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On tau functions for orthogonal polynomials and matrix models.

Blower, Gordon (2011) On tau functions for orthogonal polynomials and matrix models. Journal of Physics 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: Article
      Journal or Publication Title: Journal of Physics A: Mathematical and General
      Uncontrolled Keywords: Inverse scattering ; random matrices
      Subjects: Q Science > QA Mathematics
      Departments: Faculty of Science and Technology > Mathematics and Statistics
      ID Code: 40861
      Deposited By: Professor Gordon Blower
      Deposited On: 13 Jun 2011 11:33
      Refereed?: Yes
      Published?: Published
      Last Modified: 09 Oct 2013 13:14
      Identification Number:
      URI: http://eprints.lancs.ac.uk/id/eprint/40861

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