On tau functions for orthogonal polynomials and matrix models

Abstract

Let v be a real polynomial of even degree, and let ρbe the equilibrium probability measure for v with support S; so that v(x)\geq 2\int \log |x-y| ρ(dy)+C_v for some constant C_v with support S. Then S is the union of finitely many bounded intervals with endpoints delta_j, and ρis given by an algebrais weight w(x) on S. The system of orthogonal polynomials for w gives rise to the Magnus--Schlesinger differential equations. This paper identifies the tau function of this system with the Hankel determinant det[\in x^{j+k}ρ(dx)] of ρ. The solutions of the Magnus--Schlesinger equations are realised by a linear system, which is used to compute the tau function in terms of a Gelfand--Levitan equaiton. The tau function is associated with a potential q and a scattering problem for the Schrodinger operator with potential q. For some algebro-geometric potentials, the paper solves the scattering problem in terms of linear systems. The theory extends naturally to elliptic curves and resolves the case where S has exactly two intervals.