Generalized Lam� Operators

Abstract

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.

Quantum integrability of the deformed elliptic Calogero–Moser problem

“…This proves integrability of the system for all n and m and due to a general recent result by Chalykh, Etingof and Oblomkov [4] this also implies the algebraic integrability for integer values of the parameter m.…”

“…The complete family of the commuting quantum integrals for arbitrary m is given by the previous theorems. The algebraic integrability in the case when m is integer follows from the general result due to Chalykh, Etingof and Oblomkov (see Theorem 3.8 in [4]). …”

Bethe Ansatz for the Ruijsenaars Model of BC1-Type

“…[2,3]. We begin with two elementary results about a three-term difference operator with meromorphic coefficients:…”

“…(Notice that (3) contains eight parameters µ p , µ p , compared to the four in the Askew-Wilson operator and in (2).) Therefore, it is natural to expect that (3) and (1) should have similar properties. This is indeed the case, as we will demonstrate below.…”

Multivariable Burchnall–Chaundy theory

“…In consequence, definition 2.1 was refined into 'algebraic integrability', and in that sense, in Chalykh et al (2003) more of the conjecture is proven, by inputting techniques of differential Galois theory, which was first done in Braverman et al (1996). The main technique of proof consists in finding eigenfunctions, although most of the time inexplicitly; also not quite explicit is the construction of differential operators that commute with a given L. We try to give a flavour for this theory.…”