The Heun equation and the Calogero-Moser-Sutherland system V: generalized Darboux transformations

Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case

The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form $-d^2/dx^2+V(g;x)$, where the potential is an elliptic function depending on a coupling vector $g\in{\mathbb R}^4$. Alternatively, this operator arises from the $BC_1$ specialization of the $BC_N$ elliptic nonrelativistic Calogero-Moser system (a.k.a. the Inozemtsev system). Under suitable restrictions on the elliptic periods and on $g$, we associate to this operator a self-adjoint operator $H(g)$ on the Hilbert space ${\mathcal H}=L^2([0,ω_1],dx)$, where $2ω_1$ is the real period of …

On the diagonalization of difference Calogero-Sutherland systems

We discuss the simultaneous diagonalization of a family of commuting difference operators by Koornwinder's multivariable generalization of the Askey-Wilson polynomials. The operators constitute a complete set of quantum integrals for a difference version of the $n$-particle quantum Calogero-Sutherland system related to the root system $BC_n$.

Exact operator solution of the Calogero-Sutherland model

The wave functions of the Calogero-Sutherland model are known to be expressible in terms of Jack polynomials. A formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived. The creation operators that enter in this formula of Rodrigues-type for the Jack polynomials involve Dunkl operators.

The differential equations associated with Calogero-Moser-Sutherland particle models in the freezing regime

Multivariate Bessel processes describe Calogero-Moser-Sutherland particle models and are related with $β$-Hermite and $β$-Laguerre ensembles. They depend on a root system and a multiplicity $k$. Recently, several limit theorems for $k\to\infty$ were derived where the limits depend on the solutions of associated ODEs in these freezing regimes. In this paper we study the solutions of these ODEs which are are singular on the boundaries of their domains. In particular we prove that for a start in arbitrary boundary points, the ODEs always admit unique solutions in their domains for $t>0$.

关于冻结Calogero-Moser-Sutherland粒子模型的微分方程

On the differential equations of frozen Calogero-Moser-Sutherland particle models

本文研究冻结Calogero-Moser-Sutherland粒子模型的微分方程，证明相关常微分方程的特解吸引所有解，建立其与逆热方程的联系，并用于计算Bessel和Jacobi过程行列式公式的期望。