On tau functions for orthogonal polynomials and matrix models

TL;DR

This paper links tau functions for orthogonal polynomials to Hankel determinants and explores their relation to differential equations, scattering problems, and algebraic potentials, extending the theory to elliptic curves and multi-interval supports.

Contribution

It identifies the tau function with Hankel determinants for orthogonal polynomials and connects it to differential equations and scattering theory, including new solutions for multi-interval supports.

Findings

Tau function equals the Hankel determinant of the equilibrium measure.

Solutions are realized via a linear system and Gelfand-Levitan equation.

Extended the theory to elliptic curves and two-interval cases.

Abstract

Let v be a real polynomial of even degree, and let \rho be the equilibrium probability measure for v with support S; so that v(x)\geq 2\int \log |x-y| \rho (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 \rho is 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}\rho (dx)] of \rho. 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.