We consider the perturbed harmonic oscillator TDψ=−ψ′′+x2ψ+q(x)ψ,ψ(0)=0, in L2(R+), where q∈H+={q′,xq∈L2(R+)} is a real-valued potential. We prove that the mapping q⟼ spectral data = {eigenvalues of T D }⊕ {norming constants} is one-to-one and onto. The complete characterization of the set of spectral data which corresponds to q∈H+ is given.