Rational monodromy-free potentials and Calogero--Moser spaces

Giovanni Felder (ETH)

Tuesday 16th July 16:00-17:00 Maths 110

Abstract

 

A rational function is called monodromy-free potential if it is the
potential of a Schrödinger equation whose solutions are meromorphic for
all values of the spectral parameter. In the case of potentials with
quadratic growth at infinity it was shown by Oblomkov that monodromy-
free potentials are enumerated by partitions  via the Wronskian map for
Hermite polynomials. We show that they can also be identified with
fixed points of a circle action on the Calogero-Moser space. As a
corollary, we solve the inverse problem for the Wronskian map by
showing that the set of contents of the partition is the spectrum of
Moser's matrix evaluated at the poles of the potential. We
also prove a conjecture by Conti and Masoero by computing the weights
of the circle action at the fixed points.

In this talk I will review the classical results on the relation
between monodromy-free potentials and Calogero-Moser systems, then
explain the new results.

The talk is based on joint work with Alexander Veselov.

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