendsingle⎫vextendsinglekdy (1.55) fork>0a n d mk(x): =1 xx⎫integraldisplay 0⎫parenleftbigP(y)⎫parenrightbigkdy (1.56) fork=1 ,3 ,5 ,.... (Note that m1(x)=¯P(x)). The following results are known [85] Mk(x)→Ckxk/4,k∈[0, 9] mk(x)→ckxk/4,k=3 ,5 ,7 ,9 .(x→∞ ) (1.57) 32 1W e y l ’ s L a w 0 5 10 15 20 25 30 35 40 45 50 x-10-8-6-4-20246810N 1.e+01 1 1.00005e+011 1.0001e+01 1 x-3000-2000-10000100020003000N Figure 1.2 The remainder term P(x)of the Gaussian circle problem (respectively the ﬂuctuating part of the torus problemwith L=2π) is shown in different intervals. (C2=1 / ( 3 π2)∞⎫summationtext n=1r(n)2/n3/2[84]). It follows that the moments (1.57) are consistent with Hardy’s conjecture, P(x)=O⎫parenleftBig x1/4+ ε⎫parenrightBig , since this implies⎫parenleftbigmk(x)⎫parenrightbig1/k=O⎫parenleftBig x1/4⎫parenrightBig resp.⎫parenleftbigMk(x)⎫parenrightbig1/k=O⎫parenleftBig x1/4⎫parenrightBig , but of course they do not prove it. Nevertheless it seems justiﬁed to say that the “mean” behavior of P(x) is proportional to x1/4forx→∞ . 1.3.10 The Value Distribution of the Remainder Term in the Circle Problem In the preceding section we saw that the remainder term P(x) in the circle problem (respectively the ﬂuctuating part Nﬂ(λ) in Weyl’s law for the torus problem) is a very irregular function ﬂuctuating about zero (see Figures 1.1 and 1.2). It thus appearsnatural to consider P(x) as a random function of xand to study its statistical proper- ties in the limit x→∞ , like its moments as in Equations (1.55) and (1.56), its limit distribution (if it exists), correlations e tc., rather than to estimate its magnitude, i.e. trying to prove Hardy’s conjecture. Since the moment M 2(x), see (1.55), is the variance of P(x), an obvious quantity to study is the normalized remainder term W(x): =P(x)⎫radicalbigM2(x). Since M2(x)→C2√xforx→∞ , it turns out to be convenient to consider the function F(p): =P(p2)√p=1 π∞⎫summationdisplay n=1r(n) n3/4cos⎫parenleftbigg 2π√np–3π 4⎫parenrightbigg +O⎫parenleftBigg1 p⎫parenrightBigg (p→∞ ) (1.58) as a function of p:=√x>1a n d F(p)=0f o r p<1 .O b v i o u s l y , F(p) ﬂuctuates about zero and its mean value vanishes asymptotically for p→∞ , whereas Cramér’s result [84] implies that the second moment of F(p) exists. There now arise the following questions. i) Does F(p), where pis randomly chosen from the interval 1.3 Weyl’s Law with Remainder Term. I 33 ⎫bracketleftbig1,pm⎫bracketrightbig,h a v ef o r pm→∞ a limit distribution f(α)dαwith probability density f(α)? ii) Assuming that f(α) exists, what is its form? In view of the erratic behavior of P(p2)a n dt h u so f F(p), one may guess that the central limit theorem can be applied toF(p)a n dt h u s f(α) should be a Gaussian. The study of the distribution of F(p) was initiated by Heath-Brown [85] who proved that F(p) has indeed a distribution function f(α) in the sense that, for any interval [a,b]⊂Cwe have lim pm→∞1 pmμ⎫braceleftbigp∈⎫bracketleftbig0,pm⎫bracketrightbig:F(p)∈[a,b]⎫bracerightbig=b⎫integraldisplay af(α)dα (1.59) (here μdenotes the Lebesgue measure.) Moreover, he proved that f(α)c a nb ee x - tended to an entire function on Cand decreases faster than polynomially on the real line as |α|→∞ . The results of Heath-Brown were developed further by Bleher, Cheng, Dyson and Lebowitz [86] who proved lim pm→∞1 pmpm⎫integraldisplay 0g(F(p))ρ⎫parenleftBiggp pm⎫parenrightBigg dp=∞⎫integraldisplay –∞g(α)f(α)dα (1.60) for every piecewise continuous bounded function g(x)o nRand for an arbitrary probability density ρ(x)v0o n [0, 1]. In addition, they showed that for every ε>0 there exists α0=α0(ε) > 0 such that, on the real line α∈R, we have the upper bound 0uf(α)<e–|α|4–ε(1.61) when|α|>α0, and that the cumulative distribution C(α): =α⎫integraltext –∞f(α/prime)dα/primesatisﬁes for every α>α0the lower bound C(–α), 1 – C(α)>e–α4+ε. (1.62) These results [85, 86] came as a great surprise since they imply that f(α)d e c r e a s e s for|α|→∞ roughly as e–α4and thus faster than a Gaussian density! A numerical computation of f(α) is shown in Figure 1.3 and compared with a normal Gaussian distribution. The deviation from a Gaussi an distribution is clearly visible; more- over, one observes that f(α) is skewed towards positive values of α. In the next section we shall formulate a c onjecture which states that the non- Gaussian behavior of f(α) has its origin in the fact that the circle problem can be related to the remainder term of Weyl’s law for a quantum mechanical systemwhose corresponding classical system (i.e. the geodesic ﬂow on a torus with L=2π) is integrable and thus regular. The proof of the properties (1.60)–(1.62) is based on the fact that F(p)i sa na l m o s t periodic function of Besicovitch class B 2[86, 87], which means lim N→∞lim pm→∞1 pmpm⎫integraldisplay 0⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingleF(p)–1 πN⎫summationdisplay n=1r(n) n3/4cos⎫parenleftbigg 2π√np–3π 4⎫parenrightbigg⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle2 dp= 0 . (1.63) 34 1W e y l ’ s L a w -3 -2 -1 0 1 2 x0.00.050.10.150.20.250.30.350.4y 3 Figure 1.3 The distribution function f(α)is shown for the circle problem in comparison with a Gaussian normal distribution(dashed curve). 1.3.11 A Conjecture on the Value Distribution of the Remainder Term in Weyl’s Law for Integrable and Chaotic Systems In this section we would like to mention an application of Weyl’s law in quantum mechanics. Let us consider a bound quantum system i.e. a system whose quantumHamiltonian has a purely discrete energy spectrum⎫braceleftbigλ n⎫bracerightbig n∈N.T oh a v eas p e c i ﬁ ce x - ample in mind, think of two-dimensional quantum billiards on a bounded domain Ωwith area|Ω|, for which the time-independent Schrödinger equation reads (in appropriate units) – ΔΩun(x)=λnun(x) imposing (for example) Dirichlet or Neu- mann boundary conditions on ∂Ω(see (1.1) and (1.2)). Moreover, let us assume that Weyl’s law holds in the form N(λ)=N(λ)+Nﬂ(λ) , (1.64) where the smooth part N(λ) describes asymptotically the mean behavior of the counting function N(λ), i.e. the ﬂuctuating remainder term Nﬂ(λ)s a t i s ﬁ e s 1 λλ⎫integraldisplay λ1Nﬂ(λ/prime)dλ/prime→0f o r λ→∞ . (1.65) For “generic” two-dimensional billiards, there exists a three-term formula for N(λ) N(λ)=|Ω| 4πλ∓|∂Ω| 4π√ λ+C, (1.66) where the ﬁrst two terms correspond to Weyl’s conjecture (see (1.6) and the re- marks after (1.7)), and the constant Ctakes the curvature of ∂Ωand corner correc- tions into account (see (1.67)). 1.3 Weyl’s Law with Remainder Term. I 35 The deﬁnition of what is meant by “generic” is a very subtle subject, the dis- cussion of which is beyond the scope of this contribution. Examples of generic and nongeneric systems are discussed in [ 88]. A rigorous deﬁnition requires the introduction of geometrical concepts like “nonperiodicity” and “nonblocking”; seefor example [89]. To derive the smoothed counting function N(λ), several averag- ing procedures have been invented, in particular by Brownell [90], which are de- scribed in [91]. For a simply connected domain Ωpossessing piecewise smooth arcs of length γiand corners of angle ϕj∈(0, 2π]one obtains [91, p. 62] (1.66) with C=1 12π⎫summationdisplay i⎫integraldisplay γiκ(l)dl+1 24⎫summationdisplay j⎫parenleftBiggπ ϕj–ϕj π⎫parenrightBigg , (1.67) whereκ(l)(l∈arcγi⊂∂Ω) denotes the curvature of the arc γi.I ts h o u l db en o t e d , however, that the three-term formula (1.66) does not imply Nﬂ(λ)=O(1). On the contrary, the problem of determining α0=i n f αin the estimate Nﬂ(λ)= O(λα) is a very difﬁcult one; the circle problem discussed in Section 1.3.2 being an illustrative example. To compare the quantal spectra of different systems, one has to get rid of the system-dependent constants in N(λ), which is achieved by “unfolding” the spectra byxn:=N(λn). The unfolded spectrum {xn}n∈Nhas by construction a unit mean level spacing, and thus the corresponding counting function ˆN(x): =#{xnux} reads ˆN(x)=x+ˆNﬂ(x). Obviously, 1 x–x1x⎫integraldisplay x1ˆNﬂ(y)dy→0f o r x→∞ . (1.68) In analogy to the approach discussed in Section 1.3.10 for the circle problem, we are interested in the statistical properties of the normalized remainder term W(x): =ˆNﬂ(x)⎫radicalbigD(x), (1.69) where D(x) denotes the variance ( /ksiis a constant to be given below) D(x): =/ksi x–x1x⎫integraldisplay x1⎫parenleftBigˆNﬂ(y)⎫parenrightBig2dy. (1.70) We now consider W(x) as a random variable, where xis chosen randomly from the interval [x1,xm]and ask whether W(x) possesses in the limit xm→∞ a limit distribution. If a limit distribution exists, it has by construction a second momentof one (if the second moment exists) and a ﬁrst vanishing moment. 