Many years ago I studied the usual perturbation theory for a Sturm-Liouville problem, for practical applications. At that time we only had small calculators and big electronic machines with punched cards (computers like old IMB, but Soviet machines), so analytical formulas were quite useful for quick qualitative analysis and quantitative estimations.

We encountered divergent matrix elements and I was thinking of applying renormalization (I had just finished my studies at the University). However, I managed to reformulate the problem in better terms and obtained finite matrix elements from the very beginning. Also, I managed to construct another perturbative expansion, with even smaller terms because I figured out how to sum up exactly a part of the perturbative series into a finite function. The remaining series converged even faster.

In addition, I discovered an error in the perturbative treatment of the problem and managed to correctly derive the "matrix elements". The expansion parameter turned then from a logarithm to another function, which does not grow to infinity. I obtained some other results too.

I published a preprint and several papers in Russian, and recently I translated **some of it** in English and submitted to arXiv (sorry for my poor English).

It may be interesting and instructive for us physicists dealing with the perturbation theory.

I do not place my paper into the Review section in order to avoid downvotes from my haters.

Happy reading. Questions are welcome.