I am analyzing the sign of the eigenvalues of a singular Sturm-Liouville problem, with the function coefficient of the second order term equal to zero at the edges of the interval. The problem arises in a linear stability analysis of self similar solution of a diffusive-convective differential problem. A theorem states that for a regular SL problem the sign of the eigenvalues in some conditions is related to the sign of the coefficient functions in the interval. Is any theorem or lemma that can be used to justify for a singluar SL problem the same conclusions that can be drawn for a regular SL problem?
Are you interested in signs or signature of the eigenvalues?
If the question concerns signs of the eigenvalues than I am sure the results for many types of singular SL problems are known, although I am not an expert in the field, I guess google is the right advice.
If the question is about the signature of the eigenvalues, then check the review paper by Paul Binding (SIAM Review 1996) or more recent work by Trunk and Berhndt (multiple papers). Behrndt and Trunk are also the best experts on the topic (signature for SL problems).
Google seems to give quite a large amount of references:
singular Sturm-Liouville problem eigenvalues
Are you interested in signs or signature of the eigenvalues?
If the question concerns signs of the eigenvalues than I am sure the results for many types of singular SL problems are known, although I am not an expert in the field, I guess google is the right advice.
If the question is about the signature of the eigenvalues, then check the review paper by Paul Binding (SIAM Review 1996) or more recent work by Trunk and Berhndt (multiple papers). Behrndt and Trunk are also the best experts on the topic (signature for SL problems).
I am interested in the sign of eigenvalues. I assume that something has been published for singular SL problem, but I could not find it on the web or in other dataset.
Dear researcher
unfortunately i haven't work the subject of your question, My researchs are in field of the numerical methods to solving PDE based on Sinc method.
I have attached a file with a more detailed description of the problem I am facing.
Did you give a look to the book: Theory of Ordinary Differential Equations by Coddington & Levinson (every mathematics library should have a copy of it, it's a classic), where singular SL problems are treated? Just a warning: the book deals explicitly only with the cases where the singularity is due to the unboundedness of the interval, but it clearly says that the same technique can be employed to treat the case of a finite interval at whose endpoints the coefficients show some singular behavior. Roughly speaking, if your coefficient vanishes at the endpoints a,b of the interval, you just have to consider the same SL problem in [c,d] with a
Thanks for the answer. In fact the problem is the proof of the existence of the eigenvalues and/or the completeness of the correspondent set of eigenfunctions. The sign of the eigenvalues could be determined according to the equations added in my attachment, still valid also for the case of vanishing higher order coefficient P at the edges.
You may see the following book "E. C. Titchmarsh, ‘‘Eigenfunction Expansions, Part I,’’ Oxford Univ. Press, London, 1962."
@Sandro Longo. In Coddington & Levinson's book the existence of eigenvalues and the completeness of eigenvectors in the singular case is indeed treated. There is also a paragraph on singular SL eigenvalue problems on bounded intervals in the book A First Course In partial Differential Equations by Weinberger (chapter VII section 39) where completeness is treated. If you haven't yet looked at it this might be the first book to browse (it's another very classical book that should be available in any mathematics library).
Many thanks to Dr Papini for the useful suggestions. A First Course In partial Differential Equations by Weinberger gives many of the answers
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