Results 1  10
of
11
Spectra of selfadjoint extensions and applications to solvable Schrödinger operators
, 2007
"... ..."
Spectra of Schrödinger operators on equilateral quantum graphs
, 2006
"... We consider magnetic Schrödinger operators on quantum graphs with identical edges. The spectral problem for the quantum graph is reduced to the discrete magnetic Laplacian on the underlying combinatorial graph and a certain Hill operator. In particular, it is shown that the spectrum on the quantum g ..."
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Cited by 20 (5 self)
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We consider magnetic Schrödinger operators on quantum graphs with identical edges. The spectral problem for the quantum graph is reduced to the discrete magnetic Laplacian on the underlying combinatorial graph and a certain Hill operator. In particular, it is shown that the spectrum on the quantum graph is the preimage of the combinatorial spectrum under a certain entire function. Using this correspondence we show that that the number of gaps in the spectrum of the Schrödinger operators admits an estimate from below in terms of the Hill operator independently of the graph structure.
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 ..."
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.
Magnetic Schrödinger operators on armchair nanotubes
"... We consider the Schrödinger operator with a periodic potential on a quasi 1D continuous periodic model of armchair nanotubes in R 3 in a uniform magnetic field (with amplitude B ∈ R), which is parallel to the axis of the nanotube. The spectrum of this operator consists of an absolutely continuous pa ..."
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Cited by 3 (1 self)
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We consider the Schrödinger operator with a periodic potential on a quasi 1D continuous periodic model of armchair nanotubes in R 3 in a uniform magnetic field (with amplitude B ∈ R), which is parallel to the axis of the nanotube. The spectrum of this operator consists of an absolutely continuous part (spectral bands separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all eigenfunctions with the same eigenvalue including compactly supported. We describe the spectrum as a function of B. For some specific potentials we prove an existence of gaps independent on the magnetic field. If B = 0, then there exists an infinite number of gaps Gn with the length Gn  → ∞ as n → ∞, and we determine the asymptotics of the gaps at high energy for fixed B. Moreover, we determine the asymptotics of the gaps Gn as B → 0 for fixed n. 1 Introduction and main results We consider the Schrödinger operator HB = (−i ∇ − A) 2 + Vq with a periodic potential Vq on the armchair nanotube ΓN ⊂ R3, N � 1 in a uniform magnetic field B = B(0, 0, 1) ∈ R3,
Convergence of resonances on thin branched quantum wave guides
 J. MATH. PHYS
, 2007
"... We prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family Xε of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that th ..."
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Cited by 3 (3 self)
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We prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family Xε of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances on Xε approximate those of the Laplacian with “free” boundary conditions on X0, the skeleton graph of Xε.
Equilateral quantum graphs and their decorations
, 2005
"... ABSTRACT. We consider magnetic Schrödinger operators on quantum graphs with identical edges. The spectral problem for the quantum graph is reduced to the discrete magnetic Laplacian on the corresponding combinatorial graph and a certain Hill equation. This may be viewed as a generalization of the cl ..."
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Cited by 1 (0 self)
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ABSTRACT. We consider magnetic Schrödinger operators on quantum graphs with identical edges. The spectral problem for the quantum graph is reduced to the discrete magnetic Laplacian on the corresponding combinatorial graph and a certain Hill equation. This may be viewed as a generalization of the classical spectral analysis for the Hill operator to such structures. Using this correspondence we show that that the number of gaps in the spectrum of Schrödinger operators in graphs admits an estimate from below in terms of the Hill operator independently of the graph structure. We also discuss the decoration of graphs in the context of this correspondence.
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.
unknown title
, 802
"... Semiclassical reduction for magnetic Schrödinger operator with periodic zerorange potentials and applications ..."
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Semiclassical reduction for magnetic Schrödinger operator with periodic zerorange potentials and applications