hilbert spaces and Gaussian quadrature

Hilbert Spaces and Gaussian Quadrature (GQ) PhL 1.20.05 Here we first review the Hilbert space H ( L2 norm), and then we define a new Hilbert space HN by defining its scalar product. No use of quantum mechanical bra-ket notation appears until after the basics are all established, then we talk bras and kets in items 5 and 6. Contents: 1. The Hilbert Space H 2. GQ for Polys(N-1) 3. GQ for Polys(2N-1) 4. The Hilbert Space HN : how to compute the Ak weights 5. Bra-ket Notation for H 6. Bra-ket Notation for HN 7. The Jacobi Matrix and the EN space 8. First order perturbation theory 9. Comments on sum rules and extending GQ from degree N-1 to degree 2N-1 1. The Hilbert Space H. We know that the L2-norm and associated Hilbert space H can be expressed in terms of a scalar product (f1, f2) = dx w(x) f1(x) f2(x) where f1 and f2 are square-integrable functions (and therefore have finite norms). We know that we can associate with w(x) and interval (a,b) [ not shown ] a system of orthonormal polys Pn(x) with n = 0,1.. and with this basis we can associate the following orthogonality and completeness relations: (Pn, Pm) = n,m n,m = 0,1,2... orthogonality H (x - x')/w(x) = n=0 Pn(x)Pn(x') x,x' in (a,b) completeness As a check on the completeness relation, we can apply dx w(x) Pm(x) to both sides to get Pm(x') = dx w(x) Pm(x)n=0 Pn(x)Pn(x') = n=0 Pn(x') (Pn, Pm) = n=0 Pn(x') m,n = Pm(x') 2. GQ for Polys(N-1). We know that for polys(N-1) (polys of degree N-1 or less) we can do GQ which we write as dx w(x) p(x) = i=1N Ai p(xi) where p(x) is a poly in Polys(N-1), and we have an N-point GQ. This holds for any choice of { xi }. 3. GQ for Polys(2N-1).If we choose the xi as zeros of PN(x), the degree-N ortho poly of the w(x) and (a,b) system, then we know that the above GQ is value for Polys(2N-1). We can then say, for example, that dx w(x) p1(x)p2(x) = i=1N Ai p1(xi)p2(xi) where p1 and p2 are functions in the space Polys(N-1). In this case, p1(x)p2(x) is degree 2N-2 which is in fact one less than 2N-1, so we even have room to spare. 4. The Hilbert Space HN. Item 3 suggests that we might be able to define a new scalar product for functions in Poly(N-1) which are evaluated at the N points xi as follows (p1, p2)N i=1N Ai p1(xi)p2(xi) scalar product in Hn HN Then we know from item 3 that (p1, p2)N = (p1, p2) for any p1, p2 in Polys(N-1). We can form a basis for Polys(N-1) as (x) = { x0, x1, .... xN-1 } and therefore this space has dimension N. Of course then the basis for our new space would be (xi). But we would rather use the basis '(xi) = { P0(xi), P1(xi), .... PN-1(xi) } as we shall show in a moment. For functions in Polys(N-1), we can "prove" that (p1, p2)N really is a viable scalar product ( and therefore we really have a Hilbert Space associated with it) because it is equal, for that range of functions, to something that we know really IS a scalar product, and thus has all the required properties such as linearity. Thus, we know that (p1, p2)N defines a Hilbert space which we will call HN. We shall call our original infinite Hilbert space H. What are the orthogonality and completeness relations for HN ? First, we repeat what they were for H as was noted earlier (Pn, Pm) = n,m n,m = 0,1,2... orthogonality (x - x')/w(x) = n=0 Pn(x)Pn(x') x,x' in (a,b) completeness Then for HN we have (Pn, Pm)N = (Pn, Pm) = n,m from the definition of our HN scalar product, so we can write these equations for orthogonality and completeness (see note at end regarding orthog sum rule) (Pn, Pm)N = n,m [= i=1N Ai Pn(xi)Pm(xi) ] n,m = 0,1,2...N-1 orthogonality HN k,j /Ak = n=0N-1 Pn(xk)Pn(xj) i,j = 1...N completeness As a check on the completeness relation, we can apply k=1N Ak Pm(xk) to both sides to get LHS = k=1N Ak Pm(xk)k,j /Ak = Pm(xj) RHS = k=1N Ak Pm(xk) n=0N-1 Pn(xk)Pn(xj) = n=0N-1 Pn(xj) k=1N Ak Pm(xk)Pn(xk) = n=0N-1 Pn(xj) (Pn, Pm)N = n=0N-1 Pn(xj) n,m = Pm(xj) You could imagine that within the interval (a,b) we could make the number of points xi get larger and larger. For finite N we might write the completeness in this manner n=0N-1 Pn(xk)Pn(xj) = k,j /Ak = sinc1(xk-xj)/A(xk) where sinc1 is some sinc-like function