On the inverse to the harmonic oscillator

2013/06/28 by Marco Cappiello, Cappiello, Marco, Luigi Rodino +3
Mathematics · #30Gxx #35S05 #46F05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math.AP #msc:30Gxx #msc:33C10 #msc:35Q40 #msc:35S05 #msc:46F05 #primary 35Q40 #secondary 33C10

paper · pdf · doi:10.48550/arxiv.1306.6866

24 pages. In previous versions, certain parts were managed by arguments involving the Bargmann transform. These parts are now proved in other ways. The Bargmann parts are now moved to an other arxiv preprint

openalex publication_date 2013/06/28 · arxiv created 2014/06/04 · arxiv updated 2014/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let b_d be the Weyl symbol of the inverse to the harmonic oscillator on \R^d. We prove that b_d and its derivatives satisfy convenient bounds of Gevrey and Gelfand-Shilov type, and obtain explicit expressions for b_d. In the even-dimensional case we characterize b_d in terms of elementary functions. In the analysis we use properties of radial symmetry and a combination of different techniques involving classical a priori estimates, commutator identities, power series and asymptotic expansions.

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