Abstract. We discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds. We show that if a Neumann type Laplacian on such manifolds is amended by suitable...
CONTINUITY OF THE INTEGRATED DENSITY OF STATES ON RANDOM LENGTH METRIC GRAPHS (2009)
Daniel Lenz, Norbert Peyerimhoff, Olaf Post, Ivan Veseli Ć
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...
Eigenvalue bracketing for discrete and metric graphs (2009)
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...
SUPERSYMMETRY ON DISCRETE AND METRIC GRAPHS (2009)
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 self-adjoint boundary conditions for the associated metric...
Spectral convergence of non-compact quasi-one-dimensional spaces (2008)
Abstract. We consider a family of non-compact manifolds Xε (“graph-like manifolds”) approaching a metric graph X0 and establish convergence results of the related natural operators, namely the...
On the spectra of carbon nano-structures (2008)
Abstract. An explicit derivation of dispersion relations and spectra for periodic Schrödinger operators on carbon nano-structures (including graphene and all types of single-wall nano-tubes) is...
Post: Convergence of resonances on thin branched quantum wave guides (2008)
Abstract. 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...
SPECTRAL ANALYSIS OF METRIC GRAPHS AND RELATED SPACES (2008)
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...
EQUILATERAL QUANTUM GRAPHS AND BOUNDARY TRIPLES (2008)
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...
Continuity of the integrated density of states on random length metric graphs (2008)
Lenz, Daniel, Peyerimhoff, Norbert, Post, Olaf, Veselic', Ivan
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...
We discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds. We show that if a Neumann type Laplacian on such manifolds is amended by suitable...
QUANTUM NETWORKS MODELLED BY GRAPHS (2008)
Abstract. Quantum networks are often modelled using Schrödinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the...
FIRST ORDER OPERATORS AND BOUNDARY TRIPLES (2008)
concept of boundary triples for Laplace operators. Our main application is the Laplace operator on a manifold with boundary; a case in which the ordinary concept of boundary triples does not apply...
Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case (2008)
Abstract. We consider a family of open sets Mε which shrinks with respect to an appropriate parameter ε to a graph. Under the additional assumption that the vertex neighbourhoods are small we show...
Continuity properties of the integrated density of states on manifolds (2008)
Daniel Lenz, Norbert Peyerimhoff, Olaf Post, Ivan Veseli Ć
Abstract. We first analyze the integrated density of states (IDS) of periodic Schrödinger operators on an amenable covering manifold. A criterion for the continuity of the IDS at a prescribed energy...
Eigenvalue bracketing for discrete and metric graphs (2008)
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...
Post: Convergence of resonances on thin branched quantum wave guides (2008)
Abstract. 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...
Equilateral quantum graphs and boundary triples (2007)
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...
Spectral analysis of metric graphs and related spaces (2007)
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...
Periodic manifolds with spectral gaps (2007)
Abstract. We investigate spectral properties of the Laplace operator on a class of non-compact Riemannian manifolds. For a given number N we construct periodic manifolds such that the essential...
Spectral gaps for periodic elliptic operators with high contrast: An overview (2007)
Abstract. We discuss the band-gap structure and the integrated density of states for periodic elliptic operators in the Hilbert space L 2 (R m), for m 2. We specically consider situations where high...
First order operators and boundary triples (2007)
The aim of the present paper is to introduce a first order approach to the abstract concept of boundary triples for Laplace operators. Our main application is the Laplace operator on a manifold with...
Gaps in the differential forms spectrum on cyclic coverings (2007)
Anné, Colette, Carron, Gilles, Post, Olaf
We are interested in the spectrum of the Hodge-de Rham operator on a cyclic covering $X$ over a compact manifold $M$ of dimension $n+1$. Let $\Sigma$ be a hypersurface in $M$ which does not...
First order approach and index theorems for discrete and metric graphs (2007)
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 self-adjoint boundary...
