The inverse problem for perturbed harmonic oscillator on the half-line
  We consider the perturbed harmonic oscillator $T_D\psi=-\psi''+x^2\psi+q(x)\psi$, $\psi(0)=0$ in $L^2(R_+)$, where $q\in H_+=\{q', xq\in L^2(R_+)\}$ 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 $q\in H_+$ is given. Moreover, we solve the similar inverse problem for the family of boundary conditions $\psi'(0)=b \psi(0)$, $b\in R$. 
Dmitry Chelkak, Evgeny Korotyaev