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20
Spectra of selfadjoint extensions and applications to solvable Schrödinger operators
, 2007
"... ..."
ON THE SPECTRA OF CARBON NanoStructures
, 2007
"... An explicit derivation of dispersion relations and spectra for periodic Schrödinger operators on carbon nanostructures (including graphene and all types of singlewall nanotubes) is provided. ..."
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An explicit derivation of dispersion relations and spectra for periodic Schrödinger operators on carbon nanostructures (including graphene and all types of singlewall nanotubes) is provided.
On occurrence of spectral edges for periodic operators inside the Brillouin zone”, arXiv: mathph/0702035
, 2007
"... Abstract. The article discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics (e.g., Schrödinger, Maxwell, waveguide operators, etc.). Is it true that one can obtain the correct spectrum by using the values of the quasimomentum runn ..."
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Cited by 9 (1 self)
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Abstract. The article discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics (e.g., Schrödinger, Maxwell, waveguide operators, etc.). Is it true that one can obtain the correct spectrum by using the values of the quasimomentum running over the boundary of the (reduced) Brillouin zone only, rather than the whole zone? Or, do the edges of the spectrum occur necessarily at the set of “corner ” high symmetry points? This is known to be true in 1D, while no apparent reasons exist for this to be happening in higher dimensions. In many practical cases, though, this appears to be correct, which sometimes leads to the claims that this is always true. There seems to be no definite answer in the literature, and one encounters different opinions about this problem in the community. In this paper, starting with simple discrete graph operators, we construct a variety of convincing multiplyperiodic examples showing that the spectral edges might occur deeply inside the Brillouin zone. On the other hand, it is also shown that in a “generic” case, the situation of spectral edges appearing at high symmetry points is stable under small perturbations. This explains to some degree why in many (maybe even most) practical cases the statement still holds. AMS classification scheme numbers: 35P99, 47F05, 58J50, 81Q10Spectral edges of periodic operators 2 1.
Localization on quantum graphs with random vertex couplings
 J. Statist. Phys
"... Abstract. We consider Schrödinger operators on a class of periodic quantum graphs with randomly distributed Kirchhoff coupling constants at all vertices. Using the technique of selfadjoint extensions we obtain conditions for localization on quantum graphs in terms of finite volume criteria for some ..."
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Cited by 7 (0 self)
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Abstract. We consider Schrödinger operators on a class of periodic quantum graphs with randomly distributed Kirchhoff coupling constants at all vertices. Using the technique of selfadjoint extensions we obtain conditions for localization on quantum graphs in terms of finite volume criteria for some energydependent discrete Hamiltonians. These conditions hold in the strong disorder limit and at the spectral edges.
Localization effects in a periodic quantum graph with magnetic field and spinorbit interaction
 J. Math. Phys
, 2006
"... spinorbit interaction ..."
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.
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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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.
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.
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
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.