On perturbation theory for the Sturm-Liouville problem, Part II

TL;DR

This paper develops an iterative analytical approach to solve specific Sturm-Liouville problems with step-wise coefficients, providing accurate formulas for eigenvalues and eigenfunctions, and compares these with perturbation theory results.

Contribution

It introduces a new iterative method for solving Sturm-Liouville problems with piecewise-constant coefficients, enhancing accuracy over traditional perturbation techniques.

Findings

Derived simple, accurate formulas for ground state eigenvalues and eigenfunctions.

Demonstrated high numerical precision of the method compared to perturbation theory.

Validated the approach through comparison with existing perturbation results.

Abstract

I study some possibilities of analytically solving a particular Sturm-Liouville problem with step-wise (piece-constant) coefficients with help of an iterative procedure mentioned in my previous paper (Green's function sum rules). I construct short, simple, but very accurate analytical formulae for calculating the ground state eigenvalue and eigenfunction as well as for calculating the first eigenfunction. I study numerical precision of the obtained approximations together with the perturbation theory results.