Asymptotic inverse problem for almost-periodically perturbed quantum harmonic oscillator

Let {mu(n)}(n=0)(infinity) be the spectrum of -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(0)(n) + O(n(-1/4)), where {mu(0)(n)}(n=0)(infinity) is the spectrum of the unperturbed operator. Suppose that the asymptotic approximation to the first asymptotic correction Delta mu(n) = mu n - mu(0)(n)+ o( n(-1/4)) is given. We prove the formula that recovers the frequencies and the Fourier coefficients of q in terms of Delta mu(n).