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 |
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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 |

### Available Versions of this Item

- On the tau function associated with the generalized unitary ensemble. (deposited 28 Jul 2010 15:56)
- On tau functions for orthogonal polynomials and matrix models. (deposited 13 Jun 2011 11:33)
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- On tau functions for orthogonal polynomials and matrix models. (deposited 13 Jun 2011 11:33)

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