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.     

Distant Perturbations of the Laplacian in a Multi-Dimensional Space

We consider the Laplacian in perturbed by a finite number of distant perturbations that are abstract localized operators. We study the asymptotic behaviour of the discrete spectrum as the distances between perturbations tend to infinity. The main results are a convergence theorem and the asymptotic expansions for the eigenelements. Some examples of the possible distant perturbations are given; they may be a multiplication operator, second order differential operator, magnetic Schr?dinger operator, integral operator, and δ-potential. Submitted: January 18, 2007. Accepted: March 8, 2007.     

On the spectrum of the Laplacian

In this article we prove a generalization of Weyl’s criterion for the essential spectrum of a self-adjoint operator on a Hilbert space. We then apply this criterion to the Laplacian on functions over open manifolds and get new results for its essential spectrum.     

The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to ± or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities.Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Laplacian on functions and the Dirac operator for certain spin structures have constant spectrum while the Laplacian on 1-forms and the Dirac operator for the other spin structures have nonconstant spectrum. The marked length spectrum is also constant for these families.     

Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics

We consider a magnetic Schrödinger operator in a planar infinite strip with frequently and non-periodically alternating Dirichlet and Robin boundary conditions. Assuming that the homogenized boundary condition is the Dirichlet or the Robin one, we establish the uniform resolvent convergence in various operator norms and we prove the estimates for the rates of convergence. It is shown that these estimates can be improved by using special boundary correctors. In the case of periodic alternation, pure Laplacian, and the homogenized Robin boundary condition, we construct two-terms asymptotics for the first band functions, as well as the complete asymptotics expansion (up to an exponentially small term) for the bottom of the band spectrum.     

The eigenvalues of the Laplacian on locally finite networks

We consider the continuous Laplacian on an infinite locally finite network with equal edge lengths under natural transition conditions as continuity at the ramification nodes and classical Kirchhoff conditions at all vertices. It is shown that eigenvalues of the Laplacian in a L^∞-setting are closely related to those of the adjacency and transition operator of the network. In this way the point spectrum is determined completely in terms of combinatorial quantities and properties of the underlying graph as in the finite case [2]. Moreover, the occurrence of infinite geometric multiplicity on trees and some periodic graphs is investigated.     

Properties of eigenvalues and spectral singularities for impulsive quadratic pencil of difference operators

In this paper, we investigate the spectral analysis of impulsive quadratic pencil of difference operators. We first present a boundary value problem consisting one interior impulsive point on the whole axis corresponding to the above mentioned operator. After introducing the solutions of impulsive quadratic pencil of difference equation, we obtain the asymptotic equation of the function related to the Wronskian of these solutions to be helpful for further works, then we determine resolvent operator and continuous spectrum. Finally, we provide sufficient conditions guarenteeing finiteness of eigenvalues and spectral singularities by means of uniqueness theorems of analytic functions. The main aim of this paper is demonstrating the impulsive quadratic pencil of difference operator is of finite number of eigenvalues and spectral singularities with finite multiplicities which is an uninvestigated problem proposed in the literature.     

Calculus on the Sierpinski gasket I: polynomials, exponentials and power series

We study the analog of power series expansions on the Sierpinski gasket, for analysis based on the Kigami Laplacian. The analog of polynomials are multiharmonic functions, which have previously been studied in connection with Taylor approximations and splines. Here the main technical result is an estimate of the size of the monomials analogous to xⁿ/n!. We propose a definition of entire analytic functions as functions represented by power series whose coefficients satisfy exponential growth conditions that are stronger than what is required to guarantee uniform convergence. We present a characterization of these functions in terms of exponential growth conditions on powers of the Laplacian of the function. These entire analytic functions enjoy properties, such as rearrangement and unique determination by infinite jets, that one would expect. However, not all exponential functions (eigenfunctions of the Laplacian) are entire analytic, and also many other natural candidates, such as the heat kernel, do not belong to this class. Nevertheless, we are able to use spectral decimation to study exponentials, and in particular to create exponentially decaying functions for negative eigenvalues.     

On the spectrum of magnetic Dirac operators with Coulomb-type perturbations

We carry out the spectral analysis of singular matrix valued perturbations of 3-dimensional Dirac operators with variable magnetic field of constant direction. Under suitable assumptions on the magnetic field and on the perturbations, we obtain a limiting absorption principle, we prove the absence of singular continuous spectrum in certain intervals and state properties of the point spectrum. Constant, periodic as well as diverging magnetic fields are covered, and Coulomb potentials up to the physical nuclear charge Z<137 are allowed. The importance of an internal-type operator (a 2-dimensional Dirac operator) is also revealed in our study. The proofs rely on commutator methods.     

Spectral Asymptotics Revisited

We discuss two heuristic ideas concerning the spectrum of a Laplacian, and we give theorems and conjectures from the realms of manifolds, graphs and fractals that validate these heuristics. The first heuristic concerns Laplacians that do not have discrete spectra: here we discuss the notion of ??spectral mass?? closely related to the ??integrated density of states??, an average of the diagonal of the kernel of the spectral projection operator, and show that this can serve as a substitute for the eigenvalue counting function. The second heuristic is an ??asymptotic splitting law?? that describes the proportions of the spectrum that transforms according to the irreducible representations of a finite group that acts as a symmetry group of the Laplacian. For this to be valid we require the existence of a fundamental domain with relatively small boundary. We also give a version in the case that the symmetry group is a compact Lie group. Many of our results are reformulations of known results, and some are merely conjectures, but there is something to be gained by looking at them together with a unified perspective.     

