# On tau functions for orthogonal polynomials and matrix models.

Blower, Gordon (2011) On tau functions for orthogonal polynomials and matrix models. Journal of Physics -London- a Mathematical and General, 44 (28). pp. 1-31. ISSN 0305-4470

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## Abstract

Let v be a real polynomial of even degree, and let \rho be the equilibrium probability measure for v with support S; to that v(x) greeater than or equal to \int 2log |x-y| \rho (dy) +C for some constant C with equality on S. THen S is the union of finitely many boundd intervals with endpoints \delta_j and \rho is given by an algebraic weight w(x) on S. Then the system of orthogonal polynomials for w gives rise to a system of differential equations, known as the Schlesinger equations. This paper identifies the tau function of this system with the Hankel determinant \det [\int x^{j+k}\rho (dx)]. The solutions of the Magnus--Schlesinger equation are realised by a linear system, which is used to compute the tau functions in terms of a Gelfand--Levitan equation. The tau function is associated with a potential q and a scattering problem for the Schrodinger equation with potential q. The paper describes cases where this is integrable in terms of the nonlinear Fourier transform.

Item Type: Journal Article Journal of Physics -London- a Mathematical and General Inverse scattering ; random matrices Q Science > QA Mathematics Faculty of Science and Technology > Mathematics and Statistics 40861 Professor Gordon Blower 13 Jun 2011 11:33 Yes Published 24 Jan 2018 03:11 http://eprints.lancs.ac.uk/id/eprint/40861

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