We consider the perturbed harmonic oscillator T_Dψ = -ψ″ + x² ψ + q(x)ψ, ψ(0)=0, in L² (ℝ₊), where q ∈ H₊ = {q′, xq ∈ L ² (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.