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Spectra of Random and Almost-Periodic Operators (Paperback, Softcover reprint of the original 1st ed. 1992)
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Spectra of Random and Almost-Periodic Operators (Paperback, Softcover reprint of the original 1st ed. 1992)
Series: Grundlehren der mathematischen Wissenschaften, 297
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In the last fifteen years the spectral properties of the
Schrodinger equation and of other differential and
finite-difference operators with random and almost-periodic
coefficients have attracted considerable and ever increasing
interest. This is so not only because of the subject's position at
the in tersection of operator spectral theory, probability theory
and mathematical physics, but also because of its importance to
theoretical physics, and par ticularly to the theory of disordered
condensed systems. It was the requirements of this theory that
motivated the initial study of differential operators with random
coefficients in the fifties and sixties, by the physicists
Anderson, 1. Lifshitz and Mott; and today the same theory still
exerts a strong influence on the discipline into which this study
has evolved, and which will occupy us here. The theory of
disordered condensed systems tries to describe, in the so-called
one-particle approximation, the properties of condensed media whose
atomic structure exhibits no long-range order. Examples of such
media are crystals with chaotically distributed impurities,
amorphous substances, biopolymers, and so on. It is natural to
describe the location of atoms and other characteristics of such
media probabilistically, in such a way that the characteristics of
a region do not depend on the region's position, and the
characteristics of regions far apart are correlated only very
weakly. An appropriate model for such a medium is a homogeneous and
ergodic, that is, metrically transitive, random field."
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