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21
Harmonic analysis on metrized graphs
 Canad. J. Math
"... Abstract. This paper studies the Laplacian operator on a metrized graph, and its spectral theory. 1. ..."
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Abstract. This paper studies the Laplacian operator on a metrized graph, and its spectral theory. 1.
Localization effects in a periodic quantum graph with magnetic field and spinorbit interaction
 J. Math. Phys
, 2006
"... spinorbit interaction ..."
Eigenvalue bracketing for discrete and metric graphs
 J. Math. Anal. Appl
"... Abstract. We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the “exceptional” values of the m ..."
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Cited by 4 (2 self)
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Abstract. We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the “exceptional” values of the metric graph corresponding to the Dirichlet spectrum) we carry over these estimates from the metric graph Laplacian to the discrete case. We apply the results to covering graphs and present examples where the covering graph Laplacians have spectral gaps. 1.
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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Cited by 4 (2 self)
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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.
Quantum ergodicity for graphs related to interval maps
, 2008
"... We prove quantum ergodicity for a family of graphs that are obtained from ergodic onedimensional maps of an interval using a procedure introduced by Pakónski et al (J. Phys. A, 34, 93039317 (2001)). As observables we take the L 2 functions on the interval. The proof is based on the periodic orbit ..."
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Cited by 2 (0 self)
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We prove quantum ergodicity for a family of graphs that are obtained from ergodic onedimensional maps of an interval using a procedure introduced by Pakónski et al (J. Phys. A, 34, 93039317 (2001)). As observables we take the L 2 functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a onedimensional, Lebesguemeasurepreserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the PerronFrobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question. 1
SPECTRAL ANALYSIS OF METRIC GRAPHS AND RELATED SPACES
"... Abstract. 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 g ..."
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Abstract. 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. 1.
CONTINUITY OF THE INTEGRATED DENSITY OF STATES ON RANDOM LENGTH METRIC GRAPHS
"... Abstract. We establish several properties of the integrated density of states for random quantum graphs: Under appropriate ergodicity and amenability assumptions, the integrated density of states can be defined using an exhaustion procedure by compact subgraphs. A trace per unit volume formula holds ..."
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Abstract. We establish several properties of the integrated density of states for random quantum graphs: Under appropriate ergodicity and amenability assumptions, the integrated density of states can be defined using an exhaustion procedure by compact subgraphs. A trace per unit volume formula holds, similarly as in the Euclidean case. Our setting includes periodic graphs. For a model where the edge lengths are random and vary independently in a smooth way we prove a Wegner estimate and related regularity results for the integrated density of states. These results are illustrated for an example based on the Kagome lattice. In the periodic case we characterise all compactly supported eigenfunctions and calculate the position and size of discontinuities of the integrated density of states. 1.
SUPERSYMMETRY ON DISCRETE AND METRIC GRAPHS
"... Abstract. The aim of the present paper is to introduce a unified notion of Laplacians on discrete and metric 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 decorate ..."
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Abstract. The aim of the present paper is to introduce a unified notion of Laplacians on discrete and metric 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 illustrate the abstract concept by giving classical examples throughout the article. Our approach includes infinite graphs as well. 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 for finite graphs, the corresponding index for the metric Dirac operator agrees with the discrete one. 1.
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.