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

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

**Section:**601**Time:**TR 2:20 -3:35 pm**Room:**ZACH 119D**Office hours:****MW 1 - 2 pm, no appointment is needed**.

**Additional office hours**can be**arranged by appointment**.

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.

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

- 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.
beginner text.__Outdated, but still valuable__ - E. Titchmarsh, Elgenfunction Expansions Associated With Second Order Differential Equations.
Nabu Press 2011. This is a
reprint of the 1923 edition. ISBN 978-1178509793.__still valuable__

There are quite a few other good books,. Among them I would mention__at least partially devoted to the topic__ - T. Kato, Perturbation Theory for Linear Operators, Springer 2013 (reprint of
the 2nd edition of 1980). ISBN 978-1178509793.
. Besides the perturbation theory has a lot of operator and spectral theory.__Immortal classics!__**On reserve** - Akhiezer and I. Glazman, Theory of Linear Operators in Hilbert Space, 2 volumes in one.
Dover 1993.
The second volume covers Spectral Theory.__Classics!__**On reserve** - 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
containing functional analysis and elements of spectral theory.__classical very good book__**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, see the following three great books:__spectral theory of non-selfadjoint operators__ - 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. ISBN-13: 978-0521866293
- 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.
(in French)__A unique little book__ - Fan Chung, Spectral Graph Theory, Amer. Math. Soc. 1996. ISBN 978-1178509793.
The approach comes from PDEs and geometric analysis.__A great book!__ - 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.

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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