Liouville transformation, analytic approximation of transmutation operators and solution of spectral problems

TL;DR

This paper introduces a novel numerical method combining Liouville transformation and transmutation operators to accurately solve spectral problems for Sturm-Liouville equations, enabling precise computation of eigenvalues and eigenfunctions.

Contribution

The paper develops a new approach that approximates transmutation operators using generalized wave polynomials, improving the accuracy of spectral problem solutions.

Findings

High-accuracy computation of eigenvalues and eigenfunctions achieved.

Reduction of spectral problem to function approximation involving coefficients p, q, r.

Theoretical results on the Liouville transformation's action on formal powers.

Abstract

A method for solving spectral problems for the Sturm-Liouville equation $(pv^{\prime})^{\prime}-qv+\lambda rv=0$ based on the approximation of the Delsarte transmutation operators combined with the Liouville transformation is presented. The problem of numerical approximation of solutions and of eigendata is reduced to approximation of a pair of functions depending on the coefficients $p$, $q$ and $r$ by a finite linear combination of certain specially constructed functions related to 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. Several necessary results concerning the action of the Liouville transformation on formal powers arising in the method of spectral parameter power series are obtained as well as the transmutation operator for the Sturm-Liouville operator $\frac{1}{r}\left(\frac{d}{dx}p\frac{d}{dx}-q\right)$.