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Autour de l’approximation semi-classique. (Around semiclassical approximation). (French) Zbl 0621.35001
Progress in Mathematics, Vol. 68. Boston-Basel-Stuttgart: Birkhäuser. IX, 329 p.; DM 94.00 (1987).
This book is devoted to mathematical investigation of the quasi-classical limit of quantum mechanics. In Chapter I a short introduction into the principles of non-relativistic quantum theory is presented. In Chapter II ”h-admissible operators” calculus of scalar admissible h-pseudo- differential operators and of Fourier integral operators are developed. In Chapter III ”Functional calculus for h-admissible operators” a class of selfadjoint admissible h-pseudo-differential operators is introduced and it is proved that admissible functions of operators of this class are admissible h-pseudo-differential operators too; in particular resolvents and complex powers of operators are considered. Appendix 1 ”Essentially selfadjoint operators” and Appendix 2 ”Essential spectrum” contain certain classical functional-analytic results.
In Chapter IV ”Classical trajectories and quantum evolution” a theory of the non-stationary Schrödinger equation and of similar equations is developed and it is proved that frequency sets of solutions propagate along trajectories of the Hamilton-Jacobi equation. A climax of the book is Chapter V ”Quasi-classical properties of the discrete spectrum” where the main theorem of the book is proved. Namely, if H is an appropriate selfadjoint h-pseudo-differential operator and if (C) \(E_ 1\), \(E_ 2\) are not critical values of the quasi-principal symbol \(a_ 0\) of H then \[ (*)\quad N_ I(h)=c_ 0h^{-n}+O(h^{1-n})\quad as\quad h\to +0 \] where n is a dimension and \(c_ 0=(2\pi)^{-n}mes a_ 0^{-1}(I)\), \(I=(E_ 1,E_ 2)\) and \(N_ I\) is a number of eigenvalues of H lying in this interval. Moreover, a list of problems and complements and an extensive list of references is given.
Remark of the reviewer. This book had been written in 1983 and in 1986 only one page was added. Hence no contemporary result is contained. Let us list a few of them. Under certain conditions (*) was proved for matrix h-pseudo-differential operators in domains with boundary: if measures of the sets of all the periodic trajectories on the energy levels \(E_ 1\) and \(E_ 2\) are equal to 0 then \[ (**)\quad N_ I(h)=c_ 0h^{- n}+c_ 1h^{1-n}+o(h^{1-n})\quad as\quad h\to +0. \] On the other hand if all the trajectories are periodic with the same period and a condition to quasi-sub-principal symbol of H is fulfilled then there is a cluster asymptotics of \(N_ I(h)\). Moreover, asymptotics (*) and (**) was proved for the Schrödinger operator with \(n\geq 2\) even without condition (C) and even in the case when scalar and vector potentials have singularities of a certain type. Certain more advanced results was obtained, too. Moreover, Helffer-Sjöstrand and Simon theory of tunnelling and multiple wells and resonances was developed.
Reviewer: V.Ya.Ivrij

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35S05 Pseudodifferential operators as generalizations of partial differential operators
58J40 Pseudodifferential and Fourier integral operators on manifolds
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q15 Perturbation theories for operators and differential equations in quantum theory