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

Painlevé's transcendental differential equation P VI may be expressed as the consistency condition for a pair of linear differential equations with 2 × 2 matrix coefficients with rational entries. By a construction due to Tracy and Widom, this linear system is associated with certain kernels wh...

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Veröffentlicht in:	Journal of mathematical analysis and applications Jg. 376; H. 1; S. 294 - 316
1. Verfasser:	Blower, Gordon
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Amsterdam Elsevier Inc 01.04.2011 Elsevier
Schlagworte:	Algebra Algebraic geometry Exact sciences and technology Linear and multilinear algebra, matrix theory Mathematical analysis Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Ordinary differential equations Random matrices Sciences and techniques of general use Tracy–Widom operators Tracy–Widom operators Random matrices Arithmetics Differential equation Determinant Elliptic function Linear operator Lamé function Equation system Orthogonal projection Linear system Linear equation Mathematical analysis Hilbert space Tracy-Widom operators Transfer function Matrix equation Meromorphic function
ISSN:	0022-247X, 1096-0813
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Abstract	Painlevé's transcendental differential equation P VI may be expressed as the consistency condition for a pair of linear differential equations with 2 × 2 matrix coefficients with rational entries. By a construction due to Tracy and Widom, this linear system is associated with certain kernels which give trace class operators on Hilbert space. This paper expresses such operators in terms of Hankel operators Γ ϕ of linear systems which are realised in terms of the Laurent coefficients of the solutions of the differential equations. Let P ( t , ∞ ) : L 2 ( 0 , ∞ ) → L 2 ( t , ∞ ) be the orthogonal projection; then the Fredholm determinant τ ( t ) = det ( I − P ( t , ∞ ) Γ ϕ ) defines the τ function, which is here expressed in terms of the solution of a matrix Gelfand–Levitan equation. For suitable values of the parameters, solutions of the hypergeometric equation give a linear system with similar properties. For meromorphic transfer functions ϕ ˆ that have poles on an arithmetic progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in L 2 ( 0 , ∞ ) ; so det ( I − Γ ϕ P ( t , ∞ ) ) can be expressed as a series of finite determinants. This applies to elliptic functions of the second kind, such as satisfy Lamé's equation with ℓ = 1 .
AbstractList	Painlevé's transcendental differential equation P VI may be expressed as the consistency condition for a pair of linear differential equations with 2 × 2 matrix coefficients with rational entries. By a construction due to Tracy and Widom, this linear system is associated with certain kernels which give trace class operators on Hilbert space. This paper expresses such operators in terms of Hankel operators Γ ϕ of linear systems which are realised in terms of the Laurent coefficients of the solutions of the differential equations. Let P ( t , ∞ ) : L 2 ( 0 , ∞ ) → L 2 ( t , ∞ ) be the orthogonal projection; then the Fredholm determinant τ ( t ) = det ( I − P ( t , ∞ ) Γ ϕ ) defines the τ function, which is here expressed in terms of the solution of a matrix Gelfand–Levitan equation. For suitable values of the parameters, solutions of the hypergeometric equation give a linear system with similar properties. For meromorphic transfer functions ϕ ˆ that have poles on an arithmetic progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in L 2 ( 0 , ∞ ) ; so det ( I − Γ ϕ P ( t , ∞ ) ) can be expressed as a series of finite determinants. This applies to elliptic functions of the second kind, such as satisfy Lamé's equation with ℓ = 1 .
Author	Blower, Gordon
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Keywords	Tracy–Widom operators Random matrices Arithmetics Differential equation Determinant Elliptic function Linear operator Lamé function Equation system Orthogonal projection Linear system Linear equation Mathematical analysis Hilbert space Tracy-Widom operators Transfer function Matrix equation Meromorphic function
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Snippet	Painlevé's transcendental differential equation P VI may be expressed as the consistency condition for a pair of linear differential equations with 2 × 2...
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SubjectTerms	Algebra Algebraic geometry Exact sciences and technology Linear and multilinear algebra, matrix theory Mathematical analysis Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Ordinary differential equations Random matrices Sciences and techniques of general use Tracy–Widom operators
Title	On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI
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