36 1W e y l ’ s L a w We are now in a position to formulate the following Conjencture 1.1 ([92, 93]) For bound conservative and scaling quantum systems the quantity W (x), Equation (1.69), possesses for x →∞ a limit distribution with zero mean and unit variance. This distribution is absolutely continuous with respect to Lebesgue measure on the real line with a density f (α)i.e. lim xm→∞1 xmxm⎫integraldisplay x1g(W(x))ρ⎫parenleftBiggx xm⎫parenrightBigg dx=∞⎫integraldisplay –∞g(α)f(α)dα, (1.71) where g (x)is a bounded continuous function on R,a n d ρ(x)v0a probability density on [0, 1].F u r t h e r m o r e , ∞⎫integraldisplay –∞αf(α)dα=0 ,∞⎫integraldisplay –∞α2f(α)dα= 1 . (1.72) If the corresponding classical system is strongly chaotic, having only isolated and unstable periodic orbits, then f (α)is universally a Gaussian, f(α)=1√ 2πe–α2/2. (1.73) In contrast, a classically integrable system leads to a system-dependent non-Gaussian density f (α). Here a few remarks are in order. i) Th e normalization used in the deﬁni- tion (1.69) is crucial in order for a limit distribution to exist since in all interesting cases D(x)d i v e r g e sf o r x→∞ . From Berry’s [94] semiclassical analysis one obtains for generic integrable billiards D(x)→c√x,x→∞ , (1.74) where cis some nonuniversal constant. (For rigorous results, see the discussion of the torus billiard in Section 1.3.9 and [95]). In contrast, for generic classically chaotic systems one expects D(x)→1 2π2/betatwolnx,x→∞ , (1.75) with /betatwo= 1 for systems with anti-unitary symm etry (for example time-reversal sym- metry) and /betatwo= 2 for systems without such a symmetry. ii) The constant /ksiin (1.70) takes the value /ksi=2 / 3 , i f D(x) obeys (1.74), and /ksi= 1 in the case of (1.75). iii) The conjecture is proven for some integrable systems like the torus (Gauss circle) problem, see [96] for a review. iv) The conjecture has been checked numer-ically for several integrable (like the isospectral billiard shown in Figure 1.5) and chaotic systems [88, 93] and has been foun d to hold with high statistical signiﬁ- cance. v) In Figure 1.4 we show the numerical evaluation of f(α) for the strong- ly chaotic Hadamard–Gutzwiller model [ 64] which is the quantum version of the 1.3 Weyl’s Law with Remainder Term. I 37 geodesic ﬂow on a compact Riemann surface of genus two (for details, see Sec- tion 1.4). For this system there exists the rigorous Selberg trace formula [45] (see Equation (1.95) below) which yields for the remainder term ˆNﬂ(x) the explicit ex- pression (see (1.107) below) ˆNﬂ(x)=1 πargZ⎫parenleftBigg1 2+ix⎫parenrightBigg (1.76) in terms of the Selberg zeta function Z(s) evaluated on the critical line s=1 / 2+ ix. For the numerical computation in Figure 1.4 we used the ﬁrst 6000 eigenvalues with positive parity (computed by the boun dary-element method [97]) of a generic (nonarithmetic) Riemann surface whose fundamental domain in the Poincaré-disk model for hyperbolic geometry is described in [97]. We conclude that the computedhistogram is in nice agreement with the conjecture (1.73). vi) In many respects the nontrivial zeros of the Riemann zeta function /dzeta(s) behave like the scaled eigenval- ues of a hypothetical classically chaotic s ystem without anti-unitary symmetry, see Sections 1.4.8 and 1.4.9. The analogue of (1.76) reads ˆN ﬂ(x)=1 πarg/dzeta⎫parenleftBigg1 2+ix⎫parenrightBigg (see (1.108) below) counting only the zeros {xn}n∈N,/dzeta(1/2 + ixn)=0 ,w i t hR e xn>0 and –1/2 < Im xn< 1/2 . It has been shown by Selberg’s moment method [98–100] that the corresp onding quantity W(x), with D(x)~1 / 2 π2ln ln x, has a Gaussian limit distribution in accordance with the c onjecture. For a numerical calculation off(α) using the ﬁrst 50 000 zeros and the 50 000 zeros starting from the 1020+ 143 780 420 th zero, respectively, see Figure 8 in [101], which shows that the con- vergence of the probability distribution to the proven Gaussian limit distribution is very slow. -3 -2 -1 0 1 2 x0.00.050.10.150.20.250.30.350.40.45y 3 Figure 1.4 The distribution function f(α)is shown for the strongly chaotic Hadamard-Gutzwiller model in comparisonwith the conjectured Gaussian normal distribution (dashedcurve). 38 1W e y l ’ s L a w 1.4 Weyl’s Law with Remainder Term. II 1.4.1 The Laplace–Beltrami Operator on d-Dimensional Compact Riemann Manifolds Md and the Pre-Trace Formula In many physical applications (ergodic theory, quantum mechanics, nonlinear op- tics, general relativity, string theory, and cosmology) one has to deal with the wave equation (or heat or Schrödinger equation) on non-Euclidean spaces. Important ex- amples are d-dimensional manifolds or orbifolds Mdendowed with a Riemannian metric for which the Euclidean Laplacian has to be replaced by the correspondingLaplace–Beltrami operator. For simplicity , we discuss only manifolds with constant Gaussian curvature K. Let us ﬁrst consider smooth compact Riemannian manifolds M dwithout bound- ary which are well studied and for which one can derive exact trace formulae and therefore can obtain full information on Weyl’s law and even on Carleman’slaw [102, 103] involving the eigenfunctions. The simplest case of zero curvature K=0 i . e . ﬂ a t t o r i M d Γ=Rd/Γ,w h e r e Γis a group of motions isomorphic to Zd, which are compact Riemannian manifolds, has already been discussed in Sec- tion 1.3. The case of homogeneous manifolds with constant positive curvature K=+ 1i s also well understood but will not be treated here. The case of compact manifolds with constant negative curvature K=– 1a n dd i - mension dv2 is highly nontrivial since the eigenvalues and eigenfunctions of the Laplace–Beltrami operator corresponding to the non-Euclidean (hyperbolic) metric are not known analytically. The geodesic ﬂow i.e. the free motion of a point par- ticle on these hyperbolic manifolds was already studied by Jacques Hadamard in1898 [104, 105] and has played an important role in the development of ergodic theory ever since. Hadamard proved that all trajectories in this system are unstable and that neighboring trajectories diverg e in time at an exponential rate, the most striking property of deterministic chaos . In 1980, Martin Gutzwiller drew attention to this system as a prototype example for quantum chaos [106]. Today the quantum sys- tem governed by the free Schrödinger equation i.e. the eigenvalue problem of the Laplace–Beltrami operator on these hyperbolic manifolds (or orbifolds), is known as the Hadamard–Gutzwiller model [64, 65, 107]. In dimension d=3 ,h y p e r b o l i c manifolds are possible candidates for the spatial section of the Universe and are investigated in cosmology [108]. In order to deﬁne a hyperbolic manifold, one considers Iso H d,t h eg r o u po f isometries onHd(i.e. the distance-preserving bijections on Hd), whereHdis the d-dimensional hyperbolic space. The action of an isometry γofHdis denoted by γ(z)w i t h z∈Hd. Take a discrete subgroup Γof IsoHdand identify all points of Hdwhich can be transformed into each other by an element of Γ.T h o s ep o i n t sa r e called Γ-equivalent, and we put them into an equivalence class Γ(z)=⎫braceleftbigγ(z):γ∈Γ⎫bracerightbig with z∈Hd. The set of those classes deﬁnes the hyperbolic d-manifold represented 1.4 Weyl’s Law with Remainder Term. II 39 by the quotient space Md:=Hd/Γ=⎫braceleftBig Γ(z):z∈Hd⎫bracerightBig . To visualize a given manifold, we have to take one representative from each class such that the set of all repre- sentatives yields a simply connected set in Hd, called the fundamental domain ΩΓ. Here we discuss only compact manifolds whose fundamental domain is of ﬁnitevolume,|Ω Γ|<∞.O n ec a nc o v e ra l lo f Hdwith Γ-translates of ΩΓ.T h i sp r o d u c e s a tessellation ofHdin analogy to the case discussed in Section 1.3 for ﬂat tori. The group Γis then called a hyperbolic crystallo graphic group or simply a hyper- bolic lattice. The task is then to study the eigenvalue problem of the hyperbolic Laplacian – Δu(z)=λu(z),z∈Hd,u∈L2⎫parenleftBig Hd/Γ,/khi⎫parenrightBig ,w h e r e uisautomorphic i.e. sat- isﬁes u(γ(z)) = ¯ /khi(γ)u(z)f o ra l l γ∈Γand z∈Hd.H e r e /khiis any one-dimensional unitary representation of Γ, also called a character which satisﬁes⎫vextendsingle⎫vextendsingle⎫vextendsingle/khi(γ)⎫vextendsingle⎫vextendsingle⎫vextendsingle2=1f o r allγ∈Γ. Due to the compactness of Md,t h es p e c t r u mo f– Δis discrete with 0=λ0<λ1uλ2u....( w h e t h e r λ0=0e x i s t sd e p e n d so n Md). Let us consider the resolvent kernel G Γ(z,z/prime;λ)o nHd/Γforf∈L2⎫parenleftBig Hd/Γ,/khi⎫parenrightBig ⎫bracketleftBig (–Δ–λ)–1f⎫bracketrightBig (z)=⎫integraldisplay ΩΓGΓ(z,z/prime;λ)f(z/prime)dμ(z/prime) , (1.77) where λ∈C\[0,∞). We then obtain the correlation function [107] CF(z,z/prime): =⎫summationdisplay nF(λn)en(z)¯en(z/prime)=1 π∞⎫integraldisplay 0F(λ/prime)d i s c GΓ(z,z/prime;λ/prime)dλ/prime, (1.78) where the spectral function F(λ) is assumed to obey the following sufﬁcient condi- tions: –F(λ) is holomorphic in a strip enclosing the positive real axis, –F(λ)d r o p sf a s t e rt h a n λ–d/2forλ→∞ . The last condition is imposed to ensure convergence of the above expression for all z,z/prime∈Hdincluding the diagonal z=z/prime.F o r z=/z/primeweaker conditions are sufﬁcient. Furthermore, we have introduced the discontinuity of GΓacross the cut in the λ- plane discGΓ(z,z/prime;λ): =l i m ε→0+1 2i⎫bracketleftbigGΓ(z,z/prime;λ+iε)–GΓ(z,z/prime;λ–iε)⎫bracketrightbig. Since CF(z,z/prime) is identical to the automorphic kernel of the operator F(–Δ), we obtain the pre-trace formula ⎫summationdisplay nF(λn)=T r F(–Δ)=⎫integraldisplay ΩΓCF(z,z)dμ(z). 