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Summary:

I would like to share some of my results on the perturbation theory for a Sturm-Liouville problem.

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 series terms because I figured out how to sum up exactly a part of the 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" (or the spectral analogues of matrix elements). The expansion parameter turned then from a logarithm into another function, which does not grow to infinity. I proposed a "harmonization" of the boundary conditions influencing the initial approximation fro the sake of better convergence. 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. Apart from possible practical applications, my study shows the importance of choosing the initial approximation for the convergence of the remaining series and some danger I noticed if one decides to resolve the divergence difficulties with "renormalizations".