On tau functions for orthogonal polynomials and matrix models

Let v be a real polynomial of even degree, representing an electrostatic field and w the equilibrium density for charge on a long conducting wire. The system of orthogonal polynomials for w gives rise to 2 × 2 rational matrix differential equations Y' = A_nY which satisfy a recurrence relation. Here w is algebraic with Riemann surface {\cal E}, and \tau _n(t)=\det \big[\int _{-\infty }^t x^{j+k}w(x)\,dx\big]_{j,k=0}^{n-1} belongs to a Liouvillian tower over {\cal E}. The solutions of Y' = A_nY give data for an inverse scattering problem that can be solved via the Gelfand-Levitan equation in terms of rational operator functions. Using linear systems, the paper shows that a multiple of sin x is the scattering function for Lamé's equation -f'' + 2weierpf = λf and realises elliptic potentials from periodic linear systems.