1.4.2 The Sum Rule for the Automorphic Eigenfunctions on Md In the next step, we make use of the alternative representation of the resolvent kernel which expresses the Γ-invariant kernel GΓas a sum (“method of images”) 40 1W e y l ’ s L a w over the free resolvent kernel G(d) 0(z,z/prime;λ)o nHd GΓ(z,z/prime;λ)=⎫summationdisplay γ∈Γ/khi(γ)G(d) 0(z,γ(z/prime);λ). The crucial point now is that G(d) 0is explicitly known for all dv2, see [109]. In- troduce the wave numbers pnviap0:=⎫parenleftbig(d–1 ) / 2⎫parenrightbigiand pn:=⎫radicalbig λn–(d–1 )2/4v0 fornv1. Here p0belongs to λ0= 0 (if it exists), and pn,nv1, to the eigenvalues λnv(d–1 )2/4, where we have assumed that there are no so-called “small eigenval- ues” with 0 < λn<(d–1 )2/4. It is now convenient to replace the spectral function F(λ)b yan e w spectral function h(p): =F⎫parenleftBigg p2+(d–1)2 4⎫parenrightBigg =F(λ):C→C, which has to fulﬁl the following sufﬁcient conditions •h(–p)=h(p) •h(p) is holomorphic in the strip⎫vextendsingle⎫vextendsingle⎫vextendsingleImp⎫vextendsingle⎫vextendsingle⎫vextendsingleud–1 2+ε,ε> 0 (1.79) •h(p)=O⎫parenleftBig p–d–δ⎫parenrightBig ,δ>0f o r⎫vextendsingle⎫vextendsingle⎫vextendsinglep⎫vextendsingle⎫vextendsingle⎫vextendsingle→∞ . Then the correlation function takes the ﬁnal form of a “sum rule” for the automorphic eigenfunctions e n(dv2) [107] ∞⎫summationdisplay n=0h(pn)en(z)¯en(z/prime)=2 π⎫summationdisplay γ∈Γ/khi(γ)∞⎫integraldisplay 0ph(p)ˆΦ(d)(cosh d(z,γ(z/prime));p)dp, (1.80) where d(z,z/prime) denotes the hyperbolic distance between arbitrary points z,z/prime∈Hd. d(z,z/prime) is a point-pair invariant, i.e. d(γ(z),γ(z/prime)) = d(z,z/prime)f o ra l l γ∈Γand z,z/prime∈ Hd.F o r z=z/primethe distance τγ:=d(z,γ(z)) is the length of a closed orbit, but which is in general not a periodic one. The function ˆΦ(d)(y;p)i se x p l i c i t l yg i v e nb y( yv1) ˆΦ(d)(y;p)=π (2π)d/2(y2–1 )(2–d)/4 2p⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingleΓ⎫parenleftbigip+(d–1)/2⎫parenrightbig Γ(ip)⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle2 P(2–d)/2 –1/2+ ip(y) , (1.81) where Pμ ν(y) is the associated Legendre function of the ﬁrst kind. 1.4.3 Weyl’s Law on Mdand its Generalization by Carleman At this point let us introduce the generalized counting function N(d) Γ(λ;z,z/prime): =⎫summationdisplay λnuλen(z)¯en(z/prime), (1.82) 1.4 Weyl’s Law with Remainder Term. II 41 which for z=z/primegives Carleman’s function⎫summationtext λnuλ⎫vextendsingle⎫vextendsingle⎫vextendsingleen(z)⎫vextendsingle⎫vextendsingle⎫vextendsingle2[102,103] and after integrating over ΩΓthe usual counting function N(d) Γ(λ)=⎫integraldisplay ΩΓN(d) Γ(λ;z,z)dμ(z)=⎫summationdisplay λnuλ1 (1.83) (since⎫integraltext ΩΓem(z)¯en(z)dμ(z)=δmn). We then obtain from the sum rule (1.80) the explicit formula dN(d) Γ(λ;z,z/prime)=d N(d) Γ(λ;z,z/prime)+d N(d) Γ,ﬂ(λ;z,z/prime) (1.84) with dN(d) Γ(λ;z,z/prime): =1 πˆΦ(d)⎛⎜⎜⎜⎜⎜⎜⎜⎝cosh d(z,z/prime);⎫radicalBigg λ–⎫parenleftBiggd–1 2⎫parenrightBigg2⎞⎟⎟⎟⎟⎟⎟⎟⎠dλ (1.85) and dN(d) Γ,ﬂ(λ;z,z/prime): =1 π⎫summationdisplay γ∈Γ/prime/khi(γ)ˆΦ(d)⎛⎜⎜⎜⎜⎜⎜⎜⎝cosh d(z,γ(z/prime));⎫radicalBigg λ–⎫parenleftBiggd–1 2⎫parenrightBigg2⎞⎟⎟⎟⎟⎟⎟⎟⎠dλ, where Γ/prime:=Γ\{I}(Idenotes the identity) and /khi(I) = 1 was used. From our discus- sion of the trace formula for the tori Tdwe expect that (1.85) gives the asymptotical- ly leading smooth contribution to the generalized counting function (1.82). With d(z,z) = 0 we obtain from (1.85) for z=z/prime⎫parenleftBigg p:=⎫radicalBig λ–⎫parenleftbig(d–1 ) / 2⎫parenrightbig2⎫parenrightBigg N(d) Γ(λ;z,z): =λ⎫integraldisplay ((d–1)/2)2dN(d) Γ(λ;z,z)=2 πp⎫integraldisplay 0ˆΦ(d)⎫parenleftbig1;p/prime⎫parenrightbigp/primedp/prime, which no longer depends on z!H e r e ˆΦ(d)⎫parenleftbig1;p⎫parenrightbigfollows from (1.81) ˆΦ(d)(1;p)=π (2π)d/21 2p⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingleΓ⎫parenleftbigip+(d–1)/2⎫parenrightbig Γ(ip)⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle2 lim y→1+P(2–d)/2 –1/2+ ip(y) (y2–1 )(d–2)/4 =d (4π)d/2Γ(1+d/2)·π 2p·⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingleΓ⎫parenleftbigip+(d–1)/2⎫parenrightbig Γ(ip)⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle2 (1.86) and thus N(d) Γ(λ;z,z)=d (4π)d/2Γ(1+d/2)p⎫integraldisplay 0⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingleΓ⎫parenleftbigip /prime+(d–1)/2⎫parenrightbig Γ(ip/prime)⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle2 dp/prime. (1.87) 42 1W e y l ’ s L a w Using the asymptotic expansion ⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingleΓ⎫parenleftbigip+(d–1)/2⎫parenrightbig Γ(ip)⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingle2 =pd–1⎫parenleftBigg 1+O⎫parenleftBigg1 p2⎫parenrightBigg⎫parenrightBigg (p→∞ ), we immediately obtain N(d) Γ(λ;z,z)=1 (4π)d/2Γ(1+d/2)λd/2+O⎫parenleftBig λd/2–1⎫parenrightBig (λ→∞ ) (1.88) and after integration over ΩΓthe non-Euclidean analog of Weyl’s law (dv2) N(d) Γ(λ)=|ΩΓ| (4π)d/2Γ(1+d/2)λd/2+O⎫parenleftBig λd/2–1⎫parenrightBig . (1.89) Since one can show that the remainder term satisﬁes N(d) Γ,ﬂ(λ;z,z)=O⎫parenleftBig λd/2⎫parenrightBig ,w e obtain Carleman’s law N(d) Γ(λ;z,z)=⎫summationdisplay λnuλ⎫vextendsingle⎫vextendsingle⎫vextendsingleen(z)⎫vextendsingle⎫vextendsingle⎫vextendsingle2=λd/2 (4π)d/2Γ(1+d/2)+Oz⎫parenleftBig λd/2⎫parenrightBig (λ→∞ ) , (1.90) which is a generalization of Weyl’s law since it is not only a statement about the eigenvalues but also about the eigenfunctions. Note, however, that the sum rule (1.80) – being an exact explicit expression – contains much more information. To see this, let us consider the simplest case d= 2 in more detail. 1.4.4 The Selberg Trace Formula and Weyl’s Law In the case d= 2 we consider compact Riemann surfaces M2=H2/Γof genus gv2w i t h Γa strictly hyperbolic Fuchs ian group of the ﬁrst kind, Γ∈PSL(2,R). Such groups are characterized by the fact that all their group elements γ(except the unity I) are hyperbolic. Here we choose for H2the Poincaré!upper half plane H2=⎫braceleftbigz=x+iy:x,y∈R,y>0⎫bracerightbigwith the hyperbolic metric ds2=dx2+dy2 y2, which is invariant under fractional linear transformations: z→γ(z): =az+b cz+d, where a,b,c,d∈Rand ad–bc= 1. Then the Laplace–Beltrami operator is Δ= y2⎫parenleftBig ∂2/∂x2+∂2/∂y2⎫parenrightBig . It is also invariant under the group actions γ∈Γ.W et h e n obtain from (1.81) ˆΦ(2)(y;p)=1 4tanh( πp)P–1/2+ ip(y), 1.4 Weyl’s Law with Remainder Term. II 43 where Pν(y) denotes the Legendre function of the ﬁrst kind. Then the sum rule (1.80) takes the simple form ( /khi(γ)=1∀γ∈Γ,p0=i/2, pn=⎫radicalbig λn–1 / 4 v0,nv1) [110] ∞⎫summationdisplay n=0h(pn)en(z)¯en(z/prime)=1 2π⎫summationdisplay γ∈Γˆh⎫parenleftbigcosh d⎫parenleftbigz,γ(z/prime)⎫parenrightbig⎫parenrightbig, (1.91) where the hyperbolic distance on H2is given by cosh d(z,z/prime)=1+(x–x/prime)2+y2+y/prime2 2yy/prime. Here ˆhdenotes the Mehler transform of the spectral function hwhich is deﬁned by the relations h(p)=∞⎫integraldisplay 1ˆh(y)P–1/2+ ip(y)dy (1.92) ˆh(y)=∞⎫integraldisplay 0ptanh( πp)h(p)P–1/2+ ip(y)dp. (1.93) In [107, 110] it was shown that the sum rule (1.91) can be used to compute numer- ically the eigenfunctions en(z) called nonholomorphic (or automorphic) forms or Maass waveforms , at least if the eigenvalues λnare not too large. Taking the trace of the sum rule (1.91) one gets with (1.93) and P–1/2+ ip( 1 )=1( t h eS L ( 2 , R)-invariant area element onH2is dμ(z)=d xdy/y2) ∞⎫summationdisplay n=0h(pn)=|ΩΓ| 2π∞⎫integraldisplay 0ptanh( πp)h(p)dp+1 2π⎫summationdisplay γ∈Γ/prime⎫integraldisplay ΩΓˆh⎫parenleftbigcosh d⎫parenleftbigz,γ(z)⎫parenrightbig⎫parenrightbigdμ(z). (1.94) To evaluate the sum over γ∈Γ/primeinvolving the integral over ˆhis a nontrivial task and was ﬁrst achieved by Atle Selberg [45, 46] leading to the famous Selberg trace formula ∞⎫summationdisplay n=0h(pn)=|ΩΓ| 2π∞⎫integraldisplay 0ptanh( πp)h(p)dp+∞⎫summationdisplay {γ}p∞⎫summationdisplay n=1l(γ) 2s i n h⎫parenleftbignl(γ)/2⎫parenrightbig˜h⎫parenleftbignl(γ)⎫parenrightbig, (1.95) where ˜h(x) denotes the Fourier transform of h(p) ˜h(x): =1 2π∞⎫integraldisplay –∞eipxh(p)dp. 44 1W e y l ’ s L a w The sum on the right-hand side of (1.95) runs over the length spectrum⎫braceleftbigl(γ)⎫bracerightbig pof the primitive periodic orbits of the geodesic ﬂow on the surface M2=H2/Γ.N o t i c e that the length spectrum is uniquely given by the conjugacy classes of the hyper- bolic elements in Γas can be seen as follows. The elements γ∈Γof the discrete subgroups of PSL(2, R) can be represented as 2 ~2m a t r i c e s γ=⎫parenleftBig ab cd⎫parenrightBig with real en- tries and det γ=ad–bc= 1. For a strictly hyperbolic group one has, for all γ=/±I:⎫vextendsingle⎫vextendsingle⎫vextendsingleTrγ⎫vextendsingle⎫vextendsingle⎫vextendsingle=⎫vextendsingle⎫vextendsingle⎫vextendsinglea+d⎫vextendsingle⎫vextendsingle⎫vextendsingle> 2. The Jordan form of these matrices takes the form⎫parenleftBiga0 01 /a⎫parenrightBig with |a|>1 ,a n dt h ea c t i o no f γgives z→γ(z)=a2z,w h e r e N(γ): =a2is called the norm of the element γ. Since there exists a unique relationship between the conjugacy classes in Γand the homotopy classes of closed paths on H2, one can deﬁne in each class a length l(γ) by the length of the shortest path, and then obtains N(γ)=el(γ), l(γ)>0 .T h el e n g t h l(γ)i st h e ng i v e nb yc o s h ( l(γ)/2) =⎫vextendsingle⎫vextendsingle⎫vextendsingleTrγ⎫vextendsingle⎫vextendsingle⎫vextendsingle/2. The sums and integrals in the Selberg trace formula are all absolutely convergent if the spectral function h(p) satisﬁes conditions (1.79) for d=2 .T h eS e l b e r gt r a c e formula (1.95) can be considered as a ge neralization and noncommutative ana- logue of the classical Poisson summation formula (1.33), respectively of the trace formulae (1.37) and (1.42–1.44) for ﬂat tori. From the Selberg trace formula (1.95) we can immediately read off the complete Weyl term of the counting function (see the discussion above for general dv2) NM2 Γ⎫parenleftBigg p2+1 4⎫parenrightBigg :=|ΩΓ| 2πp⎫integraldisplay 0p/primetanh( πp/prime)dp/prime, (1.96) which behaves as NM2 Γ⎫parenleftBigg p2+1 4⎫parenrightBigg =|ΩΓ| 6p3+O⎫parenleftBig p5⎫parenrightBig forp→0, and hence we obtain W e y l ’ sl a wo nc o m p a c tR i e m a n ns u r f a c e so fg e n u sg v2 NM2 Γ⎫parenleftBigg p2+1 4⎫parenrightBigg =|ΩΓ| 4π⎫parenleftBigg p2–1 12⎫parenrightBigg +O⎫parenleftBig pe–2πp⎫parenrightBig forp→∞ . (1.97) This asymptotic formula contains the standard Weyl term proportional to λand the volume |ΩΓ|, no term proportional to√ λ,s i n c eM2has no boundary, it has a constant term and then an exponentially small correction. Below we shall also derive the ﬂuctuating remainder term of the counting function. 