H-Pseudodifferential Operators and Applications: An Introduction

Abstract

The aim of these lectures1 is to present the basic techniques in semiclassical analysis based on the use of h-pseudodifferential operators. One of our aims is to prepare to the comprehension of the recent results on the Scott’s conjecture. In particular, we hope it can help for the understanding of the recent proofs of the Scott’s conjecture for molecules by Ivrii and Sigal [107]. We present also some results turning around the Gutzwiller’s formula [69] and the Schnirelman’s theorem [141].According to the short time we had, the following notes are sometimes sketchy and some arguments are oversimplified. This has the advantage to outline the basic ideas. We refer mainly to the book [135] or to [98] for precise and rigorous statements. We hope simply that this short presentation permits to see what can be done with this theory by entering rapidly in the applications.Of course, a great part of this theory was developed initially in another context (see [98]) but it seems more efficient to work directly in the semiclassical context which has in particular the interest to remain quite near the physical intuition. We have also tried to give a rather extensive list of references but we are sure that we still forget important references.Let us mention also that the most complete treatise on this subject is a project of book being written now by V. Ivrii [106] whose reading is probably rather difficult but which presents the most sophisticated developments of the theory. We heard also about different projects of books by Y. Colin de Verdi ere [32] and by Dimassi-Sjöstrand.KeywordsPseudodifferential OperatorPseudo Differential OperatorFunctional CalculusPrincipal SymbolTauberian TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.