arXiv subjects

Olaf Post

Publications and source records attributed to Olaf Post.

51 records · Page 3Linked to original sources

arXiv2008

Equilateral quantum graphs and boundary triples

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 boundary conditions on the metric graph. In case of the standard (also named ``Kirchhoff'') boundary conditions, the discrete operator is the usual combinatorial Laplacian.

math-ph↗

arXiv2007

Existence of spectral gaps, covering manifolds and residually finite groups

In the present paper we consider Riemannian coverings $(X,g) \to (M,g)$ with residually finite covering group $Γ$ and compact base space $(M,g)$. In particular, we give two general procedures resulting in a family of deformed coverings $(X,g_\eps) \to (M,g_\eps)$ such that the spectrum of the Laplacian $Δ_{(X_\eps,g_\eps)}$ has at least a prescribed finite number of spectral gaps provided $\eps$ is small enough. If $Γ$ has a positive Kadison constant, then we can apply results by Brüning and Sunada to deduce that $\spec Δ_{(X,g_\eps)}$ has, in addition, band-structure and there is an asymptotic estimate for the number $N(λ)$ of components of $\spec {\laplacian {(X,g_\eps)}}$ that intersect the interval $[0,λ]$. We also present several classes of examples of residually finite groups that fit with our construction and study their interrelations. Finally, we mention several possible applications for our results.

math-ph↗

arXiv2007

Quantum networks modelled by graphs

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 functions at the graph vertices. In this article we give a survey, technically not too heavy, of several recent results which serve this purpose. Specifically, we consider approximations by means of ``fat graphs'' -- in other words, suitable families of shrinking manifolds -- and discuss convergence of the spectra and resonances in such a setting.

math-ph↗

arXiv2007

First order operators and boundary triples

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 boundary; a case in which the ordinary concept of boundary triples does not apply directly. In our first order approach, we show that we can use the usual boundary operators also in the abstract Green's formula. Another motivation for the first order approach is to give an intrinsic definition of the Dirichlet-to-Neumann map and intrinsic norms on the corresponding boundary spaces. We also show how the first order boundary triples can be used to define a usual boundary triple leading to a Dirac operator.

math.SP↗

arXiv2007

First order approach and index theorems for discrete and metric graphs

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 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.

math.SP↗

arXiv2007

Continuity properties of the integrated density of states on manifolds

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 is given along with examples of operators with both continuous and discontinuous IDS'. Subsequently, alloy-type perturbations of the periodic operator are considered. The randomness may enter both via the potential and the metric. A Wegner estimate is proven which implies the continuity of the corresponding IDS. This gives an example of a discontinuous "periodic" IDS which is regularized by a random perturbation.

math.SP↗

arXiv2007

Convergence of resonances on thin branched quantum wave guides

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 branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances on $X_\eps$ approximate those of the Laplacian with ``free'' boundary conditions on $X_0$, the skeleton graph of $X_\eps$.

math-ph↗

arXiv2007

On the spectra of carbon nano-structures

An explicit derivation of dispersion relations and spectra for periodic Schrödinger operators on carbon nano-structures (including graphen and all types of single-wall nano-tubes) is provided.

math-ph↗

arXiv2006

Spectral convergence of non-compact quasi-one-dimensional spaces

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 (Neumann) Laplacian $\laplacian {X_\eps}$ and the generalised Neumann (Kirchhoff) Laplacian $\laplacian {X_0}$ on the metric graph. In particular, we show the norm convergence of the resolvents, spectral projections and eigenfunctions. As a consequence, the essential and the discrete spectrum converge as well. Neither the manifolds nor the metric graph need to be compact, we only need some natural uniformity assumptions. We provide examples of manifolds having spectral gaps in the essential spectrum, discrete eigenvalues in the gaps or even manifolds approaching a fractal spectrum. The convergence results will be given in a completely abstract setting dealing with operators acting in different spaces, applicable also in other geometric situations.

math-ph↗

arXiv2005

Generating spectral gaps by geometry

Motivated by the analysis of Schrödinger operators with periodic potentials we consider the following abstract situation: Let $Δ_X$ be the Laplacian on a non-compact Riemannian covering manifold $X$ with a discrete isometric group $Γ$ acting on it such that the quotient $X/Γ$ is a compact manifold. We prove the existence of a finite number of spectral gaps for the operator $Δ_X$ associated with a suitable class of manifolds $X$ with non-abelian covering transformation groups $Γ$. This result is based on the non-abelian Floquet theory as well as the Min-Max-principle. Groups of type I specify a class of examples satisfying the assumptions of the main theorem.

math-ph↗

arXiv2005

Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case

We consider a family of open sets $M_ε$ which shrinks with respect to an appropriate parameter $ε$ to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted Dirichlet spectrum of $M_ε$ converges to the spectrum of the (differential) Laplacian on the graph with Dirichlet boundary conditions at the vertices, i.e., a graph operator without coupling between different edges. The smallness is expressed by a lower bound on the first eigenvalue of a mixed eigenvalue problem on the vertex neighbourhood. The lower bound is given by the first transversal mode of the edge neighbourhood. We also allow curved edges and show that all bounded eigenvalues converge to the spectrum of a Laplacian acting on the edge with an additional potential coming from the curvature.

math-ph↗

arXiv2004

Convergence of spectra of graph-like thin manifolds

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 spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices.

math-ph↗

arXiv2002

Spectral Gaps for Periodic Elliptic Operators with High Contrast: an Overview

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 contrast in the coefficients leads to weak coupling between the period cells. Weak coupling of periodic systems frequently produces spectral gaps or spectral concentration. Our examples include Schrödinger operators, elliptic operators in divergence form, Laplace-Beltrami-operators, Schrödinger and Pauli operators with periodic magnetic fields. There are corresponding applications in heat and wave propagation, quantum mechanics, and photonic crystals.

math-ph↗

arXiv2002

Periodic Manifolds with Spectral Gaps

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 essential spectrum of the corresponding Laplacian has at least $N$ open gaps. We use two different methods. First, we construct a periodic manifold starting from an infinite number of copies of a compact manifold, connected by small cylinders. In the second construction we begin with a periodic manifold which will be conformally deformed. In both constructions, a decoupling of the different period cells is responsible for the gaps.

math-ph↗

arXiv2002

Eigenvalues in Spectral Gaps of a Perturbed Periodic Manifold

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 eigenvalues in a gap. A lower bound on the number of eigenvalue branches crossing a fixed level is established in terms of a discrete eigenvalue problem. Furthermore, we discuss examples of perturbations leading to infinitely many eigenvalue branches coming from above resp. finitely many branches coming from below.

math-ph↗