that interpolates the function k,j between the integers. Then in the limit N, this sinc1 function becomes a delta function, and A(xk) becomes w(x). Finally, let's rewrite the completeness relation in its two cases 0 = n=0N-1 Pn(xk)Pn(xj) kj i,j = 1...N completeness HN 1/Ak = n=0N-1 [Pn(xk)]2 k=j i,j = 1...N completeness This last expression gives us a new way to compute the weight for GQ. The formula shown in Scheid requires an integral of the Lagrange multiplier, but here we have a simple sum. We could if we liked use the C-D identity to evaluate the sum. Since our Pn are orthonormal, we have hn = 1 for them (p 159 E), 1/Ak = n=0N-1 [Pn(xk)]2 = (kN-1 /kN) PN-1(xk) PN'(xk) compute Ak from the polys at xk and this is then yet another formula for the GQ weight Ak. Notice that if you do the Jacobi matrix diagonalization procedure, the eigenvalues are the xk and the eigenvectors are P(xk), so you can add up the squares of the elements of the eigenvector P(xk) to get 1/Ak. That seems a good way to compute the Gaussian weights. 5. Bra-ket Notation for H. Let's go back now to our H Hilbert space discussion where we had (Pn, Pm) = n,m n,m = 0,1,2... orthogonality (x - x')/w(x) = n=0 Pn(x)Pn(x') x,x' in (a,b) completeness We now give life to a function f independent of its "representation". Normally we think of two representations of a function, coordinate space x and momentum space k (or p), which are related by Fourier transforms. The main idea is to think of a function completely in the abstract as a vector in H which we write as | f >. At the same time we define some vectors | x > which are eigenvectors of the position operator X, X | x > = x | x > We then write the function as f(x) = <x | f >. The basis functions can be written as | Pn > or just | n > for short. We might write F(p) = <p | f > in the momentum basis, where now | p> are the eigenvectors of the momentum operator P, similar to our last equation with x p everywhere. Now, the X operator is Hermitian and its eigenvectors | x> therefore form an orthonormal basis for H . We can write orthogonality and completeness in this manner for the |x> basis, < x | x' > = (x-x')/w(x) x,x' in (a,b) orthogonality H 1 = dx w(x) | x >< x| completeness (this is an "operator" equation ) As a check on this completeness, if we close with <x'| we get <x'| 1 = dx w(x) <x' | x >< x| = <x'| Since this is true for every <x'| in H, it must be true in the operator sense shown above. Alternatively, we could sandwich our Pn basis vectors <n | and | m> around 1 to get <n | m> = <n | 1 | m> = dx w(x)<n | x >< x | m> = dx w(x)Pn(x)Pm(x) = ( Pn, Pm) = n.m which is to say <n | m> = ( Pn, Pm) = another notation for the H scalar product. so everything is consistent. This bra-ket notation is useful because it allows us to think of abstract representation-independent operators in a Hilbert space H' which is isomorphic to the space H whose elements are functions f(x), and then we make no big distinction between H' and H. Just as the vectors |x> form a basis for H, so do the vectors |m>. Obviously each of these bases has infinite dimension. In the |m> basis we have the following orthogonality and completeness conditions, <n | m> = n.m n,m = 0,1,2... orthogonality H 1 = m=0 | m >< m | completeness (operator equation) We have already derived the orthogonality in the last paragraph. As a check on completeness, we can close from the left side say with <n | to get the consistent result <n | = <n |. Again, since this is true for every one of the infinite number of basis vectors < n |, the operator equation is true stand alone without projecting onto vectors. We can do the same in the other direction | n > of course. One can interpret <n | x > = Pn(x) as a matrix which changes from one basis to the other. 