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

Dmitry Chelkak·Evgeny Korotyaev
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Abstract

We consider the perturbed harmonic oscillator $T_Dψ=-ψ''+x^2ψ+q(x)ψ$, $ψ(0)=0$ 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 $ψ'(0)=b ψ(0)$, $b\in R$.

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Dmitry Chelkak, Evgeny Korotyaev. 2005-08-29. The inverse problem for perturbed harmonic oscillator on the half-line. https://arxiv.org/abs/math-ph/0508056

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