Results 1  10
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14
Heat kernels on metric graphs and a trace formula
, 2007
"... We study heat semigroups generated by selfadjoint Laplace operators on metric graphs characterized by the property that the local scattering matrices associated with each vertex of the graph are independent from the spectral parameter. For such operators we prove a representation for the heat kerne ..."
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Cited by 31 (4 self)
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We study heat semigroups generated by selfadjoint Laplace operators on metric graphs characterized by the property that the local scattering matrices associated with each vertex of the graph are independent from the spectral parameter. For such operators we prove a representation for the heat kernel as a sum over all walks with given initial and terminal edges. Using this representation a trace formula for heat semigroups is proven. Applications of the trace formula to inverse spectral and scattering problems are also discussed.
INVERSE PROBLEMS FOR QUANTUM TREES
"... Abstract. 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 TitchmarshWeyl function, response operator (dynamic DirichlettoNeumann map) and s ..."
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Cited by 11 (1 self)
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Abstract. 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 TitchmarshWeyl function, response operator (dynamic DirichlettoNeumann 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 TitchmarshWeyl 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.
Inverse problems in geometric graphs using internal measurements, arXiv:1008.2933v1
, 2010
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Contraction semigroups on metric graphs
 Analysis on Graphs and its Applications, volume 77 of Proceedings of Symposia in Pure Mathematics
, 2008
"... Dedicated to Volker Enss on the occasion of his 65th birthday ..."
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Cited by 7 (0 self)
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Dedicated to Volker Enss on the occasion of his 65th birthday
Schrödinger operators on graphs and geometry I: Essentially bounded potentials
, 2008
"... The inverse spectral problem for Schrödinger operators on finite compact metric graphs is investigated. The relations between the spectral asymptotics and geometric properties of the underlying graph are studied. It is proven that the Euler characteristic of the graph can be calculated from the spec ..."
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Cited by 5 (2 self)
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The inverse spectral problem for Schrödinger operators on finite compact metric graphs is investigated. The relations between the spectral asymptotics and geometric properties of the underlying graph are studied. It is proven that the Euler characteristic of the graph can be calculated from the spectrum of the Schrödinger operator in the case of essentially bounded real potentials and standard boundary conditions at the vertices. Several generalizations of the presented results are discussed.
SPECTRAL ANALYSIS OF METRIC GRAPHS AND RELATED SPACES
, 2008
"... The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general fo ..."
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Cited by 3 (0 self)
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The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form of a Laplacian on a metric graph, we define a new type of combinatorial Laplacian. With this generalised discrete Laplacian, it is possible to relate the spectral theory on discrete and metric graphs. Moreover, we describe a connection of metric graphs with manifolds. Finally, we comment on Cheeger’s inequality and trace formulas for metric and discrete (generalised) Laplacians.
GENERALISED DISCRETE LAPLACIANS ON GRAPHS AND THEIR RELATION TO QUANTUM GRAPHS
"... Abstract. The aim of the present paper is to analyse the spectrum of Laplace operators on graphs. Motivated by the general form of vertex conditions of a Laplacian on a metric graph, we define a new type of combinatorial Laplacian. With this generalised discrete Laplacian, it is possible to relate t ..."
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Abstract. The aim of the present paper is to analyse the spectrum of Laplace operators on graphs. Motivated by the general form of vertex conditions of a Laplacian on a metric graph, we define a new type of combinatorial Laplacian. With this generalised discrete Laplacian, it is possible to relate the spectral theory on discrete and metric graphs using the theory of boundary triples. In particular, we derive a spectral relation for equilateral metric graphs and index formulas. Moreover, we introduce extended metric graphs occuring naturally as limits of “thick ” graphs, and provide spectral analysis of natural Laplacians on such spaces. 1.
Trace formulae for quantum graphs
, 2007
"... Quantum graph models are based on the spectral theory of (differential) Laplace operators on metric graphs. We focus on compact graphs and survey various forms of trace formulae that relate Laplace spectra to periodic orbits on the graphs. Included are representations of the heat trace as well as of ..."
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Quantum graph models are based on the spectral theory of (differential) Laplace operators on metric graphs. We focus on compact graphs and survey various forms of trace formulae that relate Laplace spectra to periodic orbits on the graphs. Included are representations of the heat trace as well as of the spectral density in terms of sums over periodic orbits. Finally, a general trace formula for any self adjoint realisation of the Laplacian on a compact, metric graph is given.
Linear Algebra and its Applications 436 (2012) 3373–3391 Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications
"... journal homepage: www.elsevier.com/locate / laa ..."