1.4.5 The Trace of the Heat Kernel on M2 Choosing the spectral function h(p)=e–(p2+1/4)t,t>0 ,w eo b t a i nf o rt h e trace of the heat kernel on a compact Riemann surface M2of genus g v2 possessing the area 1.4 Weyl’s Law with Remainder Term. II 45 |ΩΓ|=4π⎫parenleftbigg–1⎫parenrightbig(Gauss–Bonnet) the explicit formula ( t> 0) [47] ΘM2(t): =∞⎫summationdisplay n=0e–λnt=∞⎫summationdisplay n=0e–(p2 n+1/4)t=ΘM2 1(t)+ΘM2 2(t), ΘM2 1(t): =|ΩΓ|e–t/4 (4πt)3/2∞⎫integraldisplay 0x sinh (x/2)e–x2/4tdx =|ΩΓ| 4πtN⎫summationdisplay n=0bntn+O⎫parenleftBig tN⎫parenrightBig , t→0+, b0=1 ,bn=(–1)n 22nn!⎡⎢⎢⎢⎢⎢⎢⎣1+2n⎫summationdisplay k=1⎫parenleftBign k⎫parenrightBig⎫parenleftBig 22k–1–1⎫parenrightBig |B2k|⎤⎥⎥⎥⎥⎥⎥⎦, n∈N, ΘM2 2(t): =e–t/4 4√ πt⎫summationdisplay {γ}p∞⎫summationdisplay n=1l(γ) sinh⎫parenleftbig⎫parenleftbignl(γ)⎫parenrightbig/2⎫parenrightbige–n2l2(γ)/4t,(1.98) where B2kare the Bernoulli numbers ( b1= –1/3, b2= 1/15). This formula is the generalization of Poisson’s transformation formula for the elliptic theta function θ3discussed in Section 1.3.6 to Riemann surfaces of genus gv2. Thus ΘM2(t)c a n be called the non-Euclidean theta function. The formula (1.98) is quite interesting since it shows that for compact Riemann surfaces of genus gv2t h ec o m p l e t e small- ta s y m p t o t i c si se x p l i c i t l yk n o w n ,s e et h et e r m ΘM2 1, and not just the leading Weyl term |ΩΓ|/4πt. Furthermore, there even exists a closed expression for this contribution as an integral which is valid for all t> 0 and is not just an asymptotic result in the limit t→0+. Moreover, the remainder term ΘM2 2also has an explicit representation as a sum over the length spectrum of periodic orbits. This term is exponentially small in the limit t→0+and is determined by the shortest periodic orbit with primitive length l(γ1)>0 ,i . e . ΘM2 2(t)=O⎫parenleftBig t–1/2e–l2(γ1)/4t⎫parenrightBig in close analogy with the behavior on the torus T2. 1.4.6 The Trace of the Resolvent on M2and Selberg’s Zeta Function In order to calculate the trace of the resolvent of – ΔonM2, one is led to substi- tute h(p)=⎫parenleftBig 1/4 + p2–λ⎫parenrightBig–1in the trace formula. This function violates, however, the asymptotic condition in Equation (1.79) for⎫vextendsingle⎫vextendsingle⎫vextendsinglep⎫vextendsingle⎫vextendsingle⎫vextendsingle→∞ ,i . e .t h er e s o l v e n ti sn o t of trace class as a consequence of Weyl’s law which tells us that the eigenvalues behave as λ n=1 / 4+ p2 n~(4π/ΩΓ)nforn→∞ . Thus the resolvent has to be regu- larized properly. A very convenient regularization is given by the following choice.(Res,R eσ>1 ) h(p)=1 p2+(s–1 / 2 )2–1 p2+(σ–1 / 2 )2, 46 1W e y l ’ s L a w which fulﬁlls all the conditions (1.79) in the trace formula. For the integral (Weyl) term in the trace formula (1.95) one then obtains |ΩΓ| 2π∞⎫integraldisplay 0ptanh( πp)h(p)dp=–|ΩΓ| 2π⎫parenleftbigψ(s)–ψ(σ)⎫parenrightbig, where ψ(s): = Γ/prime(s)/Γ(s) is the digamma function. Using the Fourier transform (Res>1 / 2 , xv0) 1 2π∞⎫integraldisplay –∞eipx p2+(s–1 / 2 )2dp=1 2s–1e–(s–1/2)x, theSelberg trace formula for the trace of the regularized resolvent reads (Re s,R eσ>1 ) ∞⎫summationdisplay n=0⎫parenleftBigg1 λn+s(s–1 )–1 λn+σ(σ–1 )⎫parenrightBigg =–|ΩΓ| 2π⎫parenleftbigψ(s)–ψ(σ)⎫parenrightbig +1 2s–1A(s)–1 2σ–1A(σ) , (1.99) where the function A(s)i sf o rR e s> 1 given by the absolutely convergent double sum A(s): =⎫summationdisplay {γ}p∞⎫summationdisplay n=1l(γ)e–(s–1/2)nl(γ) 2s i n h⎫parenleftbignl(γ)/2⎫parenrightbig. It was one of Selberg’s deep insights to realize that A(s) can be rewritten for Re s>1 as the logarithmic derivative of a kind of zeta function Z(s): A(s)=⎫summationdisplay {γ}p∞⎫summationdisplay n=1l(γ)e–(s–1/2)nl(γ) enl(γ)/2–e–nl(γ)/2=⎫summationdisplay {γ}p∞⎫summationdisplay n=1l(γ)e–snl(γ) 1–e–nl(γ) =⎫summationdisplay {γ}p∞⎫summationdisplay n=1l(γ)e–snl(γ)∞⎫summationdisplay k=0e–knl(γ)=⎫summationdisplay {γ}p∞⎫summationdisplay k=0l(γ)∞⎫summationdisplay n=1e–(s+k)nl(γ) =⎫summationdisplay {γ}p∞⎫summationdisplay k=0l(γ)e–(s+k)l(γ) 1–e–(s+k)l(γ)=⎫summationdisplay {γ}p∞⎫summationdisplay k=0d dsln⎫parenleftBig 1–e–(s+k)l(γ)⎫parenrightBig =d dsln⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎫productdisplay {γ}p∞⎫productdisplay k=0⎫parenleftBig 1–e–(s+k)l(γ)⎫parenrightBig⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=:Z/prime(s) Z(s). Here we have deﬁned the Selberg zeta function (Res> 1) [45] Z(s): =⎫productdisplay {γ}p∞⎫productdisplay k=0⎫parenleftBig 1–e–(s+k)l(γ)⎫parenrightBig , (1.100) 1.4 Weyl’s Law with Remainder Term. II 47 which is given as a generalized Euler product over the lengths of the primitive periodic orbits. It follows from Selberg’s trace formula that the inﬁnite products in (1.100) are absolutely convergent for Re s>1 .R e p l a c i n g A(s)a n d A(σ) in (1.99) by Selberg’s zeta function, we obtain an exact relation [47] which expresses thetrace of the regularized resolvent of – Δon an arbitrary compact Riemann surface of genus gv2 in terms of the well-known ψ-function and Selberg’s zeta function. On the other hand, this relation allows us to prove that Z(s)c a nb ec o n t i n u e dt ot h e left of Re s= 1. This can be seen by rewriting (1.99) as follows [47] 1 2s–1Z/prime(s) Z(s)=– 2⎫parenleftbigg–1⎫parenrightbigψ(σ)+⎫parenleftBigg1 2σ–1Z/prime(σ) Z(σ)–1 σ(σ–1 )⎫parenrightBigg (1.101) +2⎫parenleftbigg–1⎫parenrightbigψ(s)+1 s(s–1 )+∞⎫summationdisplay n=1⎫parenleftBigg1 λn+s(s–1 )–1 λn+σ(σ–1 )⎫parenrightBigg . Note that the sum over the eigenvalues no longer contains the zero mode λ0=0 . Keeping the regulator σﬁxed with Re σ> 1, we see that the right-hand side of (1.101), derived for Re s> 1, is actually meromorphic for all s∈C. Thus the left-hand side of (1.101) is also meromorphic, and so we obtain the analytic contin- uation of Z(s)o nC. In fact, further inspection shows that the Selberg zeta function is an entire function of sof order 2 whose “trivial” zeros are at s=–k,k∈N,w i t h multiplicity 2( g– 1)(2 k+ 1). Furthermore, s= 1 is a simple zero, and s=0i saz e r o of multiplicity 2 g– 1. In addition Z(s) can have a ﬁnite number of zeros on the real axis between 0 and 1 located at s=1 / 2±⎫radicalbig 1/4 – λncorresponding to the so-called “small” eigenvalues 0 < λn< 1/4. For surfaces of genus g>2 ,o n eh a sa tm o s t 4g– 3 small eigenvalues [111, 112], while in the case of g=2t h e r ei sa tm o s to n e small eigenvalue [113]. More importantly, Z(s) has an inﬁnite number of “nontrivial” zeros located at s= 1/2±ipn,pnv0, i.e. lying on the critical line Res=1 / 2 ,a n dt h u so n ec a ns a yt h a t theRiemann hypothesis is valid for Z(s), a very remarkable result! One therefore has the exact quantization condition (pn∈R) Z⎫parenleftBigg1 2+ipn⎫parenrightBigg = 0 (1.102) for the quantal eigenvalues λn=p2 n+1 / 4 v1/4 of the Schrödinger equation, which are completely determined by the lengths of the classical periodic orbits of the corresponding classical Hamiltonian system. The reason behind the validity of the Riemann hypothesis in this case is obvious- ly that s(s–1) is an eigenvalue of a self-adjoint operator, and hence is real, whenever s is a zero of Z(s) within the critical strip. The question of whether something sim- ilar holds for the nontrivial zeros of the Riemann zeta function, will be discussed below. The information on the zeros of Z(s) enables us now to eliminate the regulator σ in (1.101) by taking the limit σ→1. With ψ(1) = – γ,w h e r e γis Euler’s constant, we deﬁne the generalized Euler constant γΔ γΔ:= 2⎫parenleftbigg–1⎫parenrightbigγ+B 48 1W e y l ’ s L a w with B:= lim σ→1⎫parenleftBigg1 2σ–1Z/prime(σ) Z(σ)–1 σ(σ–1 )⎫parenrightBigg =1 2Z/prime/prime(1) Z/prime(1)–1. Since Z(s) possesses a simple zero at s=1 ,o n eh a s Z/prime(1) =/ 0 (actually Z/prime(1) > 0 holds) and thus the constant Bi sw e l ld e ﬁ n e d .W et h e no b t a i nf o rt h e trace of the regularized resolvent of –ΔonM2=H2/Γthe ﬁnal result [47] 1 s(s–1 )+∞⎫summationdisplay n=1⎫parenleftBigg1 λn+s(s–1 )–1 λn⎫parenrightBigg =1 2s–1Z/prime(s) Z(s)–γΔ–2 (g–1 )ψ(s) . (1.103) 1.4.7 The Functional Equation for Selberg’s Zeta Function Z(s) To derive the functional equation for Z(s), we notice that s(s– 1) is invariant under s→1–sand 2 s– 1 changes sign. If we then subtract (1.103) evaluated at 1 – sfrom the same expression evaluated at s,w eo b t a i n 1 2s–1d dslnZ(s) Z(1 –s)=2 (g–1 )⎫parenleftbigψ(s)–ψ(1 –s)⎫parenrightbig. Using the functional equation ψ⎫parenleftBigg1 2+z⎫parenrightBigg –ψ⎫parenleftBigg1 2–z⎫parenrightBigg =πtan(πz) for the digamma function this then leads, with z=s–1 / 2 ,tothe functional equation for Z (s) Z(s)=e x p⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝|ΩΓ|s–1/2⎫integraldisplay 0xtan(πx)dx⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠Z(1 –s) . (1.104) Evaluating the functional equation on the critical line i.e. choosing s=1 / 2 + ip, p∈R,w eg e t Z⎫parenleftBigg1 2+ip⎫parenrightBigg =e–2πiNM2 Γ(p2+1/4)Z⎫parenleftBigg1 2–ip⎫parenrightBigg , (1.105) where the smooth term NM2 Γof the counting function given in (1.96) enters as a phase. It follows that the function /ksi(p): =Z⎫parenleftBigg1 2+ip⎫parenrightBigg eiπNM2 Γ(p2+1/4) satisﬁes the simple functional equation /ksi(p)=/ksi(–p), and furthermore that /ksi(p)i s real if p∈R, i.e. on the critical line. 1.4 Weyl’s Law with Remainder Term. II 49 1.4.8 An Explicit Formula for the Remainder Term in Weyl’s Law on M2 and the Hilbert–Polya Conjecture on the Riemann Zeros Using the argument principle, one can derive the exact Weyl formula (pv0,p=/pn) for the counting function NM2 Γ⎫parenleftBigg p2+1 4⎫parenrightBigg =NM2 Γ⎫parenleftBigg p2+1 4⎫parenrightBigg +1 πargZ⎫parenleftBigg1 2+ip⎫parenrightBigg , (1.106) which proves that the ﬂuctuating term (remainder term) of the counting function is determined by the Selberg zeta function on the critical line NM2 Γ,ﬂ⎫parenleftBigg p2+1 4⎫parenrightBigg =1 πargZ⎫parenleftBigg1 2+ip⎫parenrightBigg . (1.107) The derivation of (1.106) is completely analogous to the well-known calculation leading to the counting function NR(t)f o rt h e nontrivial Riemann zeros NR(t)=NR(t)+1 πarg/dzeta⎫parenleftBigg1 2+it⎫parenrightBigg , (1.108) which counts the number of zeros of the Riemann zeta function /dzeta(s)i nt h er e g i o n 0<R e s<1 ,0<I m sut. Here the smooth term NR(t)i sg i v e nb yt h ef a m o u s Riemann–von Mangoldt formula [114] NR(t)=t 2πlnt–1+l n2 π 2πt+7 8+O⎫parenleftBigg1 t⎫parenrightBigg (t→∞ ) . (1.109) Note that Selberg introduced his zeta function Z(s) around 1950 in analogy with the Riemann zeta function /dzeta(s) with the intention to shed some light on the properties of the nontrivial Riemann zeros and the Riemann hypothesis . He noticed the striking similarities between his trace formula (1. 