Titchmarsh-Weyl Theory and Inverse Problems for Elliptic Differential Operators

Beschreibung

The Dirichlet-to-Neumann map associated with an elliptic differential expression on a bounded or unbounded domain plays a major role in many applications such as, e.g., in the electrical impedance tomography. In the present thesis its relation to the selfadjoint operator realizations of the dierential expression is investigated. It is proved that the partial knowledge of the Dirichlet-to-Neumann map determines the selfadjoint operator subject to a Dirichlet boundary condition uniquely up to unitary equivalence. Additionally, in the case of a bounded domain a reconstruction formula for the eigenvalues and corresponding eigenfunctions and, thus, for the Dirichlet operator itself is provided. These results can be viewed as a generalized variant of Calderon‘s inverse conductivity problem. Moreover, it is shown in the present thesis that also in the case of an unbounded domain the complete spectral information of the Dirichlet operator can be recovered from the knowledge of the Dirichlet-to-Neumann map. Explicit formulas for the isolated and embedded eigenvalues and for the absolutely continuous spectrum are provided. This spectral characterization is a multidimensional analog of well-known results from the classical Titchmarsh–Weyl theory for singular Sturm–Liouville operators. In addition, analogous results are proved for selfadjoint elliptic differential operators subject to Neumann and generalized Robin boundary conditions.

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Jonathan Rohleder

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Elliptischer Differentialoperator Inverses Problem Selbstadjungierter Operator Spektraltheorie Titchmarsh-Weyl-Theorie

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