Can someone, please, suggest a reference or what steps should I take to prove the following theorem:

The set of eigenvectors of a self-adjoint differential operator, defined over a finite interval such that the operator has no singularities in the interval, spans the domain of the operator.

I've seen this stated in a couple of references but always without a proof. I've been struggling with this for quite a while just to realize that I can't prove it...

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    $\begingroup$ What kind of background do you have? Do you know any functional analysis at all? This is a spectral theorem en.m.wikipedia.org/wiki/Spectral_theorem $\endgroup$ – Sharkos Apr 16 '13 at 23:01
  • $\begingroup$ @Sharkos Well, only the one usually taught to physicists in University. But I've been reading Principles and Techniques of Applied Mathematics from Bernard Friedman which, I believe, has given me a solid basis on functional analysis. I realize this is a spectral theorem but instead of just believing in it I would like to have a notion of how to prove it just to be fully aware of its limitation of applicability... $\endgroup$ – PML Apr 16 '13 at 23:17

Try chapter 7 of Theory of Ordinary Differential Equations by Coddington and Levinson.

Rather than look at the spectrum of the differential operator, the result is recast in terms of the Green's function $G(x,\xi)$ for the differential equation. For self-adjoint, 2nd-order ODE with no singularities on a finite interval it's not too hard to show that a Green's function exists and is bounded; you can construct it more or less explicitly too. Then you look at the operator

$\mathscr{G}u(x) = \int_a^b G(x,\xi)u(\xi)d\xi$.

This operator has the same eigenfunctions as the differential operator, only it's much easier to work with because it's compact and the eigenvalues go to zero instead of infinity.

Pretty sure it's also somewhere in Courant and Hilbert but I forget which volume.

EDIT: Almost forgot, if you can find Einar Hille's Lectures on Ordinary Differential Equations, it's in chapter 8. It has lots of the proofs of things that people cite everywhere but never actually demonstrate; for example, why is it that the eigenfunctions of a differential operator get more oscillatory as the eigenvalue increases? There's some really nice special function lore in there too, e.g. deriving the properties of the elliptic functions, Mathieu functions, Bessel functions, you name it.

  • $\begingroup$ I'll check it out. Thank you for the hint. $\endgroup$ – PML Apr 16 '13 at 23:23
  • $\begingroup$ Your reference and direction pointing were spot on. Thank you very much. Unfortunately , I can't seem to find an online preview of the Einar's book which from TOC that you indicated of chapter 8 would be very helpful specially the treatment on the Mathieu functions... Well, Thank once again! $\endgroup$ – PML Apr 17 '13 at 0:05

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