An introduction to semiclassical and microlocal analysis
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. \textit{J. Sjöstrand} [Astérisque 95, 1-166 (1982; Zbl 0524.35007)], \textit{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.
scientific article; zbMATH DE number 1716498