The inverse problem for perturbed harmonic oscillator on the half-line

Abstract

We consider the perturbed harmonic oscillator $T_D\psi =-{\psi }^{\prime \prime }+x^2\psi +q(x)\psi$, $\psi (0)=0$ in $L^2(R_{+})$, where $q\in H_{+}=\{q^{\prime },xq\in L^2(R_{+})\}$ 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 ${\psi }^{\prime }(0)=b\psi (0)$, $b\in R$.