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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Vydáno v:Journal of mathematical analysis and applications Ročník 376; číslo 1; s. 294 - 316
Hlavní autor: Blower, Gordon
Médium: Journal Article
Jazyk:angličtina
Vydáno: Amsterdam Elsevier Inc 01.04.2011
Elsevier
Témata:
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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Popis
Shrnutí: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 .
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2010.10.052