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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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up vote 4 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

A full course on Sturm-Liouville is worthy of being given in my humble opinion. As already mentioned, the Sturm-Liouville problem is connected with many problems from pure and applied mathematics. For examples,

  • PDE, boundary problems, notion of Green function (as mentioned above)
  • some special functions and orthogonal polynomials
  • problems with ordinary differential equations (see the relevant chapters in Coddington-Levinson or in Whittaker): existence, uniqueness, explosion
  • problems with eigenvalues, their asymptotics, introduction to the theory of self-adjoint operators, WKB approximations, ...
  • the Weyl-Titchmarsh theory and its relation with complex analysis and Green's functions
  • numerical computations of the eigenvalue problems or of the solution (shooting methods, ...), as a way to introduce numerical analysis.
  • parabolic equations and links between semi-groups and resolvent (see e.g. ``The abstract Cauchy problem and Cauchy's problem for parabolic differential equations'' by E. Hille)
  • Feller theory of one-dimensional stochastic processes is also strongly related to SL problems
  • the Krein theory of the string (already mentioned), harmonic analysis, ...
  • and of course the many applications of the SL equation in physics and other... (on the history, see ``Sturm and Liouville's work on ordinary linear differential equations. The emergence of Sturm-Liouville theory'' by Lützen).

Actually, problems related to the SL equations are simple to write, because Sturm-Liouville operators only involve differential operators), yet their resolutions involve a great deal of ideas that have been generalized and extended in many directions. For me, dealing with SL problems could be a "smooth way" to learn and understand key ideas without relying on abstract settings. For example, I was recently very happy to read the elegant proof of Weyl on the limit circle and point circle, and to have a glimpse of its relations with the "resolution of identity", as it really enlightens it.

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You could also include the symplectic interpretation of Sturm–Liouville theory (via of the Maslov index) given in Arnold's paper The Sturm theorems and symplectic geometry. This may be of interest to students who are more geometrically/topologically inclined.

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I would recommend the book of E-C. Titchmarsh "Eigenfunction Expansions Associated with Second-order Differential Equations".

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