Kian, Y; Morancey, M; Oksanen, L; (2019) Application of the boundary control method to partial data Borg-Levinson inverse spectral problem. Mathematical Control and Related Fields , 9 (2) pp. 289-312. 10.3934/mcrf.2019015.

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Abstract

We consider the multidimensional Borg-Levinson problem of determining a potential q , appearing in the Dirichlet realization of the Schrödinger operator A q = − Δ + q on a bounded domain Ω ⊂ R n , n ≥ 2 , from the boundary spectral data of A q on an arbitrary portion of ∂ Ω . More precisely, for γ an open and non-empty subset of ∂ Ω , we consider the boundary spectral data on γ given by B S D ( q , γ ) := { ( λ k , ∂ ν φ k | γ ) : k ≥ 1 } , where { λ k : k ≥ 1 } is the non-decreasing sequence of eigenvalues of A q , { φ k : k ≥ 1 } an associated orthonormal basis of eigenfunctions, and ν is the unit outward normal vector to ∂ Ω . Our main result consists of determining a bounded potential q ∈ L ∞ ( Ω ) from the data B S D ( q , γ ) . Previous uniqueness results, with arbitrarily small γ , 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.

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