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 \(q \in {\mathbf{H}}_{+} = \{q^{\prime}, xq \in L^{2}(\mathbb{R}_{+})\}\) is a real-valued potential. We prove that the mapping \(q \longmapsto\) spectral data = {eigenvalues of T D }\(\oplus\) {norming constants} is one-to-one and onto. The complete characterization of the set of spectral data which corresponds to \(q \in {\mathbf{H}}_{+}\) is given.
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Communicated by Christian Gérard.
Dedicated to Vladimir Buslaev on the occasion of his 70th birthday
Submitted: September 27, 2006. Accepted: January 9, 2007.
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Chelkak, D., Korotyaev, E. The Inverse Problem for Perturbed Harmonic Oscillator on the Half-Line with a Dirichlet Boundary Condition. Ann. Henri Poincaré 8, 1115–1150 (2007). https://doi.org/10.1007/s00023-007-0330-z
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DOI: https://doi.org/10.1007/s00023-007-0330-z