Construction of transmutation operators and hyperbolic pseudoanalytic functions

TL;DR

This paper develops a new representation for transmutation operator kernels using hyperbolic pseudoanalytic functions, providing theoretical insights and numerical methods for spectral problems involving differential operators.

Contribution

It introduces a novel hyperbolic pseudoanalytic function framework to explicitly construct transmutation kernels and proves related expansion and Runge-type theorems.

Findings

Explicit formulas for transmutation kernels as functional series

Representation of kernels via hyperbolic pseudoanalytic functions

Proposed numerical approaches for spectral problems

Abstract

A representation for integral kernels of Delsarte transmutation operators is obtained in the form of a functional series with the exact formulas 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 $B=-\frac{d^2}{dx^2}$ results 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 the 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 obtain the new representation for their kernels. Several examples are given. Moreover, based on the presented results approaches for numerical computation of the transmutation kernels and for numerical solution of spectral problems are proposed.