Asymptotic Inverse Problem for Almost-Periodically Perturbed Quantum Harmonic Oscillator

Abstract

Let $\{\mu_n\}_{n=0}^\infty$ be the spectrum of $-\frac{d^2}{dx^2}+x^2+q(x)$ in L ²(ℝ), where q is an even almost-periodic complex-valued function with bounded primitive and derivative. It is known that $\mu_n=\mu_n^0+O(n^{-\frac{1}{4}})$, where $\{\mu_n^0\}_{n=0}^\infty$ is the spectrum of the unperturbed operator. Suppose that the asymptotic approximation to the first asymptotic correction $\Delta\mu_n=\mu_n-\mu_n^0+o(n^{-\frac{1}{4}})$ is given. We prove the formula that recovers the frequencies and the Fourier coefficients of q in terms of Δμ _n .