Home > Research > Publications & Outputs > On tau functions for orthogonal polynomials and...

Research

Associated organisational unit

Electronic data

Links

Text available via DOI:

Keywords

View graph of relations

On tau functions for orthogonal polynomials and matrix models.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Published

Standard

On tau functions for orthogonal polynomials and matrix models. / Blower, Gordon.
In: Journal of Physics A: Mathematical and Theoretical, Vol. 44, No. 28, 15.07.2011, p. 1-31.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Blower, G 2011, 'On tau functions for orthogonal polynomials and matrix models.', Journal of Physics A: Mathematical and Theoretical, vol. 44, no. 28, pp. 1-31. https://doi.org/10.1088/1751-8113/44/28/285202

APA

Blower, G. (2011). On tau functions for orthogonal polynomials and matrix models. Journal of Physics A: Mathematical and Theoretical, 44(28), 1-31. https://doi.org/10.1088/1751-8113/44/28/285202

Vancouver

Blower G. On tau functions for orthogonal polynomials and matrix models. Journal of Physics A: Mathematical and Theoretical. 2011 Jul 15;44(28):1-31. doi: 10.1088/1751-8113/44/28/285202

Author

Blower, Gordon. / On tau functions for orthogonal polynomials and matrix models. In: Journal of Physics A: Mathematical and Theoretical. 2011 ; Vol. 44, No. 28. pp. 1-31.

Bibtex

@article{871742eb6ec944d2ae3fc5c8203a4509,
title = "On tau functions for orthogonal polynomials and matrix models.",
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.",
keywords = "Inverse scattering, random matrices",
author = "Gordon Blower",
year = "2011",
month = jul,
day = "15",
doi = "10.1088/1751-8113/44/28/285202",
language = "English",
volume = "44",
pages = "1--31",
journal = "Journal of Physics A: Mathematical and Theoretical",
issn = "1751-8113",
publisher = "IOP Publishing Ltd.",
number = "28",

}

RIS

TY - JOUR

T1 - On tau functions for orthogonal polynomials and matrix models.

AU - Blower, Gordon

PY - 2011/7/15

Y1 - 2011/7/15

N2 - 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.

AB - 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.

KW - Inverse scattering

KW - random matrices

U2 - 10.1088/1751-8113/44/28/285202

DO - 10.1088/1751-8113/44/28/285202

M3 - Journal article

VL - 44

SP - 1

EP - 31

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 28

ER -