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 $\{Δμ_n\}_{n=0}^\infty$ ($Δμ_n=μ_n-μ_n^0+o(n^{-1/4})$, where $μ_n^0$ and $μ_n$ 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.