An introduction to semiclassical and microlocal analysis. (English) Zbl 0994.35003

Universitext. New York, NY: Springer. viii, 190 p. (2002).
The contents of the book correspond to a course at Ph. D. level, given by the author at the Universities of Bologna and Paris-Nord. The subject is semiclassical analysis, but the book can be read as well as introduction to standard microlocal analysis. Namely, the book treats operators of the form \[ Op_h(a)u(x;h)= (2\pi h)^{-n}\int e^{i(x-y)\xi/h} a(x,y,\xi) u(y)dy d \xi \] depending on the parameter \(h\to 0\), relevant examples being semi-classical differential operators \[ \sum_{|\alpha|\leq m}b_\alpha(x) (hD_x)^\alpha, \] in particular the celebrated Schrödinger operator \(-(h^2/2m) \Delta+V(x)\). The corresponding asymptotic properties, principally spectral properties as \(h\to 0\), allow to prove mathematically some typical phenomena in quantum mechanics. Roughly, this is semi-classical analysis, whereas taking \(h=1\) in the previous expressions, we have standard pseudo-differential operators and related concepts, i.e. microlocal analysis.
Moving in this twofold frame, the presentation has a pedagogical character, specific contents being the following: semilinear pseudo-differential calculus, microlocalization, applications to the solutions of analytic PDE’s, symplectic aspects. As useful appendix, the book presents a list of formulae.
Peculiarity of the exposition is a careful treatment of the so-called FBI transform, with applications to the microlocal exponential estimates, cf. J. Sjöstrand [Astérisque 95, 1-166 (1982; Zbl 0524.35007)], A. Martinez [in: Microlocal Analysis and Spectral Theory, NATO ASI Ser., C, Math. Phys. Sci. 490, 349-376 (1997; Zbl 0890.35120)].
Summing up, the book collects in an original way standard results and new aspects of semiclassical microlocal analysis; the reading is suggested to non-specialists as well.
Reviewer: L.Rodino (Torino)


35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35S05 Pseudodifferential operators as generalizations of partial differential operators
58J40 Pseudodifferential and Fourier integral operators on manifolds
35Q40 PDEs in connection with quantum mechanics