Asymptotics for the Eigenvalues of the Harmonic Oscillator with a Quasi-Periodic Perturbation. arXiv:math/0312110
Preprint, arXiv:math/0312110 [math.SP] (2003).
Summary: 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 \lambda_n(H+V)=\lambda_n(H)+W(\sqrt n)n^{-1/4}+O(n^{-1/2}\ln(n)) as n\to+\infty, where W is a quasi-periodic function which can be defined explicitly in terms of V.
MSC:
| 34L20 | Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
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