Blower, Gordon (2011) On tau functions for orthogonal polynomials and matrix models. Journal of Physics A: Mathematical and Theoretical, 44 (28). pp. 131. ISSN 17518113
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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 xy \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 MagnusSchlesinger equation are realised by a linear system, which is used to compute the tau functions in terms of a GelfandLevitan 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.
Available Versions of this Item

On the tau function associated with the generalized unitary ensemble. (deposited 28 Jul 2010 14:56)
 On tau functions for orthogonal polynomials and matrix models. (deposited 13 Jun 2011 10:33) [Currently Displayed]