Oleg A. Chalykh

Commutative rings of partial differential operators and Lie algebras

We give examples of finite gap Schrödinger operators in the two-dimensional case.

New integrable generalizations of Calogero–Moser quantum problem

A one-parameter deformation of Calogero–Moser quantum problem is introduced. It is shown that corresponding Schrodinger operator is integrable for any value of the parameter and algebraically integrable in case of integer value.

Integrability in the theory of Schrödinger operator and harmonic analysis

The algebraic integrability for the Schrödinger equation in ℝn and the role of the quantum Calogero-Sutherland problem and root systems in this context are discussed. For the special values of the parameters in the potential the explicit formula for the eigenfunction of the corresponding Sutherland operator is found. As an application the explicit formula for the zonal spherical functions on the symmetric spacesSU2n*/Spn (type A II in Cartan notations) is presented.

Algebraic integrability for the Schrödinger equation and finite reflection groups

Algebraic integrability of ann-dimensional Schrödinger equation means that it has more thann independent quantum integrals. Forn=1, the problem of describing such equations arose in the theory of finite-gap potentials. The present paper gives a construction which associates finite reflection groups (in particular, Weyl groups of simple Lie algebras) with algebraically integrable multidimensional Schrödinger equations. These equations correspond to special values of the parameters in the generalization of the Calogero—Sutherland system proposed by Olshanetsky and Perelomov. The analytic properties of a joint eigenfunction of the corresponding commutative rings of differential operators are described. Explicit expressions are obtained for the solution of the quantum Calogero—Sutherland problem for a special value of the coupling constant.

Bispectrality for the quantum Ruijsenaars model and its integrable deformation

An elementary construction of the eigenfunctions for the quantum rational Ruijsenaars model with integer coupling parameter is presented. As a by-product, we establish the bispectral duality between this model and the trigonometric Calogero–Moser model. In particular, this gives a new way for calculating Jack polynomials. We propose also a certain one-parameter deformation of the Ruijsenaars model, proving its integrability and bispectrality. The generalizations related to other root systems and difference operators by Macdonald are considered.

EXPLICIT FORMULAS FOR SPHERICAL FUNCTIONS ON SYMMETRIC SPACES OF TYPE AII

Let X = ,G/K be a symmetric space and let Dk(G ) denote the ring of left-invariant differential operators on G which are right-invariant under K. A spherical function of X is a K-biinvariant eigenfunction ,p(g) of the ring Dk(G ) with the normalization ~o(e) = 1 [1, 2, 31. For noncompact symmetric spaces, i.e., for a noncompact connected real semisimple Lie group G with finite center and for its maximal compact subgroup K, the following Harish-Chandra formula gives all spherical functions on G/K as integrals over K:

Multidimensional integrable Schrödinger operators with matrix potential

The Schrodinger operators with matrix rational potential, which are D-integrable, i.e., can be intertwined with the pure Laplacian, are investigated. Corresponding potentials are uniquely determined by their singular data which are a configuration of the hyperplanes in Cn with prescribed matrices. We describe some algebraic conditions (matrix locus equations) on these data, which are sufficient for D-integrability. As the examples some matrix generalizations of the Calogero–Moser operators are considered.