95) and the so-called explicit formulae in the theory of prime numbers [115], whose most general form is André Weil’s explicit formula [116]. Weil’s explicit formula establishes a deep relation between the nontrivial zeros ρn=1 / 2+ iτn,τn∈C,o f/dzeta(s) and the prime numbers p: ∞⎫summationdisplay n=1h(τn)=1 4π∞⎫integraldisplay –∞ψ⎫parenleftBigg1 4+iτ 2⎫parenrightBigg h(τ)dτ+h⎫parenleftBiggi 2⎫parenrightBigg –˜h(0)lnπ 2–⎫summationdisplay p∞⎫summationdisplay n=1lnp pn/2˜h(nlnp), (1.110) where the “test function” h(τ) satisﬁes the same conditions (1.79) as the spectral function in the Selberg trace formula for d=2 ,a n d ˜h(x) is again its Fourier trans- form. Here the sum on the right-hand side runs over all primes p.C o m p a r i n g Weil’s formula (1.110) with Selberg’s trace formula (1.95), one is tempted to inter-pret the nontrivial zeros of /dzeta(s) as eigenvalues of a hypothetical “Riemann operator” 50 1W e y l ’ s L a w and the logarithm of the prime numbers as the “lengths” l(p): =l n pof the primi- tive “periodic orbits” of the corresponding hypothetical geodesic ﬂow. The term on the right-hand side of (1.110) involving the summation over the primes then reads –⎫summationdisplay p∞⎫summationdisplay n=1l(p) enl(p)/2˜h(nl(p)) , (1.111) which is strikingly similar to the corresp onding term in the Selberg trace formu- la (1.95) involving the summation over periodic orbits. Note, however, the differ- ence between the denominator +2 sinh⎫parenleftBiggnl(γ) 2⎫parenrightBigg =enl(γ)/2–e–nl(γ)/2 in (1.95) which has a dynamical interpretation in terms of the linearized Poincaré recurrence map for unstable hyperbolic periodic orbits, see for example [82,92], andthe corresponding denominator – e nl(p)/2in (1.111), for which no dynamical inter- pretation has been found until now; see, however, the paper by Alain Connes [117] who has devised a hermitian operator whose eigenvalues are the nontrivial Rie-mann zeros. His operator is the Perron–Frobenius operator (called the transfer op- erator in physics) of a classical dynamic al system. In his framework he has found an explanation for the minus sign in (1.111). At ﬁrst sight it seems that there is another obstruction to the interpretation of the Riemann zeros as the eigenvalues of a dynamical system since the smoothcounting function NR(t) (1.109) goes asymptotically as λ/(2π)l nλ,i fw ep u t t=λ, which differs from the standard behavior according to Weyl’s law in dimension 2. It will be seen, however, in Section 1.5.2 that such logarithmic modiﬁcations toWeyl’s law can occur, for example in membrane problems, for which the domain Ω is unbounded. Mathematical wisdom has usually attributed the formulation of the idea of a hy- pothetical Riemann operator to Hilbert and Polya, independently, some time in the 1910s. (See Odlyzko’s correspondence with Polya [118].) There is another difference between the Riemann and the Selberg case. In the deﬁnition of Z(s) in (1.100) one has a double product, whereas /dzeta(s) involves only a single one. Furthermore, the “Euler factor” occurs in Z(s) with the (+1) in the exponent, and in the case of /dzeta(s) with a (–1). It turns out that, when one generalizes the Selberg zeta function to spaces of high er rank, the natural exponents are certain Euler characteristics which can take positive or negative values [119]. To get rid ofthe second product in (1.100), one simply considers the ratio R(s): =Z(s) Z(s+1 )=⎫productdisplay {γ}p⎫parenleftBig 1–e–sl(γ)⎫parenrightBig , (1.112) and ends up with the Ruelle zeta function R (s) [120], which is now a meromorphic function. R(s)o rr a t h e r1 / R(s) has been discussed in terms of Beurling’s general- ized prime numbers and in connection with a generalized prime number theo-rem [121]. 1.4 Weyl’s Law with Remainder Term. II 51 1.4.9 The Prime Number Theorem vs. th e Prime Geodesic Theorem on M2 The famous prime number theorem states that the number of primes up to x, π(x): =#⎫braceleftbigp:pux⎫bracerightbig, is asymptotically equal to the logarithmic integral, given for x>1b y(/fIntegraltext means the Cauchy principal value of the integral) li(x): =x/fIntegraldisplay 0dt lnt=x lnx+x (lnx)2+... (x→∞ ). The fact that the density of primes near xis about 1/ ln xwas already conjectured by Gauss in 1792 at the age of 15. To derive a formula for π(x) was Riemann’s main goal in his famous paper from 1859, and it was for this purpose that he studied/dzeta(s) which had been introduced for integer argument already in 1735 by Euler who discovered among several other relations the formula /dzeta(2) = π 2/6 and in 1737 established the Euler product for /dzeta(m),mv2. The prime number theorem was proved in 1896 independently by Hadamard and de la Vallée Poussin by using the Riemann zeta function. It is worthwhile noticing that the ﬁrst “elementary” proofwas found by Selberg in 1949, see for example [46]. If one associates the prime numbers with the “lengths” l(p): =l n p,t h ec o u n t i n g functionN(l): =#⎫braceleftbigp:l(p)ul⎫bracerightbigcounts the number of hypothetical “periodic orbits” with length up to l. The prime number theorem is then converted into N(l)==π⎫parenleftBig e l⎫parenrightBig ~l i⎫parenleftBig el⎫parenrightBig ~el l,( l→∞ ) . (1.113) It is this result which gives perhaps the strongest support to the Hilbert–Polya con- jecture, since it turns out that the counting function NM2 Γ(l): =#⎫braceleftbigγ∈Γ:l(γ)ul⎫bracerightbigof the genuine periodic orbits of the geodesic ﬂow on M2obeys Huber’s law [122] NM2 Γ(l)=l i⎫parenleftBig el⎫parenrightBig +O⎫parenleftBigge(3/4)l l⎫parenrightBigg ,( l→∞ ) . (1.114) This is a special case of the general prime geodesic theorem valid for the counting function of the lengths of the unstable p eriodic orbits of chaotic systems with a topological entropy τ> 0. In the general case, one has as leading term eτl/τl.T h u s Huber’s law is consistent with the we ll-known fact that the geodesic ﬂow on M2 is strongly chaotic, i.e. ergodic, mixing, possesses the Bernoulli property, and has topological entropy τ= 1. (Actually, all periodic orbits on M2are unstable and possess the same Lyapunov exponent λ(γ)=1 . ) Comparing (1.113) with (1.114), one concludes that the hypothetical dynamical system associated with the Riemann zeros s hould be chaotic, should have topologi- cal entropy τ= 1, and should possess a length spectrum of primitive periodic orbits exactly given by the logarithm of the primes, l(p)=l n p! The validity of Huber’s law (1.114) can be seen as follows. Due to the existence of the zero mode λ0= 0 with multiplicity one, ΘM2(t)=1+ O⎫parenleftBig e–λ1t⎫parenrightBig ,t→∞ ,h o l d s 52 1W e y l ’ s L a w for the trace of the heat kernel on M2. Furthermore, one infers from (1.98) that the complete Weyl term ΘM2 1(t) satisﬁes lim t→∞ΘM2 1(t) = 0, and thus the remainder term ΘM2 2(t) in (1.98) must satisfy lim t→∞ΘM2 2(t) = 1. One therefore obtains the condition lim t→∞e–t/4 2√πt∞⎫integraldisplay l1le–l2/4t–l/2dNM2 Γ(l)=1, which yields d NM2 Γ(l)=el/ldl+...forl→∞ in complete agreement with Huber’s law (1.114). In [123] an explicit formula for d NM2 Γ(l) was derived including an oscillating remainder term. The derivation starts from Selberg’s trace formula (1.95) and uses the Möbius inversion formula in complete analogy with Riemann’s explicit formula forπ(x). The formula was used to compute the lowest part of the length spectrum for the most symmetric compact Riemann surface of genus g=2u s i n gt h eﬁ r s t 200 eigenvalues, see Figure 1 in [123]. 1.5 Generalizations of Weyl’s Law 1.5.1 Weyl’s Law for Robin Boundary Conditions In Equations (1.66) and (1.67) we have given the three-term formula for the smooth term N(λ) for simply connected and bounded two-dimensional domains Ωwith smooth boundary for Dirichlet as well as for Neumann boundary conditions. A gen- eralization encountered in a nuclear physics context [124–126] are mixed or so- called Robin boundary conditions α(x)u(x)+∂nu(x)=0 (x∈∂Ω), (1.115) which leaves the problem self-adjoint when αis real. The Dirichlet and Neumann boundary conditions are recovered in the limit α→∞ andα→0, respectively. For constant αv0 and excluding corners, Sieber et al. [127] derived the three-term Weyl formula N(λ)=|Ω| 4πλ–|∂Ω| 4π⎡⎢⎢⎢⎢⎢⎣1–2⎛⎜⎜⎜⎜⎜⎝⎫radicalbigg 1+α2 λ–α√ λ⎞⎟⎟⎟⎟⎟⎠⎤⎥⎥⎥⎥⎥⎦√ λ +⎡⎢⎢⎢⎢⎢⎣1–3√ λ α⎫radicalbig 1+α2/λ–1⎫radicalbig 1+α2/λ⎤⎥⎥⎥⎥⎥⎦1 12π⎫integraldisplay ∂Ωκdl.