2009/02/18 by Yves Colin de Verdìère, Yves Colin De Verdière, De Verdière, Yves Colin
Mathematics · Physics and Astronomy · #35S30 #53D22 #53D55 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #math-ph #math.AP #math.MP #msc:35S30 #msc:53D22 #msc:53D55
paper · pdf · doi:10.48550/arxiv.0902.3084
arxiv created 2009/02/18 · openalex publication_date 2009/02/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
Motivated by many recent works (by L. Charles, V. Guillemin, T. Paul, J. Sjöstrand, A. Uribe, S. Vu Ngoc, S. Zelditch and others) on the semi-classical Birkhoff normal forms, we investigate the structure of the group of automorphisms of the graded semi-classical Weyl algebra which is used to get the normal forms. The answer is quite similar to the Theorem of Duistermaat and Singer for the usual algebra of pseudo-differential operators where all automorphisms are given by conjugation by an elliptic Fourier Integral Operator (a FIO). Here what replaces general non-linear symplectic diffeomeorhisms is just linear complex symplectic maps, because everything is localized at a single point.