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

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

2009-06-18

arxiv.org/abs/0906.3504v7

Physics

Mathematical Physics

Original study, 19 pages, 10 figures, corrected and improved text and formulas, enlarged Appendix 4 and added Appendix 5. (The

Scientific paper

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 a simple but very accurate analytical formula for calculating the lowest eigenvalue. 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.

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