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Spectra of random and almost-periodic operators. (English) Zbl 0752.47002
Grundlehren der Mathematischen Wissenschaften. 297. Berlin etc.: Springer-Verlag. viii, 587 p. (1992).
This book is a comprehensive modern account, from the Soviet School, of Spectral theory for metrically transitive operators. Important examples arise in theories of amorphous media with randomly distributed inpurities and quasi-crystals. Starting from rather general considerations of basic theory of metrically transitive operators, the book develops an increasingly detailed treatment of available results as more is assumed about the potentials and culminates in a treatment of quasi-periodic operators where the most detailed results are available. The methodology is a subtle amalgam of ideas from classical probability and potential theory with deep considerations of the origins of problems in theoretical physics. This account goes beyond the confines of results for which complete proofs are known, and includes consideration of ideas and results which are expected to hold in circumstances where proofs are not so far available. In some cases known proofs are omitted or reduced to a discussion of the principles involved to avoid cumbersome technical development and to facilitate a comprehensive treatment of all the main themes in this area. The result is a book with a vigorous style which as light as might be expected of one with hard mathematical analysis at its core. The contents are arranged in the following chapters:
I. Metrically transitive operators.
II. Asymptotic properties of metrically transitive matrix and differential operators.
III. Integrated density of states in one dimensional problems of second order.
IV. Asymptotic behaviour of the integrated density of states at spectral boundaries of one dimensional operators.
V. Lyapunov exponents and the spectrum in one dimension.
VI. Random operators.
VII. Almost-periodic operators.
Appendix A. Nevanlinna functions.
Appendix B. Distribution of eigenvalues of large random matrices.
The authors warn that those wishing to digest their material and gain further insight will require sympathy towards non-rigorous, but often deep statements and a considerable and unusual effort in reading related physics literature. The reward however will be great. They give a list of important open questions and avenues for further research. Basic questions, such as whether pure point spectra and absolutely continuous spectra can intersect in many important situations seem to be open. (If they do not intersect what can be said about the eigenfunctions for points on the boundary between them?)
To a non-expert the writing is clear and the material is presented in a vital and stimulating style. This is both a book for reference (though the index is poor) and one for private study.
Reviewer: J.F.Toland (Bath)

47-02 Research exposition (monographs, survey articles) pertaining to operator theory
35J10 Schrödinger operator, Schrödinger equation
47N20 Applications of operator theory to differential and integral equations
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)