Analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems

TL;DR

This paper introduces a novel method leveraging Delsarte transmutation operators and generalized wave polynomials to accurately approximate solutions and eigenvalues of Sturm-Liouville spectral problems.

Contribution

It presents an analytic approximation technique that significantly improves the accuracy of spectral problem solutions using transmutation operators and wave polynomial approximations.

Findings

Enables highly accurate computation of eigenvalues and eigenfunctions.

Reduces the spectral problem to approximating a potential primitive with wave polynomials.

Achieves extreme accuracy in numerical solutions of Sturm-Liouville problems.

Abstract

A method for approximate solution of spectral problems for Sturm-Liouville equations based on the construction of the Delsarte transmutation operators is presented. In fact the problem of numerical approximation of solutions and eigenvalues is reduced to approximation of a primitive of the potential by a finite linear combination of generalized wave polynomials introduced in arXiv:1208.5984, arXiv:1208.6166. The method allows one to compute both lower and higher eigendata with an extreme accuracy.