We study the operator theory associated with such infinite graphs $G$ as occur in electrical networks, in fractals, in statistical mechanics, and even in internet search engines. Our emphasis is on the determination of spectral data for a natural Laplace operator associated with the graph in question. This operator $Δ$ will depend not only on $G$, but also on a prescribed positive real valued function $c$ defined on the edges in $G$. In electrical network models, this function $c$ will determine a conductance number for each edge. We show that the corresponding Laplace operator $Δ$ is automatically essential selfadjoint. By this we mean that $Δ$ is defined on the dense subspace $\mathcal{D}$ (of all the real valued functions on the set of vertices $G^{0}$ with finite support) in the Hilbert space $l^{2}% (G^{0})$. The conclusion is that the closure of the operator $Δ$ is selfadjoint in $l^{2}(G^{0})$, and so in particular that it has a unique spectral resolution, determined by a projection valued measure on the Borel subsets of the infinite half-line. We prove that generically our graph Laplace operator $Δ=Δ_{c}$ will have continuous spectrum. For a given infinite graph $G$ with conductance function $c$, we set up a system of finite graphs with periodic boundary conditions such the finite spectra, for an ascending family of finite graphs, will have the Laplace operator for $G$ as its limit.
We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic incompleteness.
We study Laplacians on graphs and networks via regular Dirichlet forms. We give a sufficient geometric condition for essential selfadjointness and explicitly determine the generators of the associated semigroups on all $\ell^p$, $1\leq p < \infty$, in this case. We characterize stochastic completeness thereby generalizing all earlier corresponding results for graph Laplacians. Finally, we study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph.
We study Laplacians associated to a graph and single out a class of such operators with special regularity properties. In the case of locally finite graphs, this class consists of all selfadjoint, non-negative restrictions of the standard formal Laplacian and we can characterize the Dirichlet and Neumann Laplacians as the largest and smallest Markovian restrictions of the standard formal Laplacian. In the case of general graphs, this class contains the Dirichlet and Neumann Laplacians and we describe how these may differ from each other, characterize when they agree, and study connections to e…
We consider the problem of essential self-adjointness of the spatial part of the Klein-Gordon operator in stationary spacetimes. This operator is shown to be a Laplace-Beltrami type operator plus a potential. In globally hyperbolic spacetimes, essential selfadjointness is proven assuming smoothness of the metric components and semi-boundedness of the potential. This extends a recent result for static spacetimes to the stationary case. Furthermore, we generalize the results to certain non-globally hyperbolic spacetimes.
In this paper we characterize emptiness of the essential spectrum of the Laplacian under a hyperbolicity assumption for general graphs. Moreover we present a characterization for emptiness of the essential spectrum for planar tessellations in terms of curvature.
We address the Laplacian on a perturbed periodic graph which might not be a periodic graph. We present a class of perturbed graphs for which the essential spectra of the Laplacians are stable even when the graphs are perturbed by adding and removing infinitely many vertices and edges. Using this result, we demonstrate how to determine the spectra of cone-like graphs, the upper-half plane, and graphs obtained from $\mathbb{Z}^2$ by randomly adding vertices.
We construct Riemannian manifolds with singular continuous spectrum embedded in the absolutely continuous spectrum of the Laplacian. Our manifolds are asymptotically hyperbolic with sharp curvature bounds.
We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we construct such manifolds with dense embedded point spectrum and sharp curvature bounds.
We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp's measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This result holds under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points.
In 1983, Klaus studied a class of potentials with bumps and computed the essential spectrum of the associated Schr{ö}dinger operator with the help of some localisations at infinity. A key hypothesis is that the distance between two consecutive bumps tends to infinity at infinity. In this article, we introduce a new class of graphs (with patterns) that mimics this situation, in the sense that the distance between two patterns tends to infinity at infinity. These patterns tend, in some way, to asymptotic graphs. They are the localisations at infinity. Our result is that the essential spectrum of…
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 unifo…
We introduce the weighted graph Laplacian and the notion of Schrödinger operator on a locally finite weighted graph. Concerning essential self-adjointness, we extend Wojciechowski's and Dodziuk's results for graphs with vertex constant weight. The main result in this work states that on any metrically complete weighted graph with bounded degree, the weighted graph Laplacian is essentially self-adjoint and the same holds for the Schrödinger operator provided the associated quadratic form is bounded from below. We construct for the proof a strictly positive and harmonic function which allows us …
We consider open manifolds which are interiors of a compact manifold with boundary, and Riemannian metrics asymptotic to a conformally cylindrical metric near the boundary. We show that the essential spectrum of the Laplace operator on functions vanishes under the presence of a magnetic field which does not define an integral relative cohomology class. It follows that the essential spectrum is not stable by perturbation even by a compactly supported magnetic field. We also treat magnetic operators perturbed with electric fields. In the same context we describe the essential spectrum of the $k$…
We describe the spectrum of the $k$-form Laplacian on conformally cusp Riemannian manifolds. The essential spectrum is shown to vanish precisely when the $k$ and $k-1$ de Rham cohomology groups of the boundary vanish. We give Weyl-type asymptotics for the eigenvalue-counting function in the purely discrete case. In the other case we analyze the essential spectrum via positive commutator methods and establish a limiting absorption principle. This implies the absence of the singular spectrum for a wide class of metrics. We also exhibit a class of potentials $V$ such that the Schrödinger operator…
We introduce the notion of discrete cusp for a weighted graph. In this context, we provethat the form-domain of the magnetic Laplacian and that of thenon-magnetic Laplacian can be different. We establish the emptiness of the essential spectrum and compute theasymptotic of eigenvalues for the magnetic Laplacian.
