The asymptotics of spectral data of the harmonic oscillator perturbed by a potential

The asymptotics of spectral data of the perturbed harmonic oscillator $-y''+x^2y+q(x)y$ are obtained for the potentials $q(x)$ such that $q',xq\in L^2(\mathbb R)$. These results are important for the solution of the corresponding inverse spectral problem.