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Dear Math community,

               I have some general questions on Sturm-Liouville theory. We are planning to introduce a graduate course on Sturm-Liouville theory and every one has been asked to propose topics which might be suitable for the course. 

I would like to know the following.

  1. Is it worth to have a course exclusively on just Sturm-Liouville theory?

  2. If we have a course like that what are the topics that could be introduced in the course?

  3. Would it be possible for an exclusive course on sturm-liouville theory without much background in functional analysis.

Any suggestions would be really appreciated.

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2 Answers 2

up vote 1 down vote accepted

A more advanced/comprehensive course can be based on Atkinson's book Discrete and continuous boundary problems. No functional analysis is required, neither for Hilbert-Courant nor for Atkinson. (When Courant wrote the first volume of HC, functional analysis did not exist yet:-)

Another book which studies some of these questions in depth (and without any functional analysis) is Gantmakher-Krein, Oscillation matrices and kernels. (Gantmaher, Gantmacher...)

Is it worth offering such a course? The answer depends on whether you can find enough students who will enroll. This depends on your local conditions. Of course this is a beautiful and useful subject and worth learning.

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Thanks for your comments and suggestions. I do have one more question. I think the students who might be taking this course are more oriented toward algebra. Is it possible to have a little bit of an algebraic flavor to this course? –  user8974 Apr 12 '13 at 3:00
    
Yes, and Atkinson's book is good for this. It has more algebraic flavor on my opinion, because he considers discrete and continuous problems together. Same applies to Krein-Gantmakher book. –  Alexandre Eremenko Apr 12 '13 at 12:38

The classic "Methods of Mathematical Physics" by Courant and Hilbert has a wealth of material on the Sturm-Liouville problem and its connections to various themes of mathematical physics. In fact, one could almost say that the treatise is constructed around this topic. Little or no knowledge of Functional Analysis is required. It is, of course, not the most up to date version but can be recommended as a first introduction--- it contains a multitude of explicit examples (Green functions, eigenfunction expansions, connections with the classical pde's of mathematical physics via separation of variables, special functions etc.) and is a joy to read.

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I know that the students who will be taking this class are more oriented toward algebra. Is there any possibility of adding a bit of algebraic flavor to the course? –  user8974 Apr 11 '13 at 18:24

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