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

Abstract

It is proved that if the Schrödinger equation $L ψ= λψ$ of Calogero-Moser-Sutherland type with $$L = -Δ+ \sum\limits_{α\in{\cal A}_{+}} \frac{m_α(m_α+1) (α,α)}{\sin^{2}(α,x)}$$ has a solution of the product form $ψ_0 = \prod_{α\in {\cal {A}_+}} \sin^{-m_α}(α,x),$ then the function $F(x) =\sum\limits_{α\in \cal {A}_{+}} m_α (α,x)^2 {\rm log} (α,x)^2$ satisfies the generalised WDVV equations.