On tau functions arising from linear systems

Abstract: 

The motivation for this work is from the theory of random matrices, and the scattering theory of differential equations with rational matrix coefficients. Associated to a linear system is a matrix scattering function ϕ∈L2(0,∞)ϕ∈L2(0,∞) and a Hankel integral operator ΓϕΓϕ on L2(0,∞)L2(0,∞) defined by

Γϕf(t)=∫∞0ϕ(t+y)f(y)dy.Γϕf(t)=∫0∞ϕ(t+y)f(y)dy.

For ϕϕ of the form ϕ(x)(t)=Ce−(t+2x)ABϕ(x)(t)=Ce−(t+2x)AB we define a `tau function' τ(x)=det(I+Γϕ(x))τ(x)=det(I+Γϕ(x)). This tau function generalizes the notion of the θθ function of an algebraic curve. In this talk we will explore these relationships, focussing on two special cases:

1. (2,2)(2,2)-admissible linear systems which give scattering class potentials;
2. periodic linear systems which give periodic potentials as in Hill's equation.