##
**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.

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)

### MSC:

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 |