The inverse problem for perturbed harmonic oscillator on the half-line with a dirichlet boundary condition

We consider the perturbed harmonic oscillator T-D psi = -psi'' + x(2) psi + q(x)psi, psi(0) = 0, in L-2(R+), where q is an element of H+ = {q', xq is an element of L-2(R+)} is a real-valued potential. We prove that the mapping q -> spectral data characterization of the set of spectral data which corresponds to q is an element of H+ is given.