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Syllabus of Math 664, Section 601
Seminar in Applied Mathematics: Spectral Theory and its Applications,
Instructor Peter Kuchment
Office Rm. Blocker 614A,
kuchment AT math DOT tamu DOT edu,
Home Page: /~kuchment
Many topics of mathematical physics, PDEs, numerical analysis, applied
mathematics, ergodic theory, complex analysis, and other areas (and lately even
graph theory and discrete groups) require a detailed knowledge of spectral
theory of bounded and unbounded self-adjoint (and sometimes non-self-adjoint)
operators. This need goes far beyond the standard minimum usually provided
(e.g., analytic functional calculus) and requires detailed study of the
structure of spectra (e.g., absolute continuous, pure point, singular continuous)
and more advanced topics such as limiting absorption principle and eigenfunction behavior.
Spectral theory of bounded operators and analytic functional calculus. Spectral
theory of bounded and unbounded self-adjoint operators (non-self-adjoint operators
might also be touched upon). Classification of types of spectra and behavior
of eigenfunctions. Generalized eigenfunctions. Detailed examples of difference
(e.g., graph Laplacians) and differential (e.g., Dirichlet and Neumann Laplacian,
Schroedinger) operators. Elements of perturbation theory.
Real analysis with elements of Banach and Hilbert space theory and complex
analysis in one variable, or instructor's consent.
A mid-term and a final take-home projects
Although a variety of sources will be used (including the instructor's notes),
the following two nice small books will be the closest to our class:
- B. Helffer, Spectral Theory and its Applications, Cambridge Univ. Press, 2013.
ISBN 978-1-107-03230-9 Available electronically at our library.
- E. B. Davies, Spectral Theory and Differential Operators, Cambridge Univ. Press
1995. ISBN 0-521-47250-4. On reserve
Some more suggestions for your (possibly future) use.
Here are some more comprehensive books on the subject of spectral theory:
- P. Hislop, I. Sigal, Introduction to Spectral Theory. With applications to Schrödinger operators,
Springer 1996. ISBN 0-387-94501-6. On reserve
- M. Birman, M. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Spaces, Reidel Publ.1987. ISBN 90-277-2179-3
Here are some books devoted specifically to the spectral theory of differential operators:
- M. Naimark, Linear Differential Operators, George G.Harrap & Co Ltd; 1968.
ISBN 978-0245592683. A classical source on ordinary differential operators. On reserve
- Cycon et al., Schrödinger Operators: With Applications to Quantum Mechanics
and Global Geometry, Springer 2007. ISBN 978-3540167587.
A rather comprehensive source on Schrödinger operators,
- F. Berezin, M. Shubin, The Schrödinger Equation, Springer 1991. ISBN-13: 978-0792312185.
A very good text on the properties of Schrödinger equation.
- M. Schechter, Spectra of Partial Differential Operators, North-Holland 1987. ISBN
978-0444878229. Outdated, but still valuable beginner text.
- E. Titchmarsh, Elgenfunction Expansions Associated With Second Order Differential Equations.
Nabu Press 2011. This is a still valuable reprint of the 1923 edition. ISBN 978-1178509793.
There are quite a few other good books, at least partially devoted to the topic.
Among them I would mention
- T. Kato, Perturbation Theory for Linear Operators, Springer 2013 (reprint of
the 2nd edition of 1980). ISBN 978-1178509793. Immortal classics!. Besides the perturbation
theory has a lot of operator and spectral theory. On reserve
- Akhiezer and I. Glazman, Theory of Linear Operators in Hilbert Space, 2 volumes in one.
Dover 1993. Classics! The second volume covers Spectral Theory.
- M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis.
Revised and enlarged edition (although older editions are also very good), Acad. Press, 1980.
This is a classical very good book containing functional analysis
and elements of spectral theory. On reserve
- P. Lax, Functional Analysis, Wiley-Interscience 2002. On reserve
The two books below (as well as the Lax's book above), treat the spectral theory from the very useful angle of C* algebras and Gelfand-Neimark approach:
- W. Arveson, A Short Course on Spectral Theory, Springer 2001. ISBN-13: 978-0387953007. A graduate textbook.
- N. Bourbaki, Théories spectrales: Chapitres 1 et 2. Springer 1967, reprint 2006.
ISBN-13: 978-3540353300. A great little book (if you can read French or Russian, or able to find an English edition (then let me know)).
For the spectral theory of non-selfadjoint operators, see the following three great books:
- I. Gohberg and M. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space,
Amer. Math. Soc. 1969. ISBN-13: 978-0821815687. Classics!
- E. B. Davies, Linear Operators and their Spectra, Cambridge Univ. Press 2007.
- Lloyd N. Trefethen, Mark Embree, Spectra and Pseudospectra: The Behavior of
Nonnormal Matrices and Operators, Princeton Univ. Press 2005. ISBN-13: 978-0691119465.
With numerical discussions.
Some books on spectral graph theory:
- Y. Colin de Verdiere, Spectres de Graphes, Soc. Math. France 1998.
ISBN 978-2856290682. A unique little book (in French)
- Fan Chung, Spectral Graph Theory, Amer. Math. Soc. 1996. ISBN 978-1178509793. A great book! The
approach comes from PDEs and geometric analysis.
- A. Brouwer, W. Haemers, Spectra og Graphs, Springer 2011. Universitext. ISBN 978-1461419389
- G. Berkolaiko, P. Kuchment, Introduction to Quantum Graphs, Amer. Math. Soc.
2012. ISBN 978-0-8218-9211-4. The spectral theory of quantum graphs is addressed.
Class Announcements, E-Mail Policy and Communications:
Class announcements will be posted on my homepage and in most important cases
e-mailed to your NEO accounts (it is your responsibility to check your NEO
E-mail (kuchment AT math DOT tamu DOT edu) is the preferred way of contacting
me. When writing to me, please include your full name and "Math 220".
Use your NEO e-mail account to send me e-mails.
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This syllabus is subject to change at the
Last revised August 18th, 2014