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Application of the boundary control method to partial data Borg-Levinson inverse spectral problem

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  • We consider the multidimensional Borg-Levinson problem of determining a potential $q$, appearing in the Dirichlet realization of the Schrödinger operator $A_q = -\Delta+q$ on a bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, from the boundary spectral data of $A_q$ on an arbitrary portion of $\partial\Omega$. More precisely, for $\gamma$ an open and non-empty subset of $\partial\Omega$, we consider the boundary spectral data on $\gamma$ given by ${\rm BSD}(q, \gamma): = \{(\lambda_{k}, {\partial_\nu \varphi_{k}}_{|\gamma}):\ k \geq1\}$, where $\{ \lambda_k:\ k \geq1\}$ is the non-decreasing sequence of eigenvalues of $A_q$, $\{ \varphi_k:\ k \geq1 \}$ an associated orthonormal basis of eigenfunctions, and $\nu$ is the unit outward normal vector to $\partial\Omega$. Our main result consists of determining a bounded potential $q\in L^\infty(\Omega)$ from the data ${\rm BSD}(q, \gamma)$. Previous uniqueness results, with arbitrarily small $\gamma$, assume that $q$ is smooth. Our approach is based on the Boundary Control method, and we give a self-contained presentation of the method, focusing on the analytic rather than geometric aspects of the method.

    Mathematics Subject Classification: 35R30, 35J10, 35L05.

    Citation:

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  • Figure 1.  Geometric condition (3.10)

    Figure 2.  Sets $A_{x, \varepsilon}$

    Figure 3.  Support of the geometric optics solution

    Figure 4.   

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