Absence Of Singular Continuous Spectrum Research Articles

The absence of singular continuous spectrum for perturbed Jacobi operators

The absence of singular continuous spectrum for perturbed Jacobi operators

Absence of singular continuous spectrum for perturbed discrete Schrödinger operators

We show that the spectral measure of discrete Schr\"odinger operators $ (Hu)(n)= u({n+1})+u({n-1})+V(n)u(n)$ does not have singular continuous component if the potential $V(n)=O(n^{-1})$.

Quantum walks with an anisotropic coin I: spectral theory

We perform the spectral analysis of the evolution operator U of quantum walks with an anisotropic coin, which include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. In particular, we determine the essential spectrum of U, we show the existence of locally U-smooth operators, we prove the discreteness of the eigenvalues of U outside the thresholds, and we prove the absence of singular continuous spectrum for U. Our analysis is based on new commutator methods for unitary operators in a two-Hilbert spaces setting, which are of independent interest.

Titchmarsh–Weyl theory for Schrödinger operators on unbounded domains

In this paper it is proved that the complete spectral data of selfadjoint Schrödinger operators on unbounded domains can be described with an associated Dirichlet-to-Neumann map. In particular, a characterization of the isolated and embedded eigenvalues, the corresponding eigenspaces, as well as the continuous and absolutely continuous spectrum in terms of the limiting behaviour of the Dirichlet-to-Neumann map is obtained. Furthermore, a sufficient criterion for the absence of singular continuous spectrum is provided. The results are natural multidimensional analogs of classical facts from singular Sturm-Liouville theory.

Spectral theory of higher order differential operators by examples

Asymptotic integration has turned out to be a powerful method to determine the deficiency indices and spectra of higher order differential operators. Since the general method is by now well established we shall only outline this method and illustrate typical results and properties via examples. In addition to the calculation of deficiency indices, the location and multiplicity of the absolutely continuous spectrumwill be found as well as showing the absence of singular continuous spectrum. Finite singular pointswill also be considered. For unbounded coefficients new results arise from competing terms of the operators.

The Spectral Structure of the Electronic Black Box Hamiltonian

We give results on the absence of singular continuous spectrum of the one-particle Hamiltonian underlying the electronic black box model.

Schrödinger operators with slowly decaying Wigner–von Neumann type potentials

We consider Schrödinger operators with potentials satisfying a generalized bounded variation condition at infinity and an L^p decay condition. This class of potentials includes slowly decaying Wigner–von Neumann type potentials \sin(ax)/x^b with b>0 . We prove absence of singular continuous spectrum and show that embedded eigenvalues in the continuous spectrum can only take values from an explicit finite set. Conversely, we construct examples where such embedded eigenvalues are present, with exact asymptotics for the corresponding eigensolutions.

Scattering theory for Riemannian Laplacians

We introduce a notion of scattering theory for the Laplace–Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another condition is certain bounds of derivatives up to order one of the trace of this quantity. These conditions are shown to be optimal for existence and completeness of a wave operator. Our theory does not involve prescribed asymptotic behavior of the metric at infinity (like asymptotic Euclidean or hyperbolic metrics studied previously in the literature). A consequence of the theory is spectral theory for the Laplace–Beltrami operator including identification of the continuous spectrum and absence of singular continuous spectrum.

Mourre estimates for compatible Laplacians on complete manifolds with corners of codimension 2

We apply Mourre theory to compatible Laplacians on complete manifolds with corners of codimension 2 in order to prove absence of singular continuous spectrum, that non-threshold eigenvalues have finite multiplicity and can accumulate only at thresholds or infinity. It turns out that we need Mourre estimates on manifolds with cylindrical ends where the results are both expected and consequences of more general theorems. In any case, we also provide a description, interesting in its own, of Mourre theory in this context that makes our text complete and suggests generalizations to manifolds with corners of higher order codimension. We use theorems of functional analysis that are suitable for these geometric applications.

