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16
Heat kernels on metric graphs and a trace formula.” mathph/0701009
, 2007
"... Dedicated to JeanMichel Combes on the occasion of his 65th birthday ABSTRACT. 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 spe ..."
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Cited by 16 (2 self)
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Dedicated to JeanMichel Combes on the occasion of his 65th birthday ABSTRACT. 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. 1.
The inverse scattering problem for metric graphs and the traveling salesman problem
, 2006
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Stollmann: Eigenfunction expansion for Schrödinger operators on metric graphs (Preprint arXiv:0801.1376
"... Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices. ..."
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Cited by 8 (5 self)
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Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.
NONWEYL RESONANCE ASYMPTOTICS FOR QUANTUM GRAPHS
"... Abstract. We consider the resonances of a quantum graph G that consists of a compact part with one or more infinite leads attached to it. We discuss the leading term of the asymptotics of the number of resonances of G in a disc of a large radius. We call G a Weyl graph if the coefficient in front of ..."
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Cited by 4 (2 self)
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Abstract. We consider the resonances of a quantum graph G that consists of a compact part with one or more infinite leads attached to it. We discuss the leading term of the asymptotics of the number of resonances of G in a disc of a large radius. We call G a Weyl graph if the coefficient in front of this leading term coincides with the volume of the compact part of G. We give an explicit topological criterion for a graph to be Weyl. In the final section we analyze a particular example in some detail to explain how the transition from the Weyl to the nonWeyl case occurs. 1.
Vacuum Energy and Closed Orbits in Quantum Graphs
 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS
, 2008
"... The vacuum (Casimir) energy of a quantized scalar field in a given geometrical situation is a certain moment of the eigenvalue density of an associated selfadjoint differential operator. For various classes of quantum graphs it has been calculated by several methods: (1) Direct calculation from the ..."
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Cited by 3 (2 self)
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The vacuum (Casimir) energy of a quantized scalar field in a given geometrical situation is a certain moment of the eigenvalue density of an associated selfadjoint differential operator. For various classes of quantum graphs it has been calculated by several methods: (1) Direct calculation from the explicitly known spectrum is feasible only in simple cases. (2) Analysis of the secular equation determining the spectrum, as in the Kottos–Smilansky derivation of the trace formula, yields a sum over periodic orbits in the graph. (3) Construction of an associated integral kernel by the method of images yields a sum over closed (not necessarily periodic) orbits. We show that for the Kirchhoff and other scaleinvariant boundary conditions the sum over nonperiodic orbits in fact makes no contribution to the total energy, whereas for more general (frequencydependent) vertex scattering matrices it can make a nonvanishing contribution, which, however, is localized near vertices and hence can be “indexed ” a posteriori by truly periodic orbits. For the scaleinvariant cases complete calculations have been done by both methods (2) and (3), with identical results. Indeed, applying the image method to the resolvent kernel provides an alternative derivation of the trace formula.
The modulus of continuity of Wegner estimates for random Schrdinger operators on metric graphs
, 2007
"... We consider an alloy type potential on an infinite metric graph. We assume a covering condition on the single site potentials. For random Schrödingers operator associated with the alloy type potential restricted to finite volume subgraphs we prove a Wegner estimate which reproduces the modulus of ..."
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Cited by 2 (1 self)
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We consider an alloy type potential on an infinite metric graph. We assume a covering condition on the single site potentials. For random Schrödingers operator associated with the alloy type potential restricted to finite volume subgraphs we prove a Wegner estimate which reproduces the modulus of continuity of the single site distribution measure. The Wegner constant is independent of the energy.
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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Dedicated to Volker Enss on the occasion of his 65th birthday
FINITE PROPAGATION SPEED AND CAUSAL FREE QUANTUM FIELDS ON NETWORKS
, 907
"... ABSTRACT. Laplace operators on metric graphs give rise to KleinGordon and wave operators. Solutions of the KleinGordon equation and the wave equation are studied and finite propagation speed is established. Massive, free quantum fields are then constructed, whose commutator function is just the Kl ..."
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ABSTRACT. Laplace operators on metric graphs give rise to KleinGordon and wave operators. Solutions of the KleinGordon equation and the wave equation are studied and finite propagation speed is established. Massive, free quantum fields are then constructed, whose commutator function is just the KleinGordon kernel. As a consequence of finite propagation speed Einstein causality (local commutativity) holds. Comparison is made with an alternative construction of free fields involving RTalgebras. PACS: 03.65.Nk, 03.70.+k, 73.21.Hb 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.