(1.116) Since∂nu=O⎫parenleftBig√ λ⎫parenrightBig in the limit λ→∞ ,t h et e r m∂nuis asymptotically dominant in the boundary condition (1.115), and hence the mean spectrum will for ﬁxed αal- ways tend to the Neumann case. Therefore in the derivation of (1.116), λandα/√ λ 1.5 Generalizations of Weyl’s Law 53 have been considered as independent parameters. One observes that the general- ized Weyl law (1.116) interpolates neatly between the law (1.66), (1.67) for Dirichlet and Neumann boundary conditions. Formula (1.116) has been checked [127] in the case of the circle billiard, where 1/(12 π)⎫integraltext ∂Ωκdl= 1/6, for which the exact resolvent kernel is known in closed form. Apart from applications in nuclear physics, it was shown in [127] that the para- metric dependence of the spectrum on t he boundary condition is a very useful diagnostic tool in the analysis of spectra. 1.5.2 Weyl’s Law for Unbounded Quantum Billiards In Section 1.4.8 we have observed that the smooth term NR(λ)o ft h ec o u n t i n g function of the nontrivial zeros of the Rie mann zeta function grows asymptotical- ly as λlnλwhich contradicts the classical eigen value asymptotics given by Weyl’s law. Thus it appears that the interpreta tion of the nontrivial Riemann zeros as eigenvalues of the Laplacian is ruled out. It was pointed out, however, by BarrySimon [128, 129] that an asymptotic behavior of the form λlnλcan occur for the eigenvalues of the two-dimensional Dirichlet Laplacian for certain unbounded re- gions which have a purely discrete spectrum. Since this nonclassical Weyl asymp-totics again opens the possibility of iden tifying the nontrivial Riemann zeros with the eigenvalues of a hypothetical Riemann operator, it is important to determine also the nonleading terms of the countin g function for such unbounded systems. As a representative example we here quote only the result for the so-called hyper- bola billiard which is deﬁned by the two-dimensional Euclidean Dirichlet Laplacian in the “horn-shaped” region Ω=⎫braceleftBig⎫parenleftbigx,y⎫parenrightbig∈R 2 +:0ux·yu1⎫bracerightBig . It was shown by Simon [128] that this quantum system possesses a purely discrete spectrum although the corresponding cla ssical billiard has a continuous spectrum. In [130] the following asymptotic expansion for the trace of the heat kernel of the hyperbola billiard was derived ( t→0+) Θ(t): =T r etΔ=–lnt 4πt–a/prime 4πt+b 8√πt+O⎫parenleftBig t–1/4⎫parenrightBig , (1.117) where a/prime=2 l n ( 2 π)–1– γ= 2.0985...,b=4π3/2/Γ2(1/4)= 1.6944....U s i n g the Karamata–Tauberian theorem in the form [129]: lim t→0+⎫bracketleftBig –(tr/l nt)TretΔ⎫bracketrightBig =cif and only if lim λ→∞⎫bracketleftbig(λ–r/l nλ)N(λ)⎫bracketrightbig=c/Γ(r+ 1), one derives from (1.117) the leading term for the counting function N(λ)=1 4πλlnλ+...(λ→∞ ). To obtain the next terms one uses a theorem by Brownell [90] which allows to obtain a smoothed counting function N(λ). Form (1.117) one then obtains the mean 54 1W e y l ’ s L a w asymptotic growth of the number of eigenvalues of the hyperbola billiard [130] N(λ)=1 4πλlnλ–a 4πλ+b 4π√ λ+O⎫parenleftBig λ1/4lnλ⎫parenrightBig (λ→∞ ) , (1.118) where a=2⎫parenleftbigln(2π)–γ⎫parenrightbig= 2, 5213.... While the leading term in the last expression coincides with the ﬁrst term of NR(λ), Equation (1.109), the second and third terms are different. The hyperbola billiard has been extensively investigated in classical and quan- tum mechanics as a model for quantum chaos [131–133]. It turns out that the clas- sical periodic orbits can be effectively enumerated using symbolic dynamics with a ternary code, and thus the length spectrum together with the Lyapunov expo-nents can be calculated with high precision. The topological entropy of this system isτW0.6. Using the boundary-element method, a large number of eigenvalues could be calculated. The statistics of the eigenvalues is found to be consistent withthe predictions of random matrix theory for the Gaussian orthogonal ensemble. Using the semiclassical Gutzwiller trace formula, one can deﬁne a dynamical zeta function deﬁned by an Euler product over the classical periodic orbits in analogy with the Selberg zeta function (1.100). This zeta function satisﬁes an approximate functional equation and thus can be effectively used as a semiclassical quantizationcondition in analogy to the exact quantization condition (1.102). 1.6 A Proof of Weyl’s Formula Only for very special geometries of Ωis it possible to give an explicit formula for the eigenvalues of the Dirichlet Laplacian. Such a situation had been considered in theprevious sections, another is given by rectangles and cubes. Weyl’s original proof for Jordan measurable domains consisted in exhausting the domain by rectangles. This proof needs technical computations which we do not want to cover here. Thereis another more structured proof which uses properties of the heat equation and reveals an interesting connection between the heat kernel and the eigenvalues. LetΩ⊂/CANbe open and bounded with boundary ∂Ω.W ew a n tt oi m p o s eam i l d regularity condition on Ω, namely we assume that for each ϕ∈C(∂Ω)t h eD i r i c h l e t problem ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩u∈C(Ω)∩C2(Ω) Δu=0 u|∂Ω=ϕ(D(ϕ)) has a unique solution; i.e. we assume that ΩisDirichlet regular . The Dirichlet prob- lem is a classical subject of Potential Theor y with physical interpretation in electro- statics. There is a beautiful mathematical theory on the Dirichlet problem, and precise conditions on the boundary are known which imply Dirichlet regularity. It is a mild 1.6 A Proof of Weyl’s Formula 55 regularity condition on the boundary. If Ωhas C1-boundary or if Ωis a polygon, then Ωis Dirichlet regular. More generally, Lipschitz continuity of the boundary sufﬁces. In dimension 2 each simply conne c t e dd o m a i n( i . e .e a c ho p e ns e tw i t h o u t holes) is Dirichet regular. Dirichlet regularity implies that all eigenfunctions of the Dirichlet Laplacian are continuous up to the boundary i.e. they lie in the space C0(Ω)/colonequal⎫braceleftBig u∈C(Ω):u|∂Ω=0⎫bracerightBig . Thus we may describe the Dirichlet Laplaci an very simply by its spectral decompo- sition. We consider the Hilbert space L2(Ω) with respect to the Lebesgue measure. Then there exists an orthonormal basis {en:n∈ /C6}ofL2(Ω)s u c ht h a t en∈C∞(Ω)∩C0(Ω), –Δen=λnen, where 0 < λ1uλ2u···uλn→∞ .W ec a l l λnthentheigenvalue of the Dirichlet Laplacian .N o wW e y l ’ sl a ws a y st h a t lim λ→∞N(λ) λN/2=ωN (4π)N/2|Ω| (1.119) where|Ω|is the volume of Ωand ωN=πN/2Γ(1+N/2)is the volume of the unit ball in /CAN.B yN(λ)=#⎫braceleftbign:λnuλ⎫bracerightbigwe denote the counting function. Forf∈L2(Ω)w el e t etΔD Ωf=∞⎫summationdisplay n=1e–λnt⎫parenleftbigf|en⎫parenrightbigen, (1.120) where⎫parenleftbigf|g⎫parenrightbig=⎫integraltext Ωfgdxdenotes the scalar product in L2(Ω). Then etΔD Ωis a compact, self-adjoint operator on L2(Ω). We call the family of operators⎫parenleftBig etΔD Ω⎫parenrightBig tv0thesemigroup generated by the Dirichlet Laplacian . This semigroup is positive and dominated by the Gaussian semigroup ( G(t))tv0,i . e .f o r0 uf∈L2(Ω)w eh a v e 0uetΔD ΩfuG(t)f,( t> 0) (1.121) where ⎫parenleftbigG(t)f⎫parenrightbig(x)/colonequal⎫integraldisplay Ωk0 t(x,y)f(y)dy, k0 t(x,y)/colonequal(4πt)–N/2e–|x–y|2/4t, |x–y|2/colonequalN⎫summationdisplay j=1⎫parenleftBig xj–yj⎫parenrightBig2,x,y∈ /CAN. The domination property (1.121) implies also that etΔD Ωis deﬁned by a measurable kernel ˜kt(x,y)s u c ht h a t 0u˜kt(x,y)uk0 t(x,y)f o r a l l x,y∈Ω. (1.122) 56 1W e y l ’ s L a w We will express the kernel ˜ktin terms of the eigenfunctions in (1.124). But here we recall that those operators SonL2(Ω)g i v e nb y (Sf)(x)=⎫integraldisplay Ωq(x,y)f(y)dy for some q∈L2(Ω~Ω) are called Hilbert Schmidt operators .S u c haH i l b e r tS c h m i d t operator Sis always compact. And if Sis self-adjoint, then its eigenvalues ( μn)n∈ /C6 satisfy⎫summationtext∞ n=1μ2 n<∞.H e n c ei no u rc a s e ∞⎫summationdisplay n=1e–2tλn<∞ for all t>0. Replacing tbyt/4 we deduce that ∞⎫summationdisplay n=1e–tλn/2<∞ for all t> 0 . (1.123) Note that (1.122) implies that ⎫vextendsingle⎫vextendsingle⎫vextendsinglee–λnten⎫vextendsingle⎫vextendsingle⎫vextendsingle=⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsinglee tΔD Ωen⎫vextendsingle⎫vextendsingle⎫vextendsingle⎫vextendsingleuG(t)|e n|. Since/bardblen/bardblL2= 1, it follows from the Cauch y Schwarz inequality that ⎫parenleftbigG(t)|en|⎫parenrightbig(x)uct–N/4,w h e r e c=π–N/42–(3/4)N. Thus ⎫vextendsingle⎫vextendsingle⎫vextendsingleen(x)⎫vextendsingle⎫vextendsingle⎫vextendsingleuct–N/4eλnt. Letting t=1 /λnwe obtain ⎫vextendsingle⎫vextendsingle⎫vextendsingleen(x)⎫vextendsingle⎫vextendsingle⎫vextendsingleu˜cλN/4 n (x∈Ω,n∈ /C6) where ˜c=c·e. In view of (1.123), this estimate asserts that for each t>0 ,t h es e r i e s kt(x,y)/colonequal∞⎫summationdisplay n=1e–λnten(x)en(y) (1.124) converges uniformly on the set Ω~Ωand deﬁnes a continuous, bounded function kt:Ω~Ω→ /CAsuch that kt(x,y)=0w h e n e v e r x∈∂Ωory∈∂Ω. Note that ⎫parenleftBig etΔD Ωf⎫parenrightBig (x)=⎫integraldisplay Ωkt(x,y)f(y)dy, (1.125) whenever f∈{en:n∈ /C6}.S i n c et h e enform an orthonormal basis of L2(Ω)i tf o l - lows that (1.125) remains true for all f∈L2(Ω). We have shown that the function kt is the kernel of the operator etΔD Ωi.e.