Generalized Lame operators

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.

38 pages, latex; in the new version a reference was added

References in corpus (8)

• Spin generalization of the Ruijsenaars-Schneider model, non-abelian 2D Toda chain and representations of Sklyanin algebra
• Elliptic quantum many-body problem and double affine Knizhnik-Zamolodchikov equation
• Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg-de Vries Equation
• Krichever correspondence for algebraic varieties
• Locus configurations and $\vee$-systems
• The perturbation of the quantum Calogero-Moser-Sutherland system and related results
• Multidimensional Baker-Akhiezer functions and Huygens' Principle
• Cherednik algebras and differential operators on quasi-invariants

Cited by in corpus (11)

• Deformed quantum Calogero-Moser problems and Lie superalgebras
• On Darboux-Treibich-Verdier potentials
• Quantum integrability of the deformed elliptic Calogero-Moser problem
• On Algebraically Integrable Differential Operators on an Elliptic Curve
• Deformed Calogero--Moser operators and ideals of rational Cherednik algebras
• Intertwining operator for $A{G}_2$ Calogero-Moser-Sutherland system
• Bethe Ansatz for the Ruijsenaars Model of BC_1-Type
• Baker-Akhiezer function for the deformed root system $BC(l,1)$ and bispectrality
• Complex crystallographic reflection groups and Seiberg-Witten integrable systems: rank 1 case
• Coupling and particle number intertwiners in the Calogero model
• Towards Finite-Gap Integration of the Inozemtsev Model