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**On tau functions for orthogonal polynomials and...**227 KB, PDF-document

265 KB, PDF-document

10/03/13

Research output: Contribution to journal › Journal article

Published

<mark>Journal publication date</mark> | 15/07/2011 |
---|---|

<mark>Journal</mark> | Journal of Physics -London- a Mathematical and General |

Issue | 28 |

Volume | 44 |

Number of pages | 31 |

Pages | 1-31 |

<mark>Original language</mark> | English |

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.