  We consider the perturbed harmonic oscillator
$T_D\psi=-\psi''+x^2\psi+q(x)\psi$, $\psi(0)=0$, in $L^2(\R_+)$, where
$q\in\bH_+=\{q', xq\in L^2(\R_+)\}$ is a real-valued potential. We prove that
the mapping $q\mapsto{\rm spectral data}={\rm \{eigenvalues of\}T_D{\rm
\}}\oplus{\rm \{norming constants\}}$ is one-to-one and onto. The complete
characterization of the set of spectral data which corresponds to $q\in\bH_+$
is given.
