New Integrable Systems from Unitary Matrix Models

Abstract

We show that the one dimensional unitary matrix model with potential of the form $aU+b{U}^2+h.c.$ is integrable. By reduction to the dynamics of the eigenvalues, we establish the integrability of a system of particles in one space dimension in an external potential of the form $a\cos (x+\alpha )+b\cos (2x+\beta )$ and interacting through two-body potentials of the inverse sine square type. This system constitutes a generalization of the Sutherland model in the presence of external potentials. The positive-definite matrix model, obtained by analytic continuation, is also integrable, which leads to the integrability of a system of particles in hyperbolic potentials interacting through two-body potentials of the inverse hypebolic sine square type.

13 pages

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