An integral representation for eigenfunctions of linear ordinary differential operators

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: $\text{Lφ = k}{}^4\text{φ, L =}\text{d}{}^4\text{dx}{}^4\text{+}\text{d}\text{dx}\text{(q}\text{d}\text{dx}\text{) + r}$φ(0) = 1, φ′(0) = 0, φ″(0) = ?k², 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.