2009/02/26 by Alexei Iantchenko, Iantchenko, Alexei
Mathematics · Physics and Astronomy · #32A99 #35P20 #35R30 #35S99 #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics #Numerical methods in inverse problems #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math.AP #math.CV #msc:32A99 #msc:35P20 #msc:35R30 #msc:35S99
paper · pdf · doi:10.48550/arxiv.0902.4650
arxiv created 2009/02/26 · openalex publication_date 2009/02/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider semi-classical Schrödinger operator P(h)=-h2Δ+V(x) in \mathbb Rn such that the analytic potential V has a non-degenerate critical point x0=0 with critical value E0 and we can define resonances in some fixed neighborhood of E0 when h>0 is small enough. If the eigenvalues of the Hessian are \zz-independent the resonances in hδ-neighborhood of E0 (δ>0) can be calculated explicitly as the eigenvalues of the semi-classical Birkhoff normal form. Assuming that potential is symmetric with respect to reflections about the coordinate axes we show that the classical Birkhoff normal form determines the Taylor series of the potential at x0. As a consequence, the resonances in a hδ-neighborhood of E0 determine the first N terms in the Taylor series of V at x0. The proof uses the recent inverse spectral results of V. Guillemin and A. Uribe.