Asymptotic Inverse Problem for Almost-Periodically Perturbed Quantum Harmonic Oscillator

Abstract

Consider quantum harmonic oscillator, perturbed by an even almost-periodic complex-valued potential with bounded derivative and primitive. Suppose that we know the first correction to the spectral asymptotics $\{\Delta {\mu }_n{\}}_{n=0}^{\infty }$ ($\Delta {\mu }_n={\mu }_n-{\mu }_n^0+o(n^{-1/4})$, where ${\mu }_n^0$ and ${\mu }_n$ is the spectrum of the unperturbed and the perturbed operators, respectively). We obtain the formula that recovers the frequencies and the Fourier coefficients of the perturbation.

6 pages