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  Some mathematical results for a Sturm-Liouville problem

+ 3 like - 0 dislike

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.

asked Sep 11, 2014 in Chat by Vladimir Kalitvianski (102 points) [ revision history ]
edited Mar 27, 2015 by dimension10

Isn't it unfair for anyone to have a negative reputation? That is one bad thing about this site; the stackexchange websites do not allow negative reputation - a minimum reputation of 1 is required.

This paper will not get frivolous downvotes in the reviews section, I guarantee it. You don't have "haters". There are people who downvote you, but they only do so when you make incendiary comments regarding other people's methods, or else make statements which seem to contradict their prior knowledge. In this case, it is just interesting mathematical physics, you're in no danger. I am asking you to formally submit, it will make your reputation go up.

Negative rep does not silence the author. It seems like a slur or something, but it isn't, it just means you currently have more downvotes than upvotes. It doesn't influence your ability to submit papers, answer questions, although it can restrict your ability to comment freely and vote. Making sure that rep can be positive or negative is important to us for preventing uninformed voting or rigging of reviews by noncontributing political members (we require minimum rep to vote and do edits, and things like this that are sensitive). But it's totally irrelevant in this case, because on the strength of his analytic skills, Vladimir Kalitvianski will for sure move to positive rep soon enough anyway, the negative stuff is a consequence of a bunch of incendiary stuff regarding QED which led to a lot of controversy and backlash. The non-QED material is valuable and presumably entirely correct and important (I only skimmed it just now, it looks really nice).

@SanathDevalapurkar Why is it unfair? Negative reputation doesn't really affect a user's ability to ask, answer or comment. SE secretly imposes asking bans and answering bans on negative rep-users (although their rep is displayed as positive), and commenting is restricted there to users with more than 50 rep.

I agree with Ron and Dimension10 concerning the negative rep. In my opinion, what SE does by keeping the rep at 1 regardless how many negatively voted contributions a user has delivered, is introducing an unjustified positive bias in favor of people who contribute mostly negative...

BTW I think this is a rather nice paper too as far as I can tell, and it would certainly earn some upvotes and a positive review in the Reviews section.

OK, I am submitting it there.

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