Spectral Theory for Discrete Laplacians

We give the spectral representation for a class of selfadjoint discrete graph Laplacians Δ, with Δ depending on a chosen graph G and a conductance function c defined on the edges of G. We show that the spectral representations for Δ fall in two model classes, (1) tree-graphs with N-adic branching laws, and (2) lattice graphs. We show that the spectral theory of the first class may be computed with the use of rank-one perturbations of the real part of the unilateral shift, while the second is analogously built up with the use of the bilateral shift. We further analyze the effect on spectra of the conductance function c: How the spectral representation of Δ depends on c.     

Spectral asymptotics for Laplacians on self-similar sets

Given a self-similar Dirichlet form on a self-similar set, we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a ‘geometric counting function’ defined through a family of coverings of the self-similar set naturally associated with the Dirichlet space. Secondly, under (sub-)Gaussian heat kernel upper bound, we prove a detailed short time asymptotic behavior of the partition function, which is the Laplace-Stieltjes transform of the eigenvalue counting function associated with the Dirichlet form. This result can be applicable to a class of infinitely ramified self-similar sets including generalized Sierpinski carpets, and is an extension of the result given recently by B.M. Hambly for the Brownian motion on generalized Sierpinski carpets. Moreover, we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function.     

Spectral Analysis of Magnetic Laplacians on Conformally Cusp Manifolds

We consider an open manifold which is the interior of a compact manifold with boundary. Assuming gauge invariance, we classify magnetic fields with compact support into being trapping or non-trapping. We study spectral properties of the associated magnetic Laplacian for a class of Riemannian metrics which includes complete hyperbolic metrics of finite volume. When B is non-trapping, the magnetic Laplacian has nonempty essential spectrum. Using Mourre theory, we show the absence of singular continuous spectrum and the local finiteness of the point spectrum. When B is trapping, the spectrum is discrete and obeys the Weyl law. The existence of trapping magnetic fields with compact support depends on cohomological conditions, indicating a new and very strong long-range effect. In the non-gauge invariant case, we exhibit a strong Aharonov–Bohm effect. On hyperbolic surfaces with at least two cusps, we show that the magnetic Laplacian associated to every magnetic field with compact support has purely discrete spectrum for some choices of the vector potential, while other choices lead to a situation of limiting absorption principle. We also study perturbations of the metric. We show that in the Mourre theory it is not necessary to require a decay of the derivatives of the perturbation. This very singular perturbation is then brought closer to the perturbation of a potential. Submitted: February 6, 2007. Accepted: August 20, 2007.     

Spectral theory for perturbed Krein Laplacians in nonsmooth domains

We study spectral properties for HK,Ω, the Krein-von Neumann extension of the perturbed Laplacian −Δ+V defined on , where V is measurable, bounded and nonnegative, in a bounded open set Ω⊂Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r>1/2. In particular, in the aforementioned context we establish the Weyl asymptotic formula

     

Spectral multipliers for Laplacians with drift on Damek–Ricci spaces

We prove a multiplier theorem for certain Laplacians with drift on Damek–Ricci spaces, which are a class of Lie groups of exponential growth. Our theorem generalizes previous results obtained by W. Hebisch, G. Mauceri and S. Meda on Lie groups of polynomial growth.     

Second-Order Differential Equations Exhibiting Spectral Gaps

We consider the numerical solution of a linear second-order differential equation, which is an underlying equation in many engineering applications. The response of such a system is typically dominated by a relatively small number of the lowfrequency modes. Using projections on certain eigenspaces or generalized eigenspaces of the system matrix that are induced by the matrix sign function, it is shown that the original problem may be reduced to a smaller subproblem that may be handled with significantly less computational work. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Minimax Principle for the Eigenvalues in Spectral Gaps

A minimax principle is derived for the eigenvalues in the spectralgap of a possibly non-semibounded self-adjoint operator. Itallows the nth eigenvalue of the Dirac operator with Coulombpotential from below to be bound by the nth eigenvalue of asemibounded Hamiltonian which is of interest in the contextof stability of matter. As a second application it is shownthat the Dirac operator with suitable non-positive potentialhas at least as many discrete eigenvalues as the Schrödingeroperator with the same potential.     

Operator Theoretic Methods for the Eigenvalue Counting Function in Spectral Gaps

Using the notion of spectral flow, we suggest a simple approach to various asymptotic problems involving eigenvalues in the gaps of the essential spectrum of self-adjoint operators. Our approach employs some elements of the theory of the spectral shift function. Using this approach, we provide generalisations and streamlined proofs of two results in this area already existing in the literature. We also give a new proof of the generalised Birman–Schwinger principle. Submitted: March 10, 2009. Accepted: April 22, 2009.     

Spectral Gaps in the XXZ Quantum Spin Chain

We study the behavior of the finite-size energy spectrum of the HeisenbergIsing (XXZ) quantum spin chain in the momentum sector. Our numerical and perturbative results for the behavior of energy gaps are explained by considering the motion of string solutions of the Bethe-Ansatz equations. Bibliography: 13 titles.     

Wave Equation on One-Dimensional Fractals with Spectral Decimation and the Complex Dynamics of Polynomials

We study the wave equation on one-dimensional self-similar fractal structures that can be analyzed by the spectral decimation method. We develop efficient numerical approximation techniques and also provide uniform estimates obtained by analytical methods.     

Supersymmetric Dirichlet Operators, Spectral Gaps, and Correlations

In this article we construct a supersymmetric extension of the Dirichlet operator associated with a tempered Gibbs measure on $$ \user2{\mathbb{R}}^{\user2{\mathbb{Z}}^d }. $$ Under fairly general assumptions on the interaction potentials we show that the Dirichlet operator (resp. its supersymmetric extension) is essentially selfadjoint on the set of smooth, bounded cylinder functions (resp. differential forms), for all inverse temperatures. Assuming that the on-site potentials have a non-degenerated minimum and no other critical point we prove that, for sufficiently large inverse temperatures, one observes a number of subsequent gaps in the spectrum of the Dirichlet operator. For translation invariant systems with a sufficiently weak (but in general infinite range) ferromagnetic interaction, we prove the validity of a formula for the leading asymptotics of the correlation of two spin variables, as their distance and the inverse temperature tend to infinity, which has originally been derived by J. Sj?strand for finite-dimensional systems. Communicated by Bernard Helffer. Submitted: 14/05/ 2005 Accepted: 22/05/2005     

Laplacians on quantum hypergraphs

We introduce quantum hypergraphs, in analogy with the theory of quantum graphs developed over the last 15 years by many authors. We emphasize some problems that arise when one tries to define a Laplacian on a hypergraph. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Laplacians for semidiscrete surfaces

We study a Laplace operator on semidiscrete surfaces that is defined by variation of the Dirichlet energy functional. We show existence and its relation to the mean curvature normal, which is itself defined via variation of area. We establish several core properties like linear precision (closely related to the mean curvature of flat surfaces), and pointwise convergence. It is interesting to observe how a certain freedom in choosing area measures yields different kinds of Laplacians: it turns out that using as a measure a simple numerical integration rule yields a Laplacian previously studied as the pointwise limit of geometrically meaningful Laplacians on polygonal meshes.     

Hardy inequalities for Robin Laplacians

In this paper we establish a Hardy inequality for Laplace operators with Robin boundary conditions. For convex domains, in particular, we show explicitly how the corresponding Hardy weight depends on the coefficient of the Robin boundary conditions. We also study several extensions to non-convex and unbounded domains.