Inverse Problems for Graph Laplacians

This thesis is devoted to inverse spectral problems for Laplace operators on metric graphs, and it is based on the following papers: Paper I - P. Kurasov and M. Nowaczyk 2005 Inverse spectral problem for quantum graphs J. Phys. A: Math. Gen 38 4901--15 Paper II - M. Nowaczyk 2007 Inverse spectral problem for quantum graphs with rationally dependent edges Operator Theory, Analysis and Mathematical Physics Operator Theory: Advances and Applications 147 105--16 Paper III - P. Kurasov and M. Nowaczyk 2007 Geometric properties of quantum graphs and vertex scattering matrices, Preprint 2007:21 Centre for Mathematical Sciences, Lund University. Paper IV - S. Avdonin, P. Kurasov and M. Nowaczyk 2007 On the Reconstruction of the Boundary Conditions for Star Graphs, Preprint 2007:29 Centre for Mathematical Sciences, Lund University. In the first paper, we prove the trace formula and show that it can be used to reconstruct the metric graph in the case of rationally independent lengths of the edges and the Laplace operator with standard boundary conditions at the vertices. The second paper generalises this result by showing that the condition of rational independence of lengths of the edges can be weakened. In the third paper the possibility to parameterise vertex boundary conditions via the scattering matrix is investigated. The trace formula is generalised to include even arbitrary vertex boundary conditions leading to energy independent vertex scattering matrices, so-called non-resonant boundary conditions. In the last paper, we turn to the problem of recovering boundary conditions and solve it for the special case of the star graph