Picard potentials and Hill's equation on a torus

Abstract. We introduce a class of multidimensional Schrödinger operators with elliptic potential which generalize the classical Lamé operator to higher dimensions. One natural example is the Calogero-Moser operator, others are related to the root systems and their deformations. We conjecture that these operators are algebraically integrable, which is a proper generalization of the finite-gap property of the Lamé operator. Using earlier results of Braverman, Etingof and Gaitsgory, we prove this under additional assumption of the usual, Liouville integrability. In particular, this proves the Chalykh-Veselov conjecture for the elliptic Calogero-Moser problem for all root systems. We also establish algebraic integrability in all known two-dimensional cases. A general procedure for calculating the Bloch eigenfunctions is explained. It is worked out in detail for two specific examples: one is related to B 2 case, another one is a certain deformation of the A 2 case. In these two cases we also obtain similar results for the discrete versions of these problems, related to the difference operators of Macdonald-Ruijsenaars type.

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“…Notice that our approach easily extends to the case of operators of any order (cf. the remark at the end of [14]). …”

Section: Examplesmentioning

confidence: 92%

“…Now let u be elliptic, with periods 1, τ . Let us demand all the solutions of the equation Lψ = λψ to be meromorphic in C (such u are called Picard potentials in [25,14]). By the discussion above, this is equivalent to the quasi-invariance of u at each pole.…”

Section: Examplesmentioning

confidence: 99%

Generalized Lam� Operators

Chalykh

Etingof

Oblomkov

2003

Communications in Mathematical Physics

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“…In [88] using Floquet spectral theory for the Schrödinger operator it was proved that all such potentials are finite-gap. This result is an essential step in the Picard problem though its complete and effective solution is still an open problem.…”

Section: Spectral Theory Of Operators With Elliptic Coefficientsmentioning

confidence: 99%

Periodic and almost-periodic potentials in inverse problems

Krichever

Novikov

1999

Inverse Problems

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“…Later, three main approachs in the theory of elliptic solitons have been developed, including the scalar and matrix spectral problems. 1) The classical approach of Hermite-Halphen, based on the ansatz for the Ψ-function [3][4][5][6][7]; 2) An analysis of the algebraic curves of the special form -curves being a covering over tori [8,18,14]; 3) The methods, touching the Picard's theorem and hierarchies of the higher stationary nonlinear differential equations [9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%