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response1 1_12_11
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One-page email reply from Richard Price (signed RHP) to Phil, part of a correspondence on Sturm-Liouville systems and Green's functions. Price says he is unsure what Phil's question is and finds the proposed proof unclear. He notes a Green's function needs a discontinuous derivative, and that for any fixed g(x) one can find a second-order differential equation it satisfies, which may be read as a Sturm-Liouville system given suitable boundary conditions.
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RHP
Jan 13, 2011
Hi Phil,
I'm still under unconscionable pressure for reports, grant proposals, uni-
versity garbage, . . . , but here I'll try to move our conversation forward.. a
bit. I've become less sure that the answer to your question is yes. Part of
this is that I am less sure of just what your question is.
Your pro ered proof seems to me to be just a hiding behind some mystical
mathematical patterns, partly because I don't understand just what you are
writing. If you think that there's more to what you've written than has met
my eyes, please clarify.
When you ask whether any g(x) can be a Green's function, we need to
gure out just what you mean. A Green function must have a discontinuous
derivative, so any old g(x) cannot work. I assume therefore that what you
had in mind is a g(x) that has something in it that is adjustable, some
parameter or whatever.
Your question then becomes: can we nd a Sturm-Liouville di erential
equation that can be satsi ed by any particular adjustable g(x) that we
choose. One relevant statement is this: If you give me any xed g(x), I can
nd a second-order di erential equation that is satis ed by that g(x), and we
can interpret that as a Sturm-Liouville system if we have the right boundary
conditions. When I wrote to you that there exists a Green function for 1D,
this is what I had in mind.
OK.. so, exactly what is your question?
Richard