Yasushi Komori

Quantum integrability of the generalized elliptic Ruijsenaars models

The quantum integrability of the generalized elliptic Ruijsenaars models is shown. These models are mathematically related to the Macdonald operator and the Macdonald - Koornwinder operator, which appeared in the q-orthogonal polynomial theories. We construct these integrable families by using the Yang - Baxter equation and the reflection equation.

Crystallization of the Bogoyavlensky Lattice

We introduce a vertex model in two-dimension, which is associated with the Bogoyavlensky lattice. We show that in a crystallized limit ( q →0) we have a unique configuration, and that it coincides with an evolution of the soliton cellular automata which is a generalization of the system introduced by Takahashi and Satsuma.

The Perturbation of the Quantum¶Calogero–Moser–Sutherland System¶and Related Results

Abstract: The Hamiltonian of the trigonometric Calogero–Sutherland model coincides with a certain limit of the Hamiltonian of the elliptic Calogero–Moser model. In other words the elliptic Hamiltonian is a perturbed operator of the trigonometric one.In this article we show the essential self-adjointness of the Hamiltonian of the elliptic Calogero–Moser model and the regularity (convergence) of the perturbation for the arbitrary root system.We also show the holomorphy of the joint eigenfunctions of the commuting Hamiltonians w.r.t the variables (x1, …,xN) for the AN-1-case. As a result, the algebraic calculation of the perturbation is justified.

Conserved operators of the generalized elliptic Ruijsenaars models

We construct the integrable families of the generalized elliptic Ruijsenaars models by use of the Yang–Baxter equation and the reflection equation. The higher order conserved operators of these models are explicitly presented.

ELLIPTIC K-MATRIX ASSOCIATED WITH BELAVIN'S SYMMETRIC R-MATRIX

Abstract We study solutions of the reflection equation associated with Belavins R-matrix following a method of Felder and Pasquier. We construct the elliptic boundary K-matrix, which has four arbitrary parameters.

Affine R-Matrix and the Generalized Elliptic Ruijsenaars Models

Commutative elliptic difference operators associated with the affine root systems are constructed in terms of affine R-matrices. These operators describe the Ruijsenaars models with elliptic potentials and reduce to the Macdonald operators in the trigonometric limit.

Notes on the Elliptic Ruijsenaars Operators

We construct the spaces that the elliptic Ruijsenaars operators act on. It is shown that they are extensible to nonnegative selfadjoint operators on a space of square integrable functions, or preserve a finite dimensional vector space of entire functions. These facts are shown in terms of the R-operators which satisfy the Yang–lBaxter equation. The elliptic Ruijsenaars operators are considered as the elliptic analogues of the Macdonald operators or the difference analogues of the Lamé operators.

Ruijsenaars’ commuting difference operators and invariant subspace spanned by theta functions

We study a family of mutually commutative difference operators introduced by Ruijsenaars. The conjugations of these operators with an appropriate function give the Hamiltonians of some relativistic quantum systems. These operators can be regarded as elliptic analogs of the Macdonald operators and their coefficients consist of the Jacobi theta functions. We show that these operators act on the space of meromorphic functions on the Cartan subalgebra of affine Lie algebras and that the space spanned by characters of a fixed positive level is invariant under the action of these operators.

Integrability, Fusion, and Duality in the Elliptic Ruijsenaars Model

The generalized elliptic Ruijsenaars models, which are regarded as a difference analog of the Calogero–Sutherland–Moser models associated with the classical root systems are studied. The integrability and the duality using the fusion procedure of operator-valued solutions of the Yang–Baxter equation and the reflection equation are shown. In particular a new integrable difference operator of type-D is proposed. The trigonometric models are also considered in terms of the representation of the affine Hecke algebra.

Essential self-adjointness of the elliptic Ruijsenaars models

We study the elliptic Ruijsenaars models associated with arbitrary root systems, which are difference analogs of the Calogero–Moser model. We give a dense subspace in the space of square integrable functions invariant under the action of the Weyl group on a torus as a domain of its Hamiltonian and prove its essential self-adjointness by using perturbation theory. It is also clarified that these models consist of pure point spectrum.