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Syllabus of Math 664, Section 601

Seminar in Applied Mathematics: Spectral Theory and its Applications, Fall 2014

Instructor Peter Kuchment

Office Rm. Blocker 614A, Telephone (979)862-3257

E-mail: kuchment AT math DOT tamu DOT edu, Home Page: /~kuchment

Introduction

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.

Tentative content

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.

Prerequisite

Real analysis with elements of Banach and Hilbert space theory and complex analysis in one variable, or instructor's consent.

Projects

A mid-term and a final take-home projects

Recommended literature

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:
  1. B. Helffer, Spectral Theory and its Applications, Cambridge Univ. Press, 2013. ISBN 978-1-107-03230-9 Available electronically at our library.
  2. 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:
  3. P. Hislop, I. Sigal, Introduction to Spectral Theory. With applications to Schrödinger operators, Springer 1996. ISBN 0-387-94501-6. On reserve
  4. 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:
  5. M. Naimark, Linear Differential Operators, George G.Harrap & Co Ltd; 1968. ISBN 978-0245592683. A classical source on ordinary differential operators. On reserve
  6. 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,
  7. F. Berezin, M. Shubin, The Schrödinger Equation, Springer 1991. ISBN-13: 978-0792312185. A very good text on the properties of Schrödinger equation.
  8. M. Schechter, Spectra of Partial Differential Operators, North-Holland 1987. ISBN 978-0444878229. Outdated, but still valuable beginner text.
  9. 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
  10. 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
  11. Akhiezer and I. Glazman, Theory of Linear Operators in Hilbert Space, 2 volumes in one. Dover 1993. Classics! The second volume covers Spectral Theory. On reserve
  12. 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
  13. 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:
  14. W. Arveson, A Short Course on Spectral Theory, Springer 2001. ISBN-13: 978-0387953007. A graduate textbook.
  15. 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:
  16. 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!
  17. E. B. Davies, Linear Operators and their Spectra, Cambridge Univ. Press 2007. ISBN-13: 978-0521866293
  18. 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:
  19. Y. Colin de Verdiere, Spectres de Graphes, Soc. Math. France 1998. ISBN 978-2856290682. A unique little book (in French)
  20. Fan Chung, Spectral Graph Theory, Amer. Math. Soc. 1996. ISBN 978-1178509793. A great book! The approach comes from PDEs and geometric analysis.
  21. A. Brouwer, W. Haemers, Spectra og Graphs, Springer 2011. Universitext. ISBN 978-1461419389
  22. 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 accounts daily).
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.

Make-up policy:

Make-ups for missed quizzes, home assignments and exams will only be allowed for a university approved excuse in writing. Wherever possible, students should inform the instructor before an exam or quiz is missed. Consistent with University Student Rules , students are required to notify an instructor by the end of the next working day after missing an exam or quiz. If there are confirmed circumstances that do not allow this (a written confirmation is required), the student has two working days to notify the instructor. Otherwise, they forfeit their rights to a make-up.

Late work

Late work will not be accepted, unless there is an university approved excuse in writing. In the latter case student has a week to submit the work.

Grade complaints:

Sometimes the instructor might make a mistake grading your work. If you feel that this has happened, you have one week since the graded work was handed back to you to talk to the instructor. If a mistake is confirmed, the grade will be changed. No complaints after that deadline will be considered.

Students with Disabilities:

The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe you have a disability requiring an accommodation, please contact Services for Students with Disabilities (Cain Hall, Room B118, or call 845-1637).

Copyright policy:

All printed materials disseminated in class or on the web are protected by Copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.

Scholastic dishonesty:

Copying work done by others, either in class or out of class, looking on other student?s papers during exams or quizzes, having possession of unapproved information in your calculator/computer/phone, etc., and/or having someone else do your work for you are all acts of scholastic dishonesty. These acts, and other acts that can be classified as scholastic dishonesty, will be prosecuted to the full extent allowed by University policy. In this class, collaboration on graded assignments, either in class or out of class, is forbidden unless permission to do so is granted by the instructor. For more information on university policy regarding scholastic dishonesty, see University Student Rules at http://studentrules.tamu.edu/.
"An Aggie does not lie, cheat, steal, or tolerate those who do." Visit http://www.tamu.edu/aggiehonor and follow the rules of the Aggie Honor Code.

GOOD LUCK IN YOUR STUDIES!!!

This syllabus is subject to change at the instructor's discretion

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Last revised August 18th, 2014