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In this article I study different possibilities of analytically solving theSturm-Liouville problem with variable coefficients of sufficiently arbitrarybehavior with help of perturbation theory. I show how the problem can bereformulated in order to eliminate big (or divergent) corrections. I obtaincorrect formulas in case of smooth as well as in case of step-wise(piece-constant) coefficients. I build simple, but very accurate analyticalformulae for calculating the lowest eigenvalue and the ground stateeigenfunction. I advance also new boundary conditions for obtaining moreprecise initial approximations. I demonstrate how one can optimize the PTcalculation with choosing better initial approximations and thus diminishingthe perturbative corrections. Dressing, Rebuilding, and Renormalizations arediscussed in Appendices 4 and 5.

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