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

TL;DR

This paper investigates the solutions of certain singular differential equations related to Calogero-Moser-Sutherland particle models in the freezing regime, establishing existence and uniqueness of solutions starting from boundary points.

Contribution

It proves the existence and uniqueness of solutions to boundary-singular ODEs associated with these particle models, extending understanding of their freezing regimes.

Findings

Unique solutions exist for all boundary points for t>0.

Solutions are well-defined despite singularities at domain boundaries.

Results apply to multivariate Bessel processes and related ensembles.

Abstract

Multivariate Bessel processes describe Calogero-Moser-Sutherland particle models and are related with $\beta$-Hermite and $\beta$-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 $t>0$.