The inverse problem for perturbed harmonic oscillator on the half-line with Dirichlet boundary conditions

Abstract

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

References in corpus (1)

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

Cited by in corpus (2)

• The Two-Spectra Inverse Problem for Semi-Infinite Jacobi Matrices in The Limit-Circle Case
• Weyl-Titchmarsh functions of vector-valued Sturm-Liouville operators on the unit interval