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29
Mathematical Aspects of Vacuum Energy on Quantum Graphs
, 2007
"... We use quantum graphs as a model to study various mathematical aspects of the vacuum energy, such as convergence of periodic path expansions, consistency among different methods (trace formulae versus method of images) and the possible connection with the underlying classical dynamics. In our study ..."
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We use quantum graphs as a model to study various mathematical aspects of the vacuum energy, such as convergence of periodic path expansions, consistency among different methods (trace formulae versus method of images) and the possible connection with the underlying classical dynamics. In our study we derive an expansion for the vacuum energy in terms of periodic paths on the graph and prove its convergence and smooth dependence on the bond lengths of the graph. For an important special case of graphs with equal bond lengths, we derive a simpler explicit formula. With minor changes this formula also applies to graphs with rational (up to a common factor) bond lengths. The main results are derived using the trace formula. We also discuss an alternative approach using the method of images and prove that the results are consistent. This may have important consequences for other systems, since the method of images, unlike the trace formula, includes a sum over special “bounce paths”. We succeed in showing that in our model bounce paths do not contribute to the vacuum energy. Finally, we discuss the proposed possible link between the magnitude of the vacuum energy and the type (chaotic vs. integrable) of the underlying classical dynamics. Within a random matrix model we calculate the variance of the vacuum energy over several ensembles and find evidence that the level repulsion leads to suppression of the vacuum energy.
First order approach and index theorems for discrete and metric graphs
 Ann. Henri Poincaré
, 2009
"... Abstract. The aim of the present paper is to introduce the notion of first order (supersymmetric) Dirac operators on discrete and metric (“quantum”) graphs. In order to cover all selfadjoint boundary conditions for the associated metric graph Laplacian, we develop systematically a new type of discr ..."
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Abstract. The aim of the present paper is to introduce the notion of first order (supersymmetric) Dirac operators on discrete and metric (“quantum”) graphs. In order to cover all selfadjoint boundary conditions for the associated metric graph Laplacian, we develop systematically a new type of discrete graph operators acting on a decorated graph. The decoration at each vertex of degree d is given by a subspace of C d, generalising the fact that a function on the standard vertex space has only a scalar value. We develop the notion of exterior derivative, differential forms, Dirac and Laplace operators in the discrete and metric case, using a supersymmetric framework. We calculate the (supersymmetric) index of the discrete Dirac operator generalising the standard index formula involving the Euler characteristic of a graph. Finally, we show that the corresponding index for the metric Dirac operator agrees with the discrete one. 1.
Brownian Motions on Metric graphs: . . .
, 2010
"... The construction of the paths of all possible Brownian motions (in the sense of [21]) on a half line or a finite interval is reviewed. ..."
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The construction of the paths of all possible Brownian motions (in the sense of [21]) on a half line or a finite interval is reviewed.
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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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.
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
EQUILATERAL QUANTUM GRAPHS AND BOUNDARY TRIPLES
"... Abstract. The aim of the present paper is to analyse the spectrum of Laplace and Dirac type operators on metric graphs. In particular, we show for equilateral graphs how the spectrum (up to exceptional eigenvalues) can be described by a natural generalisation of the discrete Laplace operator on the ..."
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Abstract. The aim of the present paper is to analyse the spectrum of Laplace and Dirac type operators on metric graphs. In particular, we show for equilateral graphs how the spectrum (up to exceptional eigenvalues) can be described by a natural generalisation of the discrete Laplace operator on the underlying graph. These generalised Laplacians are necessary in order to cover general vertex conditions on the metric graph. In case of the standard (also named “Kirchhoff”) conditions, the discrete operator is the usual combinatorial Laplacian. 1.
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.
Relationship between scattering matrix and spectrum of quantum graphs
, 2008
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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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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.