Asymptotic Inverse Problem for Almost-Periodically Perturbed Quantum Harmonic Oscillator

TL;DR

This paper develops a method to recover the frequencies and Fourier coefficients of an almost-periodic perturbation in a quantum harmonic oscillator from spectral asymptotics, advancing inverse spectral analysis techniques.

Contribution

It provides a new formula that reconstructs the perturbation's spectral data using the first correction to spectral asymptotics in an almost-periodic setting.

Findings

Derived a formula for recovering perturbation frequencies

Reconstructed Fourier coefficients from spectral data

Extended inverse spectral analysis to almost-periodic potentials

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.