Autour de l'approximation semi-classique. (Around semiclassical approximation)
type: Item
instance of: scholarly article
title: Autour de l'approximation semi-classique. (Around semiclassical approximation)
zbMATH Open document ID: 0621.35001
published in: Progress in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: 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 boundar…
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 81Q15
zbMATH DE Number: 4006772
zbMATH Keywords: frequency set
zbMATH Keywords: quasi-classical limit
zbMATH Keywords: quantum mechanics
zbMATH Keywords: non-relativistic quantum theory
zbMATH Keywords: h-admissible operators
zbMATH Keywords: scalar admissible h-pseudo- differential operators
zbMATH Keywords: Fourier integral operators
zbMATH Keywords: Functional calculus
zbMATH Keywords: resolvents
zbMATH Keywords: complex powers
zbMATH Keywords: Essentially selfadjoint operators
zbMATH Keywords: Essential spectrum
zbMATH Keywords: Classical trajectories
zbMATH Keywords: quantum evolution
zbMATH Keywords: non-stationary Schrödinger equation
zbMATH Keywords: frequency sets of solutions
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: Quasi-classical properties of the discrete spectrum
zbMATH Keywords: quasi- principal symbol
zbMATH Keywords: periodic trajectories
zbMATH Keywords: energy levels
zbMATH Keywords: quasi-sub- principal symbol
zbMATH Keywords: cluster asymptotics
zbMATH Keywords: theory of tunnelling
zbMATH Keywords: multiple wells
zbMATH Keywords: resonances
author: Didier Robert
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4006772