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Spectral asymptotics in the semi-classical limit. (English) Zbl 0926.35002
London Mathematical Society Lecture Note Series. 268. Cambridge: Cambridge University Press. xi, 227 p. (1999).
This book is a very nice, complete, up-to-date and quick introduction to the rich field of semiclassical analysis, that is to say, the study of the relation between “quantum observables” and “classical” ones when the Planck constant tends to zero. Namely, one tries to understand properties of the evolution (time-dependent) Schrödinger equation \(i\hbar{{\partial\psi}\over{\partial t}}=(- {{\hbar^2}\over{2}} \Delta+V(x))\psi\) (where \(\Delta\) is the Laplacian in \({\mathbb R}^n\) and \(V:{\mathbb R}^n\rightarrow{\mathbb R}\) is a potential), or of the “eigenvalue equation” \((-{{\hbar^2}\over{2}}\Delta+V(x))\psi =E\psi\), in terms of the classical properties of the Hamiltonian function \(H(x,\xi)=| \xi| ^2/2+V(x)\), as \(\hbar\to 0+\). This kind of problems is dealt with by means of WKB methods (the search of “asymptotic” solutions, i.e. formal series in \(\hbar)\) and microlocal analysis, namely the calculus of \(\hbar\)-pseudodifferential operators along with their functional calculus, and the calculus of \(\hbar\)-Fourier integral operators.
The authors start by giving background in local symplectic geometry, WKB expansions, theory of selfadjoint operators, and then develop the aforementioned calculi. They then use these tools to study spectral asymptotics in the cases of non-critical Hamiltonians, of periodic trajectories that form a set of measure zero, of perturbed periodic problems, and of operators with periodic bicharacteristics.
This book should be recommended to everyone, be it student or researcher, who is interested in semiclassical analysis, or is willing to do research in the field.

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35S30 Fourier integral operators applied to PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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