6. Bra-ket Notation for HN . We now have N discrete points xi and our X operator eigenvalue equation is now this X | xi > = xi | xi > The analogous statements to those of the last section are < xj | xk > = j,k/Ak k,j = 1..N orthogonality HN 1N = i=1N Ai | xi >< xi| completeness (this is an "operator" equation ) which we compare to the results in the last section < x | x' > = (x-x')/w(x) x,x' in (a,b) orthogonality 1 = dx w(x) | x >< x| completeness (this is an "operator" equation ) Just as in H , here the | xi > form a basis, and the | m > form an alternative basis. Of course here we only have m = 0,1...N-1 instead of up to infinity. Thus, in the | m> basis we have <n | m> = n.m n,m = 0,1,2...N-1 orthogonality HN 1N = m=0N-1 | m >< m | completeness (operator equation) which again we compare to the results found for H , <n | m> = n.m n,m = 0,1,2... orthogonality 1 = m=0 | m >< m | completeness (operator equation) As before, we can make this identification <n | m> = n.m = (Pn, Pm)N = i=1N Ai Pn(xi) Pm(xi) We could consider adding a Hilbert Space label to the vectors, so here we would have |m>N to distinguish it from |m> . For now, I don't think that amount of detail is needed, because we usually know in which space we are working. Sometimes much notation tends to clutter things up and add more confusion rather than less confusion. Note: We could define vectors | i > = (1/) | xi > such that < k | i > = ki . 7. The Jacobi Matrix and the EN space. As an exercise, consider this activity in HN. Start with X | xi > = xi | xi > Close from the left with <n | < n | X | xi > = xi <n | xi > Insert 1N = m=0N-1 | m >< m | to the right of operator X, < n | X | xi > = < n | X 1N | xi > = < n | X m=0N-1 | m >< m | xi > = m=0N-1 < n | X | m >< m | xi > so we now have this equation m=0N-1 < n | X | m >< m | xi > = xi <n | xi > Now write < n | X | m > = Xnm in normal matrix notation, and we then have m=0N-1 Xnm Pm(xi) = xi Pn(xi) which in matrix and vector notation we might write as X P(xi) = xiP(xi) where P(xi) = { P0(xi), P1(xi), .... PN-1(xi) } EN The matrix X is "the Jacobi matrix" which appears in Jim's papers. That is to say, it is the X operator in the | m > representation. My friend Matthews would write this as X where = the |n> basis. Now, our matrix Xnm is just a normal N x N matrix and as such, it acts on a vector in an EN space. So now we have another N dimensional space EN that is not the same as HN. We don't need a name for this space other than EN . The scalar product in EN is the usual EN one, and is unrelated to the scalar product in HN. 8. First order perturbation theory. See my doc on this subject. We do this in EN. We start with X P(xi) = xiP(xi) where X is the Jacobi matrix. This is a standard eigenvector equation with a real symmetric matrix X acting on vectors in EN as noted above. We could just quote the result, but let's do it here anyway: We do our usual trick by applying a small variation, X P(xi) + X P(xi) = xi P(xi) + xi P(xi) => xi P(xi) = X P(xi) = + [ X - xi1N] P(xi) Now close with scalar product in EN to get xi P(xi) P(xi) = P(xi)X P(xi) + P(xi) [ X - xi1N] P(xi) As usual, we use the Hermiticity (real symmetric) of X to rewrite the last term as { [ X - xi1N]P(xi) } P(xi) = 0 P(xi) = 0, and we end up with xi P(xi) P(xi) = P(xi)X P(xi) But we can write P(xi) P(xi) = m=0N-1 [ Pm(xi) ]2 = 1/Ai Si // from section 3 above and we then get our result, as for example in B&B page 5 xi = (1/Si) P(xi)[X] P(xi) 9. Comments on sum rules and extending GQ from degree N-1 to degree 2N-1 In our item #2 above we talk about GQ for Polys(N-1). I note there that we have dx w(x) p(x) = i=1N Ai p(xi) for any distinct xi for p(x) in Polys(N-1). The truth of the above claim is derived as Theorem 1 in my Accuracy of GQ paper. Now we can of course select an Erdelyi ortho poly for p(x) of degree N-1 or less, so we have dx w(x)pk(x) = i=1N Ai pk(xi) But then LHS = (1/k0) dx w(x)p0(x) pk(x) = (h0/k0) k,0 , so we get our "sum rule" i=1N Ai pk(xi) = (h0/k0) k,0 k = 0,1,2....N-1 any distinct xi (1) In the Accuracy of GQ paper, I should how you can use this sum rule and the recursion relation to extend the previous sum rule to a larger range of k, i=1N Ai pk(xi) = (h0/k0) k,0 k = 0,1,2....2N-1 if pN(xi) = 0 (2) This then used to show that GQ is good for Polys(2N-1). Now, once we know this, we go ahead as above and develop the HN space. We then found that (Pn, Pm)N = n,m [= i=1N Ai Pn(xi)Pm(xi) ] n,m = 0,1,2...N-1 orthogonality This provides a generalization of my first sum rule above: i=1N Ai pn(xi)pm(xi) = n,m (3) If I set n = 0 in (3), we get (1).