Construction of Transmutation Operators and Hyperbolic Pseudoanalytic Functions

Abstract

A representation for integral kernels of Delsarte transmutation operators is obtained in the form of a functional series with exact formulae for the terms of the series. It is based on the application of hyperbolic pseudoanalytic function theory and recent results on mapping properties of the transmutation operators. The kernel \(K_{1}\) of the transmutation operator relating \(A=-\frac{d^{2} }{dx^{2}}+q_{1}(x)\) and \(B=-\frac{d^{2}}{dx^{2}}\) turns out to be one of the complex components of a bicomplex-valued hyperbolic pseudoanalytic function satisfying a Vekua-type hyperbolic equation of a special form. The other component of the pseudoanalytic function is the kernel of the transmutation operator relating \(C=-\frac{d^{2}}{dx^{2}}+q_{2}(x)\) and \(B\) where \(q_{2}\) is obtained from \(q_{1}\) by a Darboux transformation. We prove an expansion theorem and a Runge-type theorem for this special hyperbolic Vekua equation and using several known results from hyperbolic pseudoanalytic function theory together with the recently discovered mapping properties of the transmutation operators we obtain a new representation for their kernels. Several examples are given. Moreover, approaches for numerical computation of the transmutation kernels and for numerical solution of spectral problems are proposed.