Detecting the birth and death of finite-time coherent sets

Finite-time coherent sets (FTCSs) are distinguished regions of phase space that resist mixing with the surrounding space for some finite period of time; physical manifestations include eddies and vortices in the ocean and atmosphere, respectively. The boundaries of FTCSs are examples of Lagrangian coherent structures (LCSs). The selection of the time duration over which FTCS and LCS computations are made in practice is crucial to their success. If this time is longer than the lifetime of coherence of individual objects then existing methods will fail to detect the shorter-lived coherence. It is of clear practical interest to determine the full lifetime of coherent objects, but in complicated practical situations, for example a field of ocean eddies with varying lifetimes, this is impossible with existing approaches. Moreover, determining the timing of emergence and destruction of coherent sets is of significant scientific interest. In this work we introduce new constructions to address these issues. The key components are an inflated dynamic Laplace operator and the concept of semi-material FTCSs. We make strong mathematical connections between the inflated dynamic Laplacian and the standard dynamic Laplacian, showing that the latter arises as a limit of the former. The spectrum and eigenfunctions of the inflated dynamic Laplacian directly provide information on the number, lifetimes, and evolution of coherent sets.     

The laplace operator on normal homogeneous Riemannian manifolds

The article presents an information about the Laplace operator defined on the real-valued mappings of compact Riemannian manifolds, and its spectrum; some properties of the latter are studied. The relationship between the spectra of two Riemannian manifolds connected by a Riemannian submersion with totally geodesic fibers is established. We specify a method of calculating the spectrum of the Laplacian for simply connected simple compact Lie groups with biinvariant Riemannian metrics, by representations of their Lie algebras. As an illustration, the spectrum of the Laplacian on the group SU(2) is found.     

Which subnormal Toeplitz operators are either normal or analytic?

We study subnormal Toeplitz operators on the vector-valued Hardy space of the unit circle, along with an appropriate reformulation of P.R. Halmos?s Problem 5: Which subnormal block Toeplitz operators are either normal or analytic? We extend and prove Abrahamse?s theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two bounded analytic functions), whose analytic and co-analytic parts have the “left coprime factorization”, is normal or analytic. We also prove that the left coprime factorization condition is essential. Finally, we examine a well-known conjecture, of whether every subnormal Toeplitz operator with finite rank self-commutator is normal or analytic.     

Study of Volterra Integro-Differential Equations with Kernels Depending on a Parameter

We carry out spectral analysis of operator functions that are the symbols of integro-differential equations with unbounded operator coefficients in a separable Hilbert space. The structure and localization of the spectrum of operator functions which are symbols of these equations play an important role in studies of the asymptotic behavior of their solutions.     

Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows

We consider a planar waveguide modeled by the Laplacian in a straight infinite strip with the Dirichlet boundary condition on the upper boundary and with frequently alternating boundary conditions (Dirichlet and Neumann) on the lower boundary. The homogenized operator is the Laplacian subject to the Dirichlet boundary condition on the upper boundary and to the Dirichlet or Neumann condition on the lower one. We prove the uniform resolvent convergence for the perturbed operator in both cases and obtain the estimates for the rate of convergence. Moreover, we construct the leading terms of the asymptotic expansions for the first band functions and the complete asymptotic expansion for the bottom of the spectrum. Bibliography: 17 titles. Illustrations: 3 figures.     

The Quadratic Numerical Range of an Analytic Operator Function

We introduce a new concept of numerical range for analytic operator functions. This so-called quadratic numerical range is induced by a decomposition of the underlying Hilbert space. Like the classical numerical range of an operator function, it is not convex, it has the spectral inclusion property, and it provides resolvent estimates and estimates for the lengths of Jordan chains in boundary points having the exterior cone property. As the quadratic numerical range is contained in the numerical range, it yields tighter enclosures for the eigenvalues and the spectrum of analytic operator functions.     

On the extrema of band functions in periodic waveguides

In the work we construct an example of a periodic differential operator whose spectrum has gaps with edges attained by the band functions at interior points of the Brillouin zone. This example is the Laplacian on a pair of infinite parallel strips with common boundary from which a periodic system of small holes is cut out. At that, on the outer boundaries the Dirichlet condition is imposed, while on the common boundary the Neumann condition is considered.     

On the Laplacian spectrum of an infinite graph

In this paper, we introduce the notion of Laplacian spectrum of an infinite countable graph in a different way than in the papers by B. Mohar. We prove some basic properties of this type of spectrum. The approach used is in line with our approach to the limiting spectrum of an infinite graph. The technique of the Laplacian spectrum of finite graphs is essential in this approach.     

Wave Invariants of the Spectrum of the G-Invariant Laplacian and the Basic Spectrum of a Riemannian Foliation

Given a compact boundaryless Riemannian manifold upon which a compact Lie group G acts by isometries, recall that the G-invariant Laplacian is the restriction of the ordinary Laplacian on functions to the space of functions which are constant along the orbits of the action. By considering the wave trace of the invariant Laplacian and the connection between G manifolds and Riemannian foliations, invariants of the spectrum of the G-invariant Laplacian can be computed. These invariants include the lengths of certain geodesic arcs which are orthogonal to the principal orbits and contained in the open dense set of principal orbits are associated to the singularities of the wave trace of the G-invariant spectrum. If the action admits finite orbits, then the invariants also include the lengths of certain geodesics arcs connecting the finite orbit to itself. Under additional hypotheses, we obtain partial wave traces. As an application, a partial trace formula for Riemannian foliations with bundle-like metrics is also presented, as well as several special cases where better results are available.