Transmutations and spectral parameter power series in eigenvalue problems

TL;DR

This paper reviews recent advances in Sturm-Liouville theory, focusing on transmutation operators and spectral parameter power series (SPPS), highlighting their ability to analytically solve spectral problems and their applications to Darboux transformed Schrödinger operators.

Contribution

It introduces the SPPS method and transmutation operators, demonstrating their properties, applications, and how they enable solving spectral problems analytically.

Findings

SPPS provides an analytic form for dispersion equations.

Transmutation operators can be constructed even without their kernels.

Applications include Darboux transformed Schrödinger operators.

Abstract

We give an overview of recent developments in Sturm-Liouville theory concerning operators of transmutation (transformation) and spectral parameter power series (SPPS). The possibility to write down the dispersion (characteristic) equations corresponding to a variety of spectral problems related to Sturm-Liouville equations in an analytic form is an attractive feature of the SPPS method. It is based on a computation of certain systems of recursive integrals. Considered as families of functions these systems are complete in the $L_{2}$-space and result to be the images of the nonnegative integer powers of the independent variable under the action of a corresponding transmutation operator. This recently revealed property of the Delsarte transmutations opens the way to apply the transmutation operator even when its integral kernel is unknown and gives the possibility to obtain further interesting properties concerning the Darboux transformed Schr\"{o}dinger operators. We introduce the systems of recursive integrals and the SPPS approach, explain some of its applications to spectral problems with numerical illustrations, give the definition and basic properties of transmutation operators, introduce a parametrized family of transmutation operators, study their mapping properties and construct the transmutation operators for Darboux transformed Schr\"{o}dinger operators.