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Spectral Theory and its Applications

Bernard Helffer
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Studies in Advanced Mathematics 139
[Reviewed by
Fernando Q. Gouvêa
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The Studies in Advanced Mathematics series from Cambridge University Press includes a very wide range of books. Some will serve as first-year graduate textbooks, others are quite advanced. Spectral Theory and its Applications weighs in at the “quite advanced” level.

A more accurate title for the book would have been “Spectral Theory of Unbounded Operators,” because the author assumes his readers already know the standard theory for bounded operators, which is reviewed briefly in chapter 6. Also assumed is knowledge of Fourier Analysis, \(L^2\) spaces, Sobolev spaces, Distribution Theory, and so on. The applications come from mathematical physics, with Quantum Mechanics taking pride of place. Here as well the author assumes the reader arrives with background knowledge. For example, in the first chapter the author talks about “the operator of the harmonic oscillator” \( H = - \dfrac{d^2}{dx^2} + x^2\), with the reader being assumed to know how this operator relates to the physical harmonic oscillator.

Another aspect of the book’s advanced nature is that the author is occasionally casual about things that a beginner might have trouble with. Notes at the ends of chapters sometimes say things like “the material in this chapter is very basic,” which might be dispiriting. Equalities are sometimes followed by “in the sense of distributions.” Containment relations between various spaces are assumed known and used without comment.

For the reader with the right background, the book’s main virtue is that it builds the theory of unbounded operators and their spectra with physics constantly in view. Each theoretical section is followed by examples from physics, often involving extensive calculation. Graduate students and professionals interested in how spectral theory can be used in mathematical physics will learn a lot.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.

1. Introduction
2. Unbounded operators
3. Representation theorems
4. Semibounded operators
5. Compact operators
6. Spectral theory for bounded operators
7. Applications in physics and PDE
8. Spectrum for self-adjoint operators
9. Essentially self-adjoint operators
10. Discrete spectrum, essential spectrum
11. The max-min principle
12. An application to fluid mechanics
13. Pseudospectra
14. Applications for 1D-models
15. Applications in kinetic theory
16. Problems