On tau functions for orthogonal polynomials and matrix models

Abstract

Abstract. Let v be a real polynomial of even degree, and let ρ be the equilibrium probability measure for v with support S; so that, v(x) ≥ 2 log |x−y| ρ(dy)+C v for some constant C v with equality on S. Then S is the union of finitely many bounded intervals with endpoints δ j , and ρ is given by an algebraic weight w(x) on S. The system of orthogonal polynomials for w gives rise to the Magnus-Schlesinger differential equations. This paper identifies the τ function of this system with the Hankel determinant det[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 equation. The tau function is associated with a potential q and a scattering problem for the Schrödinger operator with potential q. For some algebro-geometric q, 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.MSC (2000) classification: 60B20 (37K15)

On tau functions associated with linear systems

“…By simple manipulations of (7.22), one shows that ϑ 1 is entire and elliptic of the third kind, and from the product formula it is evident that ϑ 1 has a simple zero at x = 0 and no others in the fundamental cell of C/Λ. In section 7 of [11] we likewise obtained a uniform periodic linear system with tau function ϑ 1 (x).…”