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**Inverse spectral-scattering problem with two sets of discrete spectra for the radial Schrödinger equation.**
*(English)*
Zbl 1096.34064

The paper is devoted to the inverse problem for the one-dimensional Schrödinger equation on the half-line given by

\[ -\psi''+V\psi=k^2\psi \quad \text{on }(0,\infty) \]

with the boundary conditions

\[ \sin(\alpha)\,\psi'(k,0)+\cos(\alpha)\,\psi(k,0)=0\quad \text{for some }\alpha\in(0,\pi]. \]

The potential \(V\) is assumed to belong to the Faddeev class, i.e., it is real-valued and obeys \(\int_0^\infty (1+| x| )| V(x)| dx<\infty\). Denote by \(H_\alpha\) the selfadjoint realization in \(L^2(0,\infty)\) of \(-\frac{d^2}{dx^2}+V\) with boundary condition given above. It is well known, that the spectrum of \(H_\alpha\) consists of finitely many negative eigenvalues, i.e., \(\sigma_d(H_\alpha):=\{-\kappa^2_{\alpha j}\}_{j=1}^{N_\alpha}\), and a purely absolutely continuous part \(\sigma_{ac}(H_\alpha)=[0,\infty)\). By \(F_\alpha(k)\), \(k\in\mathbb R\), we denote the Jost function associated to the Schrödinger equation above.

Assume that \(\beta \in(0,\pi)\) with \(\beta <\alpha \leq \pi\). Let \(\mathcal D\) be a given data set, which contains the Jost function \(| F_\alpha(k)| \) for \(k\in\mathbb R\), the whole set \(\{\kappa_{\alpha j}\}_{j=1}^{N_\alpha}\) and a subset of \(\{\kappa_{\beta j}\}_{j=1}^{N_\beta}\) containing \(N_\alpha\) elements. Alternatively, the set \(\mathcal D\) may include \(| F_\beta(k)| \), \(k\in\mathbb R\), and the sets \(\{\kappa_{\alpha j}\}_{j=1}^{N_\alpha}\) and \(\{\kappa_{\beta j}\}_{j=1}^{N_\beta}\). The authors give conditions under which a data set \(\mathcal D\) determines the potential \(V\) and the boundary values \(\alpha\), \(\beta\) uniquely. Moreover, methods to reconstruct the potential are presented and illustrated by means of an example.

The result presented in the paper extends the two-spectrum uniqueness theorem of Borg and Marchenko to the case, where there is also a continuous spectrum. Another extension of this type of problems is given by F. Gesztesy and B. Simon [Trans. Am. Math. Soc. 348, No. 1, 349–373 (1996; Zbl 0846.34090)], where knowledge of the spectral shift function is required.

\[ -\psi''+V\psi=k^2\psi \quad \text{on }(0,\infty) \]

with the boundary conditions

\[ \sin(\alpha)\,\psi'(k,0)+\cos(\alpha)\,\psi(k,0)=0\quad \text{for some }\alpha\in(0,\pi]. \]

The potential \(V\) is assumed to belong to the Faddeev class, i.e., it is real-valued and obeys \(\int_0^\infty (1+| x| )| V(x)| dx<\infty\). Denote by \(H_\alpha\) the selfadjoint realization in \(L^2(0,\infty)\) of \(-\frac{d^2}{dx^2}+V\) with boundary condition given above. It is well known, that the spectrum of \(H_\alpha\) consists of finitely many negative eigenvalues, i.e., \(\sigma_d(H_\alpha):=\{-\kappa^2_{\alpha j}\}_{j=1}^{N_\alpha}\), and a purely absolutely continuous part \(\sigma_{ac}(H_\alpha)=[0,\infty)\). By \(F_\alpha(k)\), \(k\in\mathbb R\), we denote the Jost function associated to the Schrödinger equation above.

Assume that \(\beta \in(0,\pi)\) with \(\beta <\alpha \leq \pi\). Let \(\mathcal D\) be a given data set, which contains the Jost function \(| F_\alpha(k)| \) for \(k\in\mathbb R\), the whole set \(\{\kappa_{\alpha j}\}_{j=1}^{N_\alpha}\) and a subset of \(\{\kappa_{\beta j}\}_{j=1}^{N_\beta}\) containing \(N_\alpha\) elements. Alternatively, the set \(\mathcal D\) may include \(| F_\beta(k)| \), \(k\in\mathbb R\), and the sets \(\{\kappa_{\alpha j}\}_{j=1}^{N_\alpha}\) and \(\{\kappa_{\beta j}\}_{j=1}^{N_\beta}\). The authors give conditions under which a data set \(\mathcal D\) determines the potential \(V\) and the boundary values \(\alpha\), \(\beta\) uniquely. Moreover, methods to reconstruct the potential are presented and illustrated by means of an example.

The result presented in the paper extends the two-spectrum uniqueness theorem of Borg and Marchenko to the case, where there is also a continuous spectrum. Another extension of this type of problems is given by F. Gesztesy and B. Simon [Trans. Am. Math. Soc. 348, No. 1, 349–373 (1996; Zbl 0846.34090)], where knowledge of the spectral shift function is required.

Reviewer: Michael Baro (Berlin)

### MSC:

34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |

81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |

34A55 | Inverse problems involving ordinary differential equations |

34L25 | Scattering theory, inverse scattering involving ordinary differential operators |

34B24 | Sturm-Liouville theory |