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

Inverse spectral problem for quantum graphs with rationally dependent edges

In this paper we study the problem of unique reconstruction of the quantum graphs. The idea is based on the trace formula which establishes the relation between the spectrum of Laplace operator and the set of periodic orbits, the number of edges and the total length of the graph. We analyse conditions under which is it possible to reconstruct simple graphs containing edges with rationally dependent lengths

Geometric properties of quantum graphs and vertex scattering matrices

Differential operators on metric graphs are investigated. It is proven that vertex matching (boundary) conditions can be successfully parameterized by the vertex scattering matrix. Two new families of matching conditions are investigated: hyperplanar Neumann and hyperplanar Dirichlet conditions. Using trace formula it is shown that the spectrum of the Laplace operator determines certain geometric properties of the underlying graph

Geometric properties of quantum graphs and vertex scattering matrices

Differential operators on metric graphs are investigated. It is proven that vertex matching (boundary) conditions can be successfully parameterized by the vertex scattering matrix. Two new families of matching conditions are investigated: hyperplanar Neumann and hyperplanar Dirichlet conditions. Using trace formula it is shown that the spectrum of the Laplace operator determines certain geometric properties of the underlying graph

Inverse spectral problem for quantum graphs

The inverse spectral problem for the Laplace operator on a finite metric graph is investigated. It is shown that this problem has a unique solution for graphs with rationally independent edges and without vertices having valence 2. To prove the result, a trace formula connecting the spectrum of the Laplace operator with the set of periodic orbits for the metric graph is established