On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI - Blower, Gordon (Journal of mathematical analysis and applications, 2011)
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 .