Spectral properties for perturbations of unitary operators

Consider a unitary operator U 0 acting on a complex separable Hilbert space H . In this paper we study spectral properties for perturbations of U 0 of the type, U β = U 0 e i K β , with K a compact self-adjoint operator acting on H and β a real parameter. We apply the commutator theory developed for unitary operators in Astaburuaga et al. (2006) [1] to prove the absence of singular continuous spectrum for U β . Moreover, we study the eigenvalue problem for U β when the unperturbed operator U 0 does not have any. A typical example of this situation corresponds to the case when U 0 is purely absolutely continuous. Conditions on the eigenvalues of K are given to produce eigenvalues for U β for both cases finite and infinite rank of K, and we give an example where the results can be applied.

Spectral Analysis of Four Term Differential Operator

By strengthening dichotomy condition and weakening decay conditions, we show that a four term 2n-th order differential operator with unbounded coefficients is nonlimit-point. Using stringent conditions we show that the deficiency index of this operator is determined by the behaviour of the coefficients themselves. Similarly, we prove the absence of singular continuous spectrum and that the absolutely continuous spectrum has multiplicity two.

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 limit 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.

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.

On perturbations of Dirac operators with variable magnetic field of constant direction

We carry out the spectral analysis of 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 pertubations, we obtain a limiting absorption principle, we prove the absence of singular continuous spectrum in certain intervals and state properties of the point spectrum. Various situations, for example when the magnetic field is constant, periodic or diverging at infinity, are covered. The importance of an internal-type operator (a 2-dimensional Dirac operator) is also revealed in our study. The proofs rely on commutator methods.

The nature of the essential spectrum in curved quantum waveguides

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 the strips embedded in abstract surfaces.

The Nelson Model with Less Than Two Photons

We study the spectral and scattering theory of the Nelson model for an atom interacting with a photon field in the subspace with less than two photons. For the free electron-photon system, the spectral property of the reduced Hamiltonian in the center of mass coordinates and the large time dynamics are determined. If the electron is under the influence of the nucleus via spatially decaying potentials, we locate the essential spectrum, prove the absence of singular continuous spectrum and the existence of the ground state, and construct wave operators giving the asymptotic dynamics.

On the spectral properties of Hamiltonians without conservation of the particle number

We consider quantum systems with variable but finite number of particles. For such systems we develop geometric and commutator techniques. We use these techniques to find the location of the spectrum, to prove absence of singular continuous spectrum, and identify accumulation points of the discrete spectrum. The fact that the total number of particles is bounded allows us to give relatively elementary proofs of these basic results for an important class of many-body systems with nonconserved number of particles.

Measures of Fermi Surfaces and Absence of Singular Continuous Spectrum for Magnetic Schrödinger Operators

Fermi surfaces are basic objects in solid state physics and in the spectral theory of periodic operators. We define several measures connected to Fermi surfaces and study their measure theoretic properties. From this we get absence of singular continuous spectrum and of singular continuous components in the density of states for symmetric periodic elliptic di .erential operators acting on vector bundles. This includes Schrodinger operators with periodic magnetic field and rational flux, as well as the corresponding Pauli and Dirac-type operators.

Spectra of Anderson type models with decaying randomness

In this paper we consider some Anderson type models, with free parts having long range tails and with the random perturbations decaying at different rates in different directions and prove that there is a.c. spectrum in the model which is pure. In addition, we show that there is pure point spectrum outside some interval. Our models include potentials decaying in all directions in which case absence of singular continuous spectrum is also shown.

Spectral and Scattering Theory¶of Spatially Cut-Off P(ϕ)2 Hamiltonians

We study spatially cut-off P( ϕ)2 Hamiltonians. We show the local finiteness of the pure point spectrum outside of thresholds, the limiting absorption principle and asymptotic completeness of scattering for such Hamiltonians. Our results imply the absence of singular continuous spectrum.