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

It is proved that if the Schrödinger equation Lψ = λψ of CalogeroMoser-Sutherland type with L = −∆ + ∑ α∈A+ mα(mα + 1)(α, α) sin(α, x) has a solution of the product form ψ0 = ∏ α∈A+ sin −mα(α, x), then the function F (x) = ∑ α∈A+ mα(α, x) 2 log (α, x) satisfies the generalised WDVV equations. This explains the relation between these equations and the deformed CMS quantum problems observed in [7]. Introduction. Let A be a finite set of vectors α in the Euclidean space R which generates the space and is invariant under the symmetry x → −x. We assume that −α is the only vector from A which is proportional to α. Let A+ be its half positive with respect to some linear form. Let us prescribe to each vector α ∈ A a real number (”multiplicity”) mα such that m−α = mα. Consider the following Schrödinger operator