˜kt=kt. 1.6 A Proof of Weyl’s Formula 57 For our purposes the following immediate consequence is crucial. ⎫integraldisplay Ωkt(x,x)dx=∞⎫summationdisplay n=1e–λnt(1.126) This formula allows us to estimate the counting function N(λ)=#⎫braceleftbign:λn<λ⎫bracerightbigwith the help of the kernel kt. For this we will make use of the following Tauberian theorem due to Karamata [134]. Theorem 1.1 Let(λn)n∈ /C6b eas e q u e n c eo fp o s i t i v er e a ln u m b e r ss u c ht h a tt h es e r i e s⎫summationtext n∈ /C6e–λntconverges for every t >0.T h e nf o rr >0and a∈ /CAt h ef o l l o w i n ga r ee q u i v a - lent. (a) lim t→0tr⎫summationdisplay n∈ /C6e–λnt=a (b) lim λ→∞λ–rN(λ)=a Γ(r+1 ) Here N denotes the counting function N (λ)=#⎫braceleftbigλnuλ⎫bracerightbig,a n d Γ(r)=⎫integraltext∞ 0xr–1e–xdxi st h e usual Gamma function. Combining formula (1.126) and Theorem 1.1 we see that Weyl’s law (1.119) is equivalent to the kernel estimate lim t→0tN/2⎫integraldisplay Ωkt(x,x)dx=|Ω| (4π)N/2. (1.127) It is easily seen that the left-hand side of (1.127) is not greater than the right-hand side as the kernel ktis bounded by the Gaussian kernel i. e.kt(x,y)uk0 t(x,y)f o r x,y∈Ω,t>0 . The lower estimate is more delicate. For this we will consider the heat equation on the inﬁnite cylinder /CA+~Ωwhose boundary we denote by Γ=({0}~Ω)∪ ((0,∞)~∂Ω). It is a remarkable fact that Dirichlet regularity of Ωalso implies that the following boundary value problem for the heat equation is well-posed. Theorem 1.2 ([135, Theorem 6.2.8], [136]) Letψ∈C(Γ). Then there exists a unique solution of ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩u∈C⎫parenleftBig/CA+~Ω⎫parenrightBig ∩C∞⎫parenleftBig (0,∞)~Ω⎫parenrightBig , ∂ ∂tu(t,x)=Δu(t,x), ( t>0 ,x∈Ω) u|Γ=ψ.(1.128) This solution satisﬁes the parabolic maximum principle , which says that for all t >0 and all 0usut, x∈Ω, u(s,x)umax Γtu where Γt/colonequalΓ∩⎫parenleftBig [0,t]~Ω⎫parenrightBig . 58 1W e y l ’ s L a w Example 1.1 Let f∈C0(Ω)and deﬁne ψ∈C(Γ)byψ(0,x)=f(x)for x∈Ω,a n d ψ(t,z)=0 for t>0,z∈∂Ω. Then the solution of (1.128) is given by u (t,x)=⎫parenleftBig etΔD Ωf⎫parenrightBig (x). Thus, the semigroup⎫parenleftBig etΔD Ω⎫parenrightBig tv0governs the homogeneous boundary value problem (1.128) . Its solution can be expressed by the kernel k t,n a m e l y , u(t,x)=⎫integraldisplay Ωkt(x,y)f(y)dy. For this reason we call k ttheheat kernel associated with the Dirichlet Laplacian . To obtain a lower bound for the kernel we formalize the idea that at some distance away from the boundary, ktbehaves just like the Gaussian kernel. Lemma 1.1 Let x∈Ωbe arbitrary, and for y ∈Ωlet t 0(y)/colonequaldist(y,∂Ω)2/2Nd e n o t e the scaled squared distance of y to the boundary of Ω.T h e n k0 t(x,y)–kt(x,y)u⎧⎪⎪⎨⎪⎪⎩(4πt)–N/2e–d i s t ( y,∂Ω)2/4t,tut0(y), ⎫parenleftbig4πt0(y)⎫parenrightbig–N/2e–N/2, t>t0(y). Proof Fixy∈Ω. Then by Theorem 1.2 there exists a unique function p(·,·,y) solving the parabolic boundary value problem ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩p(·,·,y)∈C⎫parenleftBig/CA+~Ω⎫parenrightBig ∩C∞⎫parenleftBig (0,∞)~Ω⎫parenrightBig , ∂ ∂tp(t,x,y)=Δxp(t,x,y), ( t>0 ,x∈Ω) p(t,x,y)=0, ( x∈Ω) p(t,x,y)=(4πt)–N/2e–|x–y|2/4t.( t>0 ,x∈∂Ω) Then p(t,x,y)=k0 t(x,y)–kt(x,y). In fact, let f∈C0(Ω) be arbitrary, and let u(t,x)/colonequal⎫integraldisplay Ω⎫parenleftBig k0 t(x,y)f(y)–p(t,x,y)f(y)⎫parenrightBig dy. The properties u∈C∞⎫parenleftBig (0,∞)~Ω⎫parenrightBig ,ut=Δuon (0,∞)~Ωandu(t,x)=0i f x∈∂Ω, t> 0 are obvious. Moreover, it is easy to prove that ucan be continuously extended tot=0a n d u(0,x)=f(x)f o ra l l x∈Ω.T h u s u(t,·)=etΔD Ωfaccording to Example 1.1. Since psolves a parabolic problem, we can use the parabolic maximum principle to deduce that pattains its maximum on the boundary i. e. p(t,x)usup 0usut x∈∂Ω(4πs)–N/2e–|x–y|2/4susup 0usut(4πs)–N/2e–d i s t ( y,∂Ω)2/4s. (1.129) Calculating the derivative of (4πt)–N/2e–d i s t ( y,∂Ω)2/4tas a function in the variable tone sees that the maximum is attained at time t=t0(y). We can thus simplify (1.129) accordingly which completes the proof. q 1.7 Can One Hear the Shape of a Drum? 59 We are interested in the error⎫integraltext Ω⎫parenleftBig k0 t(x,x)–k(x,x)⎫parenrightBig dxof the approximation of k0 t byktasttends to 0. Since the lemma essentially says that problems may only arise near the boundary, it is natural to decompose Ωinto a good part Ω1(t)/colonequal⎫braceleftBig x∈Ω:d i s t ( x,∂Ω)vt1/4⎫bracerightBig and a bad part Ω2(t)/colonequalΩ\Ω1(t). Note that⎫vextendsingle⎫vextendsingle⎫vextendsingleΩ2(t)⎫vextendsingle⎫vextendsingle⎫vextendsingle→0 ast→0. If tu1/4N2, then for every x∈Ω1(t)w eh a v e t0(x)v√t/2Nvt.H e n c e we can apply the lemma to obtain tN/2⎫integraldisplay Ω1(t)⎫parenleftBig k0 t(x,x)–kt(x,x)⎫parenrightBig dxu|Ω|(4π)–N/2e–√t/4t→0(t→0) . On the other hand, using the trivial estimate ktv0w es e e tN/2⎫integraldisplay Ω2(t)⎫parenleftBig k0 t(x,x)–kt(x,x)⎫parenrightBig dxu⎫vextendsingle⎫vextendsingle⎫vextendsingleΩ2(t)⎫vextendsingle⎫vextendsingle⎫vextendsingle(4π)–N/2→0(t→0) . Combining these two estimates, we have proved lim inf t→0tN/2⎫integraldisplay Ωkt(x,x)dxvlim inf t→0tN/2⎫integraldisplay Ωk0 t(x,x)dx=|Ω| (4π)N/2 This was the missing inequality required to prove (1.127). Since (1.127) has been shown to be equivalent to Weyl’s law, we have completed the proof. Weyl’s law also holds for arbitrary bounded open sets, [137, Theorem 1.11]. A simple proof by approximating an arbitrary open set by regular sets from the in- terior is given in [138, Section 6.5.2]. For further results on domain approximation we refer to the survey article [139] by Daners. T h ep r o o fg i v e nh e r ei se s s e n t i a l l yt h eo n eg i v e nb yK a c[ 9 ]w h of o u n df o r - mula (1.126) and used Karamata’s Tauberian theorem. We were also inspired by Dodzink [140] and the Diploma thesis by E. Michel [141]. However, the use ofDirichlet regularity and Theorem 1.2 in particular comes from [138, Chapter 5] where more details can be found. Conce rning the Dirichlet problem we refer to [142, 143] and the literature mentioned there. 1.7 Can One Hear the Shape of a Drum? Weyl’s law shows us in particular the following. Assume that Ω⊂ /CANis a bounded open set and we know all the eigenvalues of the Dirichlet Laplacian. Then we also know the volume of Ω. Thus the spectrum of the Dirichlet Laplacian determines the volume. It is natural to ask whether there are other properties or qualities which we may deduce from the spectrum. Those types of questions are called inverse (spec- tral) problems . Let us say that two open bounded sets Ω1andΩ2in /CANareisospectral if the corresponding Di richlet Laplacians ΔD Ω1andΔD Ω2have the same sequence of eigenvalues. We already know that isospec tral sets have the same volume. There is another result of this kind. 60 1W e y l ’ s L a w Theorem 1.3 LetΩ1,Ω2⊂ /CANbe open bounded sets with Lipschitz boundary. If Ω1 andΩ2are isospectral, then they have the same surface area. Here we use the natural measure σon the boundary ∂ΩiofΩii. e. the surface measure or (which is the same) the ( N– 1)-dimensional Hausdorff measure. The surface area of Ωiis by deﬁnition σ(∂Ωi). For a proof, we refer to [144]. The most radical inverse spectral probl em is whether the spectrum determines the domain completely. This question became famous by Marc Kac’s article [9] from 1966. We want to formulate it more precisely. Two open sets Ω1,Ω2⊂ /CANare called congruent if there exists an orthogonal matrix Band a vector bin /CANsuch that Ω2=⎫braceleftbigBx+b:x∈Ω1⎫bracerightbig. This is just congruence in the Euclidean sense. It is obvious that congruent open sets are isospectral. Question 1.1 (Kac’s Question) LetΩ1,Ω2⊂ /CA2be two bounded smooth domains which are isospectral. Are they necessarily congruent? By a domain we mean an open connected set. An open bounded set is called smooth if the boundary is of class C∞. Kac’s question became so popular because i t has a fascinating physical interpre- tation. We consider a bounded smooth domain Ω⊂ /CA2as a membrane which is ﬁxed at the boundary ΓofΩ. If it is set into motion, then the vertical displacement u(t,x)a tt i m e t>0a tt h ep o i n t x∈Ωsatisﬁes the wave equation utt=cΔu(t,x) (t>0 ,x∈Ω). We normalize physical units in such a way that c=1 . Of particular interest are solutions of the form u(t,x)=v(x)eiωtwhich are called thepure tones of the membrane. In order that such ube a solution of the wave equation it is necessary and sufﬁcient that –Δv=ω2v. Thus uis a solution if and only if vis an eigenfunction of the Dirichlet Laplacian for the eigenvalue ω2,w h e r e ωis the frequency of the displacement u.N o ww e see that the eigenvalues of the Dirichlet L aplacian correspond exactly to the pure tones of the membrane which we can hear. This lead Kac to reformulate his ques- tion by asking “Can one hear the shape of a drum?”