Quantum networks modelled by graphs (2007)
Quantum networks are often modelled using Schroedinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave...
Continuity properties of the integrated density of states on manifolds (2007)
Lenz, Daniel, Peyerimhoff, Norbert, Post, Olaf, Veselic', Ivan
We first analyze the integrated density of states (IDS) of periodic Schr\"odinger operators on an amenable covering manifold. A criterion for the continuity of the IDS at a prescribed energy is given...
Convergence of resonances on thin branched quantum wave guides (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 $X_\eps$ of...
Gaps in the differential forms spectrum on cyclic coverings (2007)
Colette Anné, Gilles Carron, Olaf Post
We are interested in the spectrum of the Hodge-de Rham operator on a Z-covering X over a compact manifold M of dimension n + 1. Let Σ be a hypersurface in M which does not disconnect M and such that...
Gaps in the differential forms spectrum on cyclic coverings, Preprint (arXiv:0708.3981 (2007)
Colette Anné, Gilles Carron, Olaf Post
Abstract. We are interested in the spectrum of the Hodge-de Rham operator on a Z-covering X over a compact manifold M of dimension n + 1. Let Σ be a hypersurface in M which does not disconnect M and...
On the spectra of carbon nano-structures (2006)
An explicit derivation of dispersion relations and spectra for periodic Schr\"{o}dinger operators on carbon nano-structures (including graphen and all types of single-wall nano-tubes) is provided.
Anderson Localization for radial tree-like random quantum graphs (2006)
We prove that certain random models associated with radial, tree-like, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model (RLM) and...
Post: Anderson localization for radial tree-like random quantum graphs (2006)
Abstract. We prove that certain random models associated with radial, tree-like, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model...
Spectral convergence of non-compact quasi-one-dimensional spaces (2005)
We consider a family of non-compact manifolds $X_\eps$ (``graph-like manifolds'') approaching a metric graph $X_0$ and establish convergence results of the related natural operators, namely the...
Existence of spectral gaps, covering manifolds and residually finite groups (2005)
In the present paper we consider Riemannian coverings $(X,g) \to (M,g)$ with residually finite covering group $\Gamma$ and compact base space $(M,g)$. In particular, we give two general procedures...
Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case (2005)
We consider a family of open sets $M_\epsilon$ which shrinks with respect to an appropriate parameter $\epsilon$ to a graph. Under the additional assumption that the vertex neighbourhoods are small...
Existence of spectral gaps, covering manifolds and residually finite groups (2005)
Abstract. In the present paper we consider Riemannian coverings (X, g) → (M, g) with residually finite covering group Γ and compact base space (M, g). In particular, we give two general procedures...
Generating spectral gaps by geometry (2004)
Motivated by the analysis of Schr\"odinger operators with periodic potentials we consider the following abstract situation: Let $\Delta_X$ be the Laplacian on a non-compact Riemannian covering...
Convergence of spectra of graph-like thin manifolds (2003)
We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace-Beltrami operator converges to the...
Abstract. We consider a non-compact Riemannian periodic manifold such that the corresponding Laplacian has a spectral gap. By continuously perturbing the periodic metric locally we can prove the...
Spectral Gaps for Periodic Elliptic Operators with High Contrast: an Overview (2002)
We discuss the band-gap structure and the integrated density of states for periodic elliptic operators in the Hilbert space $L_2(\R^m)$, for $m \ge 2$. We specifically consider situations where high...
Periodic Manifolds with Spectral Gaps (2002)
We investigate spectral properties of the Laplace operator on a class of non-compact Riemannian manifolds. For a given number $N$ we construct periodic (i.e. covering) manifolds such that the...
Eigenvalues in Spectral Gaps of a Perturbed Periodic Manifold (2002)
We consider a non-compact Riemannian periodic manifold such that the corresponding Laplacian has a spectral gap. By continuously perturbing the periodic metric locally we can prove the existence of...