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Spectra of partial differential operators. (2nd ed.). (English) Zbl 0607.35005
North-Holland Series in Applied Mathematics and Mechanics, Vol. 14. Amsterdam - New York - Oxford: North-Holland. XIII, 310 p. $ 49.00; Dfl. 160.00 (1986).
It is a second edition, after fifteen years, of the previous book [Zbl 0225.35001]. Chapters 1, 2, 3 contain various topics from functional analysis, introduce classes of function spaces and collects some results of the theory of partial differential equations which are needed in the book. Most proofs of theorems are omitted or given in the chapter 11 which contains a much added number of references, background material and discussions of related works. The chapters 4-10 are addressed to the problem of the description of the spectrum of a linear partial differential operator \(P(D)+Q(.,D)\) in \(L^ p({\mathbb{R}}^ n)\), \(1\leq p\leq \infty\), describing the effect on the spectrum of operator with constant coefficients P(D) which is produced by the addition of the operator with variable coefficients Q(.,D) and founding the conditions which insure that the spectrum of P(D) is not ”appreciably” disturbed from the perturbation Q(.,D). Chapter 4 deals with the general theory of the constant coefficients operators, while relative compactness is studied in chapter 5. In the present edition two sections are added describing the spectra of operators in \(L^ 1\) and \(L^{\infty}\) and studying the essential spectrum. Elliptic operators are considered in chapter 6, which here is rewritten to obtain much larger conditions on a class of functions V, for multiplication by V to be a continuous or a compact operator from \(H^{s,p}\) to \(L^ q\) and better conditions on V which guarantee that \(P(D)+V\) has s-extension with the same essential spectrum as P(D). Chapter 7, which has some theorems strengthened, treats the \(L^ 2\) theory for operators bounded from below and chapter 8 selfadjoint operators; two new sections show ways of extimating negative eigenvalues and obtaining bounds for integral operators. Chapter 9 gives the theory for second order operators and in it is replaced the theorem on essential self-adjointness by recent stronger results of Kato. In chapter 10 are given some applications to quantum mechanical systems of particles.
Reviewer: G.Bottaro

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35P05 General topics in linear spectral theory for PDEs
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35E20 General theory of PDEs and systems of PDEs with constant coefficients
35B20 Perturbations in context of PDEs