Asymptotics for the Eigenvalues of the Harmonic Oscillator with a Quasi-Periodic Perturbation

2003/12/04 by Daniel M. Elton, Elton, Daniel M.
Computer Science · Mathematics · #34L20 #34L40 (Primary) 34E10 #81Q10 (Secondary) #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.CA #math.SP #msc:34E10 #msc:34L20 #msc:34L40 #msc:81Q10

paper · pdf · doi:10.48550/arxiv.math/0312110

18 pages, LaTeX

arxiv created 2003/12/04 · openalex publication_date 2003/12/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider operators of the form H+V where H is the one-dimensional harmonic oscillator and V is a zero-order pseudo-differential operator which is quasi-periodic in an appropriate sense (one can take V to be multiplication by a periodic function for example). It is shown that the eigenvalues of H+V have asymptotics of the form λ_n(H+V)=λ_n(H)+W(√ n)n^-1/4+O(n^-1/2ln(n)) as n→+∞, where W is a quasi-periodic function which can be defined explicitly in terms of V.

Related

• On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator
• On Perturbations of Quasiperiodic Schroedinger Operators
• Asymptotic expansion for the eigenvalues of a perturbed anharmonic\n oscillator
• Complete Semiclassical Spectral Asymptotics for Periodic and Almost Periodic Perturbations of Constant Operators
• High energy asymptotics and trace formulae for the perturbed harmonic oscillator