Blower, Gordon (2011) On tau functions for orthogonal polynomials and matrix models. Journal of Physics London a Mathematical and General, 44 (28). pp. 131. ISSN 03054470
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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.
Item Type:  Journal Article 

Journal or Publication Title:  Journal of Physics London a Mathematical and General 
Uncontrolled Keywords:  /dk/atira/pure/researchoutput/libraryofcongress/qa 
Subjects:  
Departments:  Faculty of Science and Technology > Mathematics and Statistics 
ID Code:  40861 
Deposited By:  Professor Gordon Blower 
Deposited On:  13 Jun 2011 10:33 
Refereed?:  Yes 
Published?:  Published 
Last Modified:  16 Sep 2019 00:34 
URI:  https://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 14:56)
 On tau functions for orthogonal polynomials and matrix models. (deposited 13 Jun 2011 10:33) [Currently Displayed]
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