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Manin ends his 1978 ICM talk with this remark:

I would also like to mention I. M. Gel'fand's suggestion that the $\zeta$-functions of certain special differential operators should have an arithmetic meaning. The first class to consider is that of the Schrödinger operators $-d^2/dx^2 + u(x)$ with algebro-geometric potentials $u(x)$ arising as solutions of the Korteweg-de Vries equation, for example: $u(x) = 2\wp(x)$ where $\wp$ is the Weierstrass function of an elliptic curve over $Q$. In fact, it seems that the values of this zeta-function at negative integers, which can be calculated explicitly, admit a $p$-adic interpolation.

Simply put, I'd be grateful to anyone who can explain to me what he's talking about. In particular, has "Gel'fand's suggestion" found explicit form, either as theorem or conjecture, anywhere in the literature?

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Let me formulate a different but related question. Assume $u$ is $P$-periodic. Floquet's theory tells us that the spectrum of $-d^2/dx^2+u(x)$ over $L^2(\mathbb R)$ is the union of intervals $[a_{2j},b_{2j}]$ and $[b_{2j+1},a_{2j+1}]$ with $a_k$ and $b_k$ non-decreasing. The $a_k$'s (resp. $b_k$'s) are e.v. associated with periodic (resp. anti-periodic) e.f. If $u=2\wp$, these eigenvalues have double multiplicity, except for $a_0$. This is the only case without gaps in the spectrum. Is there any relation with the fact that $\zeta$ has good properties ? – Denis Serre Dec 24 '10 at 8:47
@Denis I would certainly benefit from more details or a reference. For example, I can't decipher "e.v." or $P$-periodic (I'll guess that P is a lattice). – David Feldman Dec 25 '10 at 18:26
"eigenvalues associated with periodic (respectively anti-periodic) eigenfunctions", I'd guess. – David Loeffler Dec 26 '10 at 12:18
Right, thanks David Loeffler. – David Feldman Dec 26 '10 at 19:07

From Introduction to “Scattering Theory for Automorphic Functions” by Lax and Phillips (1976) and some other sources I could suppose, that proper reference would be the Gelfand presentation on 1962 ICM, but I do not have access to it, and so not quite certain.

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Gelfand's article is here . See… on how to quickly find ICM articles. – Gil Kalai Apr 27 '11 at 18:02

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