SUMMARY OF DOCS IN THIS FOLDER

Summary of docs in this folder PhL 1.19.11 These summaries are in chrono order by creation date. The general subject is convincing myself that Jim's first and second proposed solutions L to L([1/r]) = 2πδ2(r) really work. These are in-house docs not ever sent to Jim. 1. Jim's First Proposed L ( 9 pages, 1.8.11) 1 2. Find L in 1D Lxs(x,y)=δ(x-y) (18 pages, 1.9.11) 1 3. Find L in 1D Lr(x) = δ(x) for the Love kernel. ( 10 pages, 1.10.11) 1 4. Dissing a claim I made in my jim email of 1_13_11 5PM.doc ( 8 pages, 1.14.11) 2 1. Jim's First Proposed L ( 9 pages, 1.8.11) Jim made this proposal by phone (which is his way to comm) on 1/8 and I then wrote this in-house doc pondering his proposal which eventually became -∂r2(r[1/r]). Section 1 was written perhaps on 1/17 after I knew more things, so is really out of place here. In Section 1 I use the Stak theorem to verify Jim's first proposal, but I did not know how to do this on 1/8. In Section 2 (back on 1/8) I ponder how one would prove the usual 3D fact without using the divergence theorem, and I don't know! Then in Section 3 I look at that Stak theorem and I search for a sequence, trying [1/r] = 1/(r+ε) and then trying [1/r] = 1/r1+ε, both acted upon by Jim's proposed L, but neither works. So I am unable to find a Stakgold sequence that works on 1/8. 2. Find L in 1D Lxs(x,y)=δ(x-y) (18 pages, 1.9.11) Here I first consider this problem for the general 1D kernel k(x,y), which I call s(x,y). My method of attack is always to expand both sides using a Fourier transform expansion. In Section 1 I first try this with an(x) coefficient functions and I get a horrible mess. Then in part (e) I try again with constant coefficients an and I learn that there is no hope (in the "local sense") unless s(x,y) = r(x-y), but even in that case if r(x) is even, there exists no solution with constant coefficients. I later repeat this argument in Section 6 and show how it applies to the Love kernel. In Section 2 I try a double Fourier on s(x,y) and this goes nowhere. In Section 3 I try starting with the r(x-y) form and using an(x) and get a horrible mess. In Section 4 I try r(x-y) = |x-y| because I know this has to work with L = ∂x2 since it is the 1D fundy solution, but this r(x) has no Fourier Transform! In Section 5 I think of Lr = δ as a matrix problem, and the problem is simple of you use basis functions φn(x) which are EFs of L. But this is a useless method, since we know neither Lx nor φn(x). At this point I sent my first email to R Price. 3. Find L in 1D Lr(x) = δ(x) for the Love kernel. ( 10 pages, 1.10.11) I show that, for the Love integral equation's kernel, r(x) = (d/π) / [ d2 +x2 ], there exists no differential operator Lx with constant coefficients (even if we allow infinite order) such that Lx r(x) = δ(x). I show this two different ways and arrive at the same conclusion twice. It follows that there is no Lx such that Lx r(x-y) = δ(x-y) and thus it follows that the Love integral equation cannot be represented as a differential equation of the form Lx[f(x) - 1] = ∓ !Syntax Error, Idt Lx r(x-t) f(t) = ∓ !Syntax Error, Idt Lx δ(x-t) f(t) = ∓ f(x) θ(x>-1)θ(x<+1) where Lx has the form just stated. To some extent this analysis is replicated in the previous doc above. The main thing is that the Love FT is e-d|k| which is not analytic in k at k=0. 4. Dissing a claim I made in my jim email of 1_13_11 5PM.doc ( 8 pages, 1.14.11) When I wrote this doc, I was pretty much convinced ( and in agreement with Jim Ball) that Jim's second proposal L= (1/r)∂(r [1/r]) was NOT true. But I had a lot of evidence that it WAS true. So here I am trying to show why each of my three pieces of evidence was wrong. In Section 1 I examined the Stakgold Theorem n=1 proof of (2.11). But this proof shows conclusively that Jim #2 works! For a while I had a factor of 1/2 discrepancy, but I then learned (Section 4) about this factor which arises only in the n=1 case. In Section 2 I examine the Stakgold Theorem itself. I review how one proves the three facts he calls (a),(b),(c) on page 13 of his book. I convince myself it is correct and there is nothing special about my application which should make it not be correct. In fact g(r) can blow up at r=0, it is only the normalization condition that matters. In Section 3 I convince myself that the correct n=1 normalization condition has 1/2 on the RHS. In Section 4 I explain away the factor of 1/2 discrepancy I was getting in Section 1. By Section 5 I have finally realized that Jim #2 really does work, and I list off the pieces of evidence. In Section 6 I do for Jim another Stakgold n=2 case limε→0 [ ε / (r2+ε2)3/2 ] = δ2(r). I prove it here, and then I note that it appears in Stakgold as an exercise. I think Jim felt that you really needed some derivatives ∂r (and preferably two as in ∂r2) in order to construct a limit expression which makes a delta function, and this Section 6 case is a counterexample. I think Jim is a little amazed by these limit forms, and I don't think despite 30 years of teaching Math Methods that he ever ran into them before in the form Stakgold gives.