Pavel Kurasov Mikael Passare
Inverse Problems & Imaging 2010, 4(4): i-iv doi: 10.3934/ipi.2010.4.4i
This volume contains the proceedings of the international conference
Integral Geometry and Tomography

held at Stockholm University, August 12-15, 2008. The meeting was dedicated to Jan Boman on the occasion of his 75-th birthday.
   We are happy that so many of the participants have contributed to these proceedings with original research articles, some of which have been presented at the conference, others resulting from inspiring discussions during the meeting. A few contributions have also been written by colleagues who were invited but could not come to Stockholm.

For more information please click the “Full Text” above.
Inverse problems for quantum trees
Sergei Avdonin Pavel Kurasov
Inverse Problems & Imaging 2008, 2(1): 1-21 doi: 10.3934/ipi.2008.2.1
Three different inverse problems for the Schrödinger operator on a metric tree are considered, so far with standard boundary conditions at the vertices. These inverse problems are connected with the matrix Titchmarsh-Weyl function, response operator (dynamic Dirichlet-to-Neumann map) and scattering matrix. Our approach is based on the boundary control (BC) method and in particular on the study of the response operator. It is proven that the response operator determines the quantum tree completely, i.e. its connectivity, lengths of the edges and potentials on them. The same holds if the response operator is known for all but one boundary points, as well as for the Titchmarsh-Weyl function and scattering matrix. If the connectivity of the graph is known, then the lengths of the edges and the corresponding potentials are determined by just the diagonal terms of the data.
keywords: inverse problems wave equation controllability Quantum graphs boundary control. Schrödinger equation
Inverse problems for quantum trees II: Recovering matching conditions for star graphs
Sergei Avdonin Pavel Kurasov Marlena Nowaczyk
Inverse Problems & Imaging 2010, 4(4): 579-598 doi: 10.3934/ipi.2010.4.579
The inverse problem for the Schrödinger operator on a star graph is investigated. It is proven that such Schrödinger operator, i.e. the graph, the real potential on it and the matching conditions at the central vertex, can be reconstructed from the Titchmarsh-Weyl matrix function associated with the graph boundary. The reconstruction is also unique if the spectral data include not the whole Titchmarsh-Weyl function but its principal block (the matrix reduced by one dimension). The same result holds true if instead of the Titchmarsh-Weyl function the dynamical response operator or just its principal block is known.
keywords: matching conditions. quantum graphs inverse problems

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