On linear systems and tau functions associated with Lame's equation and Painleve's equation VI

TL;DR

This paper explores the connection between linear systems, tau functions, and special functions related to Lame's and Painleve's equations, expressing tau functions via Hankel operators and determinants, with applications to elliptic functions.

Contribution

It provides a novel representation of tau functions associated with Painleve VI and Lame's equations using Hankel operators and matrix Gelfand–Levitan equations.

Findings

Tau functions are expressed as Fredholm determinants of Hankel operators.

Solutions of hypergeometric equations yield linear systems with similar properties.

Explicit formulas for tau functions in terms of finite determinants for meromorphic transfer functions.

Abstract

Painleve's transcendental differential equation P_{VI} may be expressed as the consistency condition for a pair of linear differential equations with 2 by 2 matrix coefficients with rational entries. By a construction due to Tracy and Widom, this linear system is associated with certain kernels which give trace class operators on Hilbert space. This paper expresses such operators in terms of the Hankel operators \Gamma_\phi of linear systems which are realised in terms of the Laurent coefficients of the solutions of the differential equations. Let P_{(t infty)}:L^2(0, \infty)\to L^2(t, \infty) be the orthogonal projection. For such, the Fredholm determinant \tau (t)=det (I-P_{(t, \infty)}\Gamma_\phi) defines the tau function, which is here expressed in terms of the solutions of a matrix Gelfand--Levitan equation. For suitable paramters, solutions of the hypergeometric equation give a linear system with similar properties. For meromorphic transfer functions \hat\phi that have poles on an arithmetic progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in L^2(0, \infty); so \tau (t) can be expressed in terms of finite determinants. This applies to elliptic functions of the second kind, such as satisfy Lame's equation with \ell=1.