Asymptotic Inverse Problem for Almost-Periodically Perturbed Quantum Harmonic Oscillator

Let {μn}n=0 be the spectrum of − d 2 dx2 +x2+q(x) in L2(R), where q is an even almostperiodic complex-valued function with bounded primitive and derivative. Suppose that the asymptotic approximation to the first asymptotic correction ∆μn = μn − μn + o(n− 1 4 ), μn = 2n + 1 is known. We prove the formula that gives the frequencies and the Fourier coefficients of q in terms of ∆μn. AMS Mathematics Subject Classifications (1991): 34L20 (Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions); 81Q15 (Perturbation theories for operators and differential equations).