An integral representation for eigenfunctions of linear ordinary differential operators

Abstract

This paper generalizes an integral representation formula for eigenfunctions of Sturm-Liouville operators, known as the Volterra transformation operator in the theory of the inverse scattering problem, to higher-order differential operators. A specific fourth-order initial value problem is considered: Lφ = k 4φ, L = d 4 dx 4 + d dx (q d dx ) + r φ(0) = 1, φ′(0) = 0, φ″(0) = − k 2, φ‴ = 0 The solution for complex k is expressed as an inverse-Laplace-Borel transform. Jump formulae are obtained relating the representing kernel directly to the coefficients of L. The result admits obvious generalization to operators of arbitrary order.