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In a Sturm Liouville problem, how can we predict that it only has simple eigenvalues or it contains multiple eigenvalues? Do the boundary conditions such as 1) zero Dirichlet 2) periodic 3) semi-periodic matter? Do regularity/singularity matter?

Also, is there a general approach that able us to approximate the eigenvalues one by one in ascending order?

Hosein Javanmardi
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    If the problem is regular on a finite interval, then separated endpoint conditions lead to one dimensional eigenspaces. Periodic conditions are not necessarily like that, which you know from the simplest case where sin and cos are solutions – Disintegrating By Parts Sep 20 '22 at 12:24
  • @DisintegratingByParts Thank you. Can you also introduce me a good reference regarding Sturm Liouville problems that provides these helpful theorems and their proofs, including one which is about numerical estimation of eigenvalues. I am an engineer and have just started this subject. many engineering references such as Kreyszig is not thorough. I should also point that my particular case can be reduced to a nonlinear eigenvalue problem $A(\lambda)*X=0$ which is itself another complicated problem. I would be glad if you share your experience. – Hosein Javanmardi Sep 20 '22 at 13:11
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    Have you looked at Zettl's book? Also, search Sturm -Liouville on Amazon. What is your problem $A(\lambda) * X= 0$? That's a little vague to me. – Disintegrating By Parts Sep 20 '22 at 13:49
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    I must read the book, although it seems difficult for engineers. In the problem I'm solving, the coefficient functions, $q(x)=0$, and $p(x)$ and $w(x)$ are piecewise constant. This allows me to solve it in separate subintervals and impose boundary conditions which can be written in matrix form. $A(\lambda)$ is a matrix whose elements depend on eigenvalues and $X$ is the eigenvectors. I am trying to solve it. If you have time and are interested, we can discuss it however you like. Thank you. – Hosein Javanmardi Sep 20 '22 at 17:59
  • I started with the classical book from E. C. Titchmarsh, which uses Advanced Calculus and some basic Complex Analysis such as Cauchy's Theorem in order to analyze and justify eigenfunction expansions including discrete and Fourier integral types of expansions. Titchmarsh was a master, and one of G. H. Hardy's favorite students. Titchmarsh's style is terse, but it's well worth learning his approach in order to be able to stick to basics instead of invoking so much abstract machinery. Probably every book written on the subject references Titchmarsh; if they don't, they're not worth reading. :) – Disintegrating By Parts Sep 20 '22 at 21:28
  • Here's an online copy from archive.org, which is legitimate. This book is from 1946. It's one of the best Math books I've ever read. https://archive.org/details/elgenfunctionexp032099mbp/mode/2up – Disintegrating By Parts Sep 20 '22 at 22:01
  • Thanks. for now, the most thing I need is the estimation of all eigenvalues. Is there any part of the book devoted for that? I couldn't find. – Hosein Javanmardi Sep 21 '22 at 04:59
  • Titchmarsh deals with asymptotics for eigenvalues. If you check the link I sent you, take a look at Chapter VII : The Distribution of Eigenvalues. The link is a safe link at archive.org. They host the wayback machine where they have been gathering snapshots for all domains. They're a legitimate non-profit organization in the US with funding to preserve historical records of the development of the internet. You can select any domain and look at various snapshots of that domain. And they offer legitimate copies of old books, etc.. – Disintegrating By Parts Sep 21 '22 at 05:59
  • About archive.org : https://archive.org/about/ – Disintegrating By Parts Sep 21 '22 at 06:03
  • Yes I saved a copy. thanks. I have another expert question: https://math.stackexchange.com/q/4535810/851785 , I would appreciate if you can help. – Hosein Javanmardi Sep 21 '22 at 06:20

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