We study the Laplacian on family preserving metric graphs. These are graphs that have a certain symmetry that, as we show, allows for a decomposition into a direct sum of one-dimensional operators whose properties are explicitly related to the structure of the graph. Such decompositions have been extremely useful in the study of Schrödinger operators on metric trees. We show that the tree structure is not essential, and moreover, obtain a direct and simple correspondence between such decompositions in the discrete and continuum case.
We discuss connections between the essential self-adjointness of a symmetric operator and the constancy of functions which are in the kernel of the adjoint of the operator. We then illustrate this relationship in the case of Laplacians on both manifolds and graphs. Furthermore, we discuss the Green's function and when it gives a non-constant harmonic function which is square integrable.
The goal of this paper is to derive estimates of eigenvalue moments for Dirichlet Laplacians and Schrödinger operators in regions having infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show how those geometric properties enter the eigenvalue bounds. The obtained inequalities reflect the essentially one-dimensional character of the cusps and we give an example showing that in an intermediate energy region they can be much stronger than the usual semiclassical bounds.
Let $Ω_-$ and $Ω_+$ be two bounded smooth domains in $\mathbb{R}^n$, $n\ge 2$, separated by a hypersurface $Σ$. For $μ>0$, consider the function $h_μ=1_{Ω_-}-μ1_{Ω_+}$. We discuss self-adjoint realizations of the operator $L_μ=-\nabla\cdot h_μ\nabla$ in $L^2(Ω_-\cupΩ_+)$ with the Dirichlet condition at the exterior boundary. We show that $L_μ$ is always essentially self-adjoint on the natural domain (corresponding to transmission-type boundary conditions at the interface $Σ$) and study some properties of its unique self-adjoint extension $\mathcal{L}_μ:=\overline{L_μ}$. If $μ\ne 1$, then $\mat…
It is known that the essential spectrum of a Schrödinger operator $H$ on $\ell^{2}\left(\mathbb{N}\right)$ is equal to the union of the spectra of right limits of $H$. The natural generalization of this relation to $\mathbb{Z}^{n}$ is known to hold as well. In this paper we generalize the notion of right limits to general infinite connected graphs and construct examples of graphs for which the essential spectrum of the Laplacian is strictly bigger than the union of the spectra of its right limits. As these right limits are trees, this result is complemented by the fact that the equality still …
Two-dimension almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane. They are generalized Riemannian structures for which the vectors of a local orthonormal frame can become parallel. Under the 2-step assumption the singular set $Z$, where the structure is not Riemannian, is a 1D embedded submanifold. While approaching the singular set, all Riemannian quantities diverge. A remarkable property of these structures is that the geodesics can cross the singular set without singularities, but the heat and the solution of the Schrödinger equation (with the Laplace-Be…
Let $(G,c)$ be an infinite network, and let $\mathcal{E}$ be the canonical energy form. Let $Δ_2$ be the Laplace operator with dense domain in $\ell^2(G)$ and let $Δ_{\mathcal{E}}$ be the Laplace operator with dense domain in the Hilbert space $\mathcal{H}_\mathcal{E}$ of finite energy functions on $G$. It is known that $Δ_2$ is essentially self-adjoint, but that $Δ_{\mathcal{E}}$ is \emph{not}. In this paper, we characterize the Friedrichs extension of $Δ_{\mathcal{E}}$ in terms of $Δ_2$ and show that the spectral measures of the two operators are mutually absolutely continuous with Radon-Nik…
We consider the Dirichlet Laplacian in tubular neighbourhoods of complete non-compact Riemannian manifolds immersed in the Euclidean space. We show that the essential spectrum coincides with the spectrum of a planar tube provided that the second fundamental form of the manifold vanishes at infinity and the transport of the cross-section along the manifold is asymptotically parallel. For low dimensions and codimensions, the result applies to the location of propagating states in nanostructures under physically natural conditions.
We study the nature of the essential spectrum of the Dirichlet Laplacian in tubes about infinite curves embedded in Euclidean spaces. Under suitable assumptions about the decay of curvatures at infinity, we prove the absence of singular continuous spectrum and state properties of possible embedded eigenvalues. The argument is based on Mourre conjugate operator method developed for acoustic multistratified domains by Benbernou and Dermenjian et al. As a technical preliminary, we carry out a spectral analysis for Schrodinger-type operators in straight Dirichlet tubes. We also apply the result to…
Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own within applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this theory has been extended to the case where the spectrum is two-dimensional, or even fractal, with di…
Motivated by potential theory on discrete spaces, we study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. These operators are discrete analogues of the classical conformal Laplacians and Hamiltonians from statistical mechanics. For an infinite discrete set $X$, we consider operators acting on Hilbert spaces of functions on $X$, and their representations as infinite matrices; the focus is on $\ell^2(X)$, and the energy space $\mathcal{H}_{\mathcal E}$. In particular, we prove that these operators are always essentially s…
Given a Hermitian vector bundle over an infinite weighted graph, we define the Laplacian associated to a unitary connection on this bundle and study the essential self-adjointness of a perturbation of this Laplacian by an operator-valued potential. Additionally, we give a sufficient condition for the resulting Schrödinger operator to serve as the generator of a strongly continuous contraction semigroup in the corresponding l^{p}-space.
