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Théorie spectrale pour des opérateurs globalement elliptiques. (French) Zbl 0541.35002
Astérisque, 112. Publié avec le concours du Centre National de la Recherche Scientifique. Paris: Société Mathématique de France. IX, 197 p. FF 115.00; $ 15.00 (1984).
The purpose of the present book is to develop the spectral theory of globally elliptic and strictly positive differential (or pseudo- differential) operators on \(R^ n\). A typical example of the operators, which are applied to the theory of this book, is the harmonic oscillator \(-\Delta +\| x\|^ 2\) on \(R^ n\). The main tools are pseudo- differential operators and Fourier integral operators on \(R^ n\). To make the book self-contained the author gives definitions and fundamental formulas of these operators.
Chapter I is devoted to develop the theory of pseudo-differential operators on \(R^ n\). Definition of operators, symbolic calculus and functional calculus of globally elliptic pseudo-differential operators, which is needed to study the spectral theory, are stated in this chapter. The development of the theory in this chapter is mainly due to M. M. Shubin. In Chapter II the author gives the definition and fundamental formulas of Fourier integral operators on \(R^ n\). He also gives some important results in the theory of Fourier integral operators (for example, Egorov’s theorem in § 2.6) to the class treated in this chapter. In Chapter III the author constructs the fundamental solution to the Cauchy problem. The construction is done by approximating it by the Fourier integral operators introduced in Chapter II. In the final Chapter IV, the author gives the asymptotic behavior of the eigenvalues for the differential operators which are globally strongly elliptic and formally self-adjoint on \(R^ n\). He also gives some results on the spectral theory to these operators. The development of the theory in Chapters II, III and IV are mainly due to the author and D. Robert.
Reviewer: M.Nagase

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35P20 Asymptotic distributions of eigenvalues in context of PDEs
47Gxx Integral, integro-differential, and pseudodifferential operators
35S05 Pseudodifferential operators as generalizations of partial differential operators
35A08 Fundamental solutions to PDEs
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type