. Following Kac, people like toformulate inverse spectral proble ms by asking which properties of Ωone can hear. For example, we already know that we can hear the volume and the surface area of a Lipschitz domain. Kac himself said in [9]: “I believe that one cannot hear the shape of a tambourine but I may be wrong and I am not prepared to bet large sums either way.” Today the question raised by Kac is still open. But much more is known about it. In fact, we may ask more generally if two bounded isospectral domains in/CAN are congruent. That is, we consider arbitrary dimensions now and give up the very restrictive smoothness hypothesis. Let us note though that some hypothesis on the 1.7 Can One Hear the Shape of a Drum? 61 boundary is needed to avoid trivialities. For instance, if we consider the disc Ω1=⎫braceleftBig x∈ /CA2:|x|<1⎫bracerightBig a n dt h ep u n c t u r e dd i s c Ω2=Ω1\{0}, then they are isospectral but not congruent. In fact, L2(Ω1)=L2(Ω2) and also the Dirichlet Laplacians with respect to these two open sets are identica l. We will describe below precisely which regularity of the boundary is needed to avoid such simple counterexamples. Here we want to impose throughout that all bounded domains have a Lipschitz boundary ,a n d we call them Lipschitz domains for short. They include all polygons in particular. Before we describe some of the results concerning Kac’s question we mention that the analogous question for compact manifolds has a negative answer as John Milnor [70] had already shown in 1964. So the challenge concerns the Euclidean case. A ﬁrst counterexample was given by Urakawa [145] in 1982 who constructed two isospectral Lipschitz domains in /CA4which are not congruent. Ten years later, Gordon, Webb and Wolpert [146] found a two-dimensional example. By putting together seven triangles th ey obtained two polygons in /CA2which are isospectral but not congruent, see Figure 1.5. These two polygons are not convex, though. It is an open question whether convex isospectral polygons in /CA2are congruent. However, in four dimensions convexity alone does not help. There are convex isospectal setswhich are not congruent. In fact, by modi fying Urakawa’s example, Gordon and Webb [147] obtained two truncated convex cones in/CA4which are isospectral but not congruent. These cones are induced by some vector space bases in /CA4.H e r ei s an explicit formulation. Example 1.2 (Gordon, Webb) Let u1/colonequal⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝0 0 1 0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,u 2/colonequal⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1 –1 0 0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,u 3/colonequal⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1 1 1 0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,u 4/colonequal⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝0 0 0 1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ be the ﬁrst basis of/CA4and v1/colonequal⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1 0 0 0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,v 2/colonequal⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1√ 3 0 0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,v 3/colonequal⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝0 0 1 0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,v 4/colonequal⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝0 0 1 1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ Figure 1.5 Isospectral polygons in two dimensions. 62 1W e y l ’ s L a w the second. Consider the corresponding positive cones C1/colonequal⎧⎪⎪⎨⎪⎪⎩4⎫summationdisplay i=1aiui:aiv0,i=1 ,...,4⎫⎪⎪⎬⎪⎪⎭,C2/colonequal⎧⎪⎪⎨⎪⎪⎩4⎫summationdisplay i=1aivi:aiv0,i=1 ,...,4⎫⎪⎪⎬⎪⎪⎭. Let B 0/colonequal⎫braceleftBig x∈ /CA4:0<|x|<1⎫bracerightBig be the punctured unit ball in /CA4with respect to the Euclidean norm |x|=⎫radicalBig⎫summationtext4 j=1x2 j.T h e n Ω1/colonequalB0∩C1andΩ2/colonequalB0∩C2are isospectral but not congruent. So far no smooth counterexample is known in any dimension. But in a very recent work Zelditch [148] showed that isospectral domains with an analytic boundary, having some symmetry, are congruent. A simple class of domains having such a symmetry are ellipses and stadiums. Th us he shows in particular that those do- mains can be distinguished by their spectra. Now we describe further positive results. We mention that two isospectral tri- angles are congruent, see [149] and references therein. Moreover, one can hear whether a Lipschitz domain in /CANis a ball. Theorem 1.4 LetΩ1⊂ /CANbe a ball and Ω2⊂ /CANa Lipschitz domain. If Ω1andΩ2 are isospectral, then they are congruent. Proof IfΩis a Lipschitz domain, then one can hear its volume |Ω|according to Weyl’s law. The Faber–Krahn inequality λΩ 1vcN|Ω|–2/N(1.130) holds for all such domains, where λΩ 1denotes the ﬁrst eigenvalue of the Dirichlet Laplacian on Ωand cNis an optimal constant which depends only on the dimen- sion N[150, Theorem 3.1]. Moreover, (1.130) is an equality if and only if Ωis a ball, see [151, Theorem 1.2]. q The above theorem can be found in Kac’s paper [9]. However, Kac uses the isoperi- metric inequality together with Theorem 1.3 instead of (1.130). For this argument one has to be able to deﬁne the surface area of the domain. The above proof on the other hand works in much more generality. The result can even be made optimalin a sense that we will describe now. For this, we need the notion of capacity which is used to describe the size of sets in/CANin terms of Sobolev norms. For a sys- tematic introduction we refer to [152]. The capacity cap( A)o fas e t A⊂ /CANmay be any number in [0, ∞], but here we only need to know whether a set has capacity 0. Sets of capacity 0 are also called polar sets . Although it is not trivial to characterize all polar sets, thinking of them as subsets of /CANof dimension at most N–2g i v e s a good impression of how they look. For example, single points in /CA2and smooth curves in /CA3are polar, but curves in /CA2and surfaces in /CA3are not polar. Moreover, subsets of polar sets and countable unions of polar sets are also polar. 1.8 Does Diffusion Determine the Domain? 63 What makes the notion of capacity paticu larly interesting is the fact that the Dirichlet Laplacian “does not see” polar sets. More precisely, if Ω1and Ω2are open subsets of /CANthat only differ by a polar set i.e. Ω2\Ω1and Ω1\Ω2are both polar, then the sets differ only by a set of Lebesgue measure zero, henceL 2(Ω1)=L2(Ω2)a ss u b s p a c e so f L2( /CAN). But in fact, ΔΩ1 D=ΔΩ2 Das operators on this space, thus they have the same spect rum. This shows that inverse spectral problems for the Dirichlet Laplacian are meaningful only up to polar sets. Thus we are lead to introduce a notion of regularity which asserts that there are no artiﬁcial “polar holes” in the set. More precisely, call an open set Ωin /CANregular in capacity if cap⎫parenleftbigB(z,r)\Ω⎫parenrightbig>0f o ra l l z∈∂Ωand all r>0 ,w h e r e B(z,r) denotes the ball of radius rcentered in z. We refer to [153] where this regularity assumption is in- troduced and discussed. Here we only mention that all Dirichlet regular sets, andhence all Lipschitz domains, are regular in capacity. Given any open set Ω⊂/CAN,t h e r ee x i s t sau n i q u eo p e ns e t Ω/primewhich is regular in capacity such that Ω⊂Ω/primeand cap( Ω/prime\Ω) = 0. Since the Laplacian does not see polar sets it is natural to consider merely open sets which are regular in capacity. An inspection of Daners’ proof [151] shows that for a bounded open set Ωwhich is regular in capacity the Faber–Krahn inequality becomes an identity if and only if Ω is a ball. Thus Theorem 1.4 remains true if we assume that Ω2is regular in capacity instead of being a Lipschitz domain. In other words, if Ω2is an arbitrary open set which is isospectral to a ball Ω1, then the regular version of Ω2is a ball of the same radius, or, what is the same, there exists a ball B⊂ /CANwhich is a translation of Ω1 such that Ω2⊂Band cap( B\Ω2)=0 . 1.8 Does Diffusion Determine the Domain? In this short section we follow a paradigm which is slightly different from that in the last section. Instead of the wave equat ion let us consider the diffusion equation ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ut(t,x)= Δu(t,x) (t>0 ,x∈Ω), u(0,x)= u0(x) (x∈Ω), u(t,z)=0 (z∈∂Ω).(D) Here again Ωis a Lipschitz domain with boundary Γ.T h es o l u t i o n uof (D) has the following interp retation. Assume that Ωis a body containing water and some dissolving liquid, for instance ink. Then u0is the initial concentration of the ink i. e. for ω⊂Ωthe amount of ink in ωis given by⎫integraltext ωu0(x)dx.T h es o l u t i o n u(t,x) gives the concentration at time t>0i . e .f o r ω⊂Ω,⎫integraltext ωu(t,x)dxis the amount of ink in ωat time t. Given u0∈L2(Ω), Equation (D) has a unique solution u: /CA+→L2(Ω), where we letu(t,x)=u(t)(x), given by u(t)=etΔD Ωu0=⎫summationdisplay n∈ /C6e–λnt(u0|en)en, 64 1W e y l ’ s L a w (compare (1.120)). In fact, since (d/d t)e–λnten=e–λnt(–λnen)=e–λntΔen,uis a solu- tion of (D). Its uniqueness follows from Theorem 1.128, the parabolic maximum principle. Thus the semigroup generated by ΔD Ω,e–tΔD Ω,i sf r e q u e n t l yc a l l e dt h e dif- fusion semigroup . Now let Ω1andΩ2be two Lipschitz domains. If Ω1andΩ2are isospectral, then we ﬁnd orthonormal bases (en)n∈ /C6ofL2(Ω1)a n d⎫parenleftbigfn⎫parenrightbig n∈ /C6ofL2(Ω2)s u c ht h a t –ΔD Ω1en=λnen and – ΔD Ω2fn=λnfn for all n∈ /C6.C o n s i d e rt h eu n i t a r yo p e r a t o r U:L2(Ω1)→L2(Ω2) satisfying Uen= fn.T h e n UetΔD Ω1=etΔD Ω2U (t> 0) , (1.131) i. e.U intertwines the two diffusion semigroups. In other words, Umaps solutions of the ﬁrst diffusion equation to solutions of the other diffusion equation. Con-versely, if we ﬁnd an intertwining invertible operator U:L 2(Ω1)→L2(Ω2), then Ω1andΩ2are isospectral. Now we remember th at for the physical interpretation only positive concentrations 0 uu0∈L2(Ω1)a r em e a n i n g f u l .I f u0(x)v0f o ra l l x∈Ω1,t h e n u(t,x)v0f o ra l l x∈Ω1and all t> 0. This is the positivity property of the diffusion equation. The physical inte rpretation motivates us to consider, in- stead of unitary operators, operators Uwhich preserve positivity. 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