Spectral theory and differential operators. (English) Zbl 0893.47004

Davies, E. B.

Cambridge Studies in Advanced Mathematics. 42. Cambridge: Cambridge Univ. Press. ix, 182 p. (1995).

This book is an introduction to the theory of partial differential operators and their spectral properties. It assumes that the reader has a basic knowledge of introductory functional analysis, up to the spectral theorem for bounded linear operators on Banach spaces. However, it also describes more advanced topics like distributions and their Fourier transforms as far as this is needed to analyze the spectrum of partial differential operators with constant coefficients. The spectral theorem for unbounded selfadjoint operators is followed by its application to a variety of second-order elliptic operators, from those with discrete spectra to Schrödinger operators acting on \(L^2\) spaces over the whole Euclidean space. The book contains a detailed account of the applications of variational methods to estimate the eigenvalues of operators with measurable coefficients defined by the use of quadratic form techniques.
The book is well written and understandable throughout. It contains not only many important results on spectra of differential operators in a form which is accessible to non-specialists, but also a number of interesting examples, exercises, and historical remarks. In the same way as the other two books by the same author on related topics [‘One-parameter semigroups’, Academic Press, London (1980; Zbl 0457.47030) and ‘Heat kernels and spectral theory’, Cambridge Univ. Press, Cambridge (1989; Zbl 0699.35006)], this book may be highly recommended to any student who is interested to get an idea of the fascinating interplay between functional analysis, operator theory, and mathematical physics.

Reviewer: J.Appell (Würzburg)

MSC:

47A10	Spectrum, resolvent
47-02	Research exposition (monographs, survey articles) pertaining to operator theory
47E05	General theory of ordinary differential operators
47F05	General theory of partial differential operators
34L05	General spectral theory of ordinary differential operators
35P05	General topics in linear spectral theory for PDEs
35J10	Schrödinger operator, Schrödinger equation

Keywords:

partial differential operators; spectral properties; spectral theorem; Fourier transforms; unbounded selfadjoint operators; second-order elliptic operators; Schrödinger operators; variational methods; operators with measurable coefficients; quadratic form techniques

Citations:

Zbl 0457.47030; Zbl 0699.35006