On Perturbation Theory for the Sturm-Liouville Problem with Variable Coefficients

TL;DR

This paper develops analytical perturbation methods for solving Sturm-Liouville problems with variable coefficients, providing formulas for eigenvalues and eigenfunctions, and introduces optimization techniques for improved accuracy.

Contribution

It introduces new reformulation and boundary conditions for perturbation theory applied to variable coefficient Sturm-Liouville problems, enhancing accuracy and convergence.

Findings

Derived accurate formulas for lowest eigenvalues and ground states

Proposed optimized initial approximations to reduce perturbation corrections

Validated methods for smooth and step-wise coefficients

Abstract

In this article I study different possibilities of analytically solving the Sturm-Liouville problem with variable coefficients of sufficiently arbitrary behavior with help of perturbation theory. I show how the problem can be reformulated in order to eliminate big (or divergent) corrections. I obtain correct formulas in case of smooth as well as in case of step-wise (piece-constant) coefficients. I build simple, but very accurate analytical formulae for calculating the lowest eigenvalue and the ground state eigenfunction. I advance also new boundary conditions for obtaining more precise initial approximations. I demonstrate how one can optimize the PT calculation with choosing better initial approximations and thus diminishing the perturbative corrections. Dressing, Rebuilding, and Renormalizations are discussed in Appendices 4 and 5.