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On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI.

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

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On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI. / Blower, Gordon.
In: Journal of Mathematical Analysis and Applications, Vol. 376, No. 1, 01.04.2011, p. 294-316.

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

Harvard

Blower, G 2011, 'On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI.', Journal of Mathematical Analysis and Applications, vol. 376, no. 1, pp. 294-316. https://doi.org/10.1016/j.jmaa.2010.10.052

APA

Blower, G. (2011). On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI. Journal of Mathematical Analysis and Applications, 376(1), 294-316. https://doi.org/10.1016/j.jmaa.2010.10.052

Vancouver

Blower G. On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI. Journal of Mathematical Analysis and Applications. 2011 Apr 1;376(1):294-316. doi: 10.1016/j.jmaa.2010.10.052

Author

Blower, Gordon. / On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI. In: Journal of Mathematical Analysis and Applications. 2011 ; Vol. 376, No. 1. pp. 294-316.

Bibtex

@article{c1b18a8a811c48b788498bf031f20fe8,
title = "On linear systems and τ functions associated with Lam{\'e}'s equation and Painlev{\'e}'s equation VI.",
abstract = "Painleve's transcendental differential equation PVI may be expressed as the consistency condition for a pair of linear differential equations with 2 by 2 matrix coefficients with rational entries. This linear system is assocaited with certain kernels which give trace class operatos on Hilbert space. This paper expresses such operators in terms of Hankel operators \Gamma_\phi of linear systems. For such, the Fredholm determinant \det (I-\Gamma_\phi P_(t, \infty )) gives rise to the tau function, which is shown to be the solution of a matrix Gelfand--Levitan equation. For meromorphic transfer functions that have poles on an arithmetric progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in L^2(0, \infty ); so \det (I-\Gamma_\phi)can be expressed as a series of finite determinants. This applies to ellliptic functios of the second kind, such as satisfy Lame's differential equation.",
keywords = "random matrices, Tracy Widom operators",
author = "Gordon Blower",
year = "2011",
month = apr,
day = "1",
doi = "10.1016/j.jmaa.2010.10.052",
language = "English",
volume = "376",
pages = "294--316",
journal = "Journal of Mathematical Analysis and Applications",
issn = "0022-247X",
publisher = "Academic Press Inc.",
number = "1",

}

RIS

TY - JOUR

T1 - On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI.

AU - Blower, Gordon

PY - 2011/4/1

Y1 - 2011/4/1

N2 - Painleve's transcendental differential equation PVI may be expressed as the consistency condition for a pair of linear differential equations with 2 by 2 matrix coefficients with rational entries. This linear system is assocaited with certain kernels which give trace class operatos on Hilbert space. This paper expresses such operators in terms of Hankel operators \Gamma_\phi of linear systems. For such, the Fredholm determinant \det (I-\Gamma_\phi P_(t, \infty )) gives rise to the tau function, which is shown to be the solution of a matrix Gelfand--Levitan equation. For meromorphic transfer functions that have poles on an arithmetric progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in L^2(0, \infty ); so \det (I-\Gamma_\phi)can be expressed as a series of finite determinants. This applies to ellliptic functios of the second kind, such as satisfy Lame's differential equation.

AB - Painleve's transcendental differential equation PVI may be expressed as the consistency condition for a pair of linear differential equations with 2 by 2 matrix coefficients with rational entries. This linear system is assocaited with certain kernels which give trace class operatos on Hilbert space. This paper expresses such operators in terms of Hankel operators \Gamma_\phi of linear systems. For such, the Fredholm determinant \det (I-\Gamma_\phi P_(t, \infty )) gives rise to the tau function, which is shown to be the solution of a matrix Gelfand--Levitan equation. For meromorphic transfer functions that have poles on an arithmetric progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in L^2(0, \infty ); so \det (I-\Gamma_\phi)can be expressed as a series of finite determinants. This applies to ellliptic functios of the second kind, such as satisfy Lame's differential equation.

KW - random matrices

KW - Tracy Widom operators

U2 - 10.1016/j.jmaa.2010.10.052

DO - 10.1016/j.jmaa.2010.10.052

M3 - Journal article

VL - 376

SP - 294

EP - 316

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -