On generalisations of Calogero-Moser-Sutherland quantum problem and WDVV equations

TL;DR

This paper proves a connection between solutions of a Calogero-Moser-Sutherland type Schrödinger equation and the satisfaction of generalized WDVV equations by a specific function, revealing a deep link between quantum integrable systems and algebraic structures.

Contribution

It establishes that a particular product-form solution to the Schrödinger equation implies the associated function satisfies generalized WDVV equations, extending known relations in integrable systems.

Findings

Solution of Schrödinger equation implies WDVV equations for a specific function.

Connects quantum integrable models with algebraic structures governed by WDVV.

Provides a new criterion for WDVV equations based on Schrödinger solutions.

Abstract

It is proved that if the Schr\"odinger equation $L \psi = \lambda \psi$ of Calogero-Moser-Sutherland type with $$L = -\Delta + \sum\limits_{\alpha\in{\cal A}_{+}} \frac{m_{\alpha}(m_{\alpha}+1) (\alpha,\alpha)}{\sin^{2}(\alpha,x)}$$ has a solution of the product form $\psi_0 = \prod_{\alpha \in {\cal {A}_+}} \sin^{-m_{\alpha}}(\alpha,x),$ then the function $F(x) =\sum\limits_{\alpha \in \cal {A}_{+}} m_{\alpha} (\alpha,x)^2 {\rm log} (\alpha,x)^2$ satisfies the generalised WDVV equations.