The semi-classical spectrum and the Birkhoff normal form

2009/02/14 by Yves Colin de Verdìère, Yves Colin De Verdière, De Verdière, Yves Colin
Computer Science · Engineering · Mathematics · Physics and Astronomy · #35P20 #35S30 #Advanced Numerical Analysis Techniques #Digital Filter Design and Implementation #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.MP #math.SP #msc:35P20 #msc:35S30

paper · pdf · doi:10.48550/arxiv.0902.2470

arxiv created 2009/02/14 · openalex publication_date 2009/02/14 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/31

Abstract

The purposes of this note are: 1) to propose a direct and "elementary" proof of the main result proved by Guillemin-Paul-Uribe [GPU], namely that the semi-classical spectrum near a global minimum of the classical Hamiltonian determines the whole semi-classical Birkhoff normal form (denoted the BNF) in the non-resonant case. 2) to present in the completely resonant case a similar problem.

Related

• Birkhoff Normal Forms in Semi-Classical Inverse Problems
• Quantum Birkhoff Normal Form in the σ-Bruno-Rüssmann non-resonant condition
• Nonresonant smooth integrable Hamiltonian systems which don't admit a local smooth Birkhoff normal form
• Canonical forms for perturbations of the harmonic oscillator
• Semiclassical Birkhoff-Gustavson normal forms and spectral asymptotics for nearly resonant Schrödinger operators