On the Finiteness of the Number of Eigenvalues of Jacobi Operators below the Essential Spectrum

We present a new oscillation criterion to determine whether the number of eigenvalues below the essential spectrum of a given Jacobi operator is finite or not. As an application we show that Kneser's criterion for Jacobi operators follows as a special case.     

Boosted Simon‐Wolff Spectral Criterion and Resonant Delocalization

Discussed here are criteria for the existence of continuous components in the spectra of operators with random potential. First, the essential condition for the Simon‐Wolff criterion is shown to be measurable at infinity. By implication, for the i.i.d. case and more generally potentials with the K‐property, the criterion is boosted by a zero‐one law. The boosted criterion, combined with tunneling estimates, is then applied for sufficiency conditions for the presence of continuous spectrum for random Schrödinger operators. The general proof strategy that this yields is modeled on the resonant delocalization arguments by which continuous spectrum in the presence of disorder was previously established for random operators on tree graphs. In another application of the Simon‐Wolff rank‐one analysis we prove the almost sure simplicity of the pure point spectrum for operators with random potentials of conditionally continuous distribution.© 2015 Wiley Periodicals, Inc.     

Effective Prüfer angles and relative oscillation criteria

We present a streamlined approach to relative oscillation criteria based on effective Prüfer angles adapted to the use at the edges of the essential spectrum.Based on this we provided a new scale of oscillation criteria for general Sturm-Liouville operators which answer the question whether a perturbation inserts a finite or an infinite number of eigenvalues into an essential spectral gap. As a special case we recover and generalize the Gesztesy-Ünal criterion (which works below the spectrum and contains classical criteria by Kneser, Hartman, Hille, and Weber) and the well-known results by Rofe-Beketov including the extensions by Schmidt.     

A criterion for the existence of the essential spectrum for beak-shaped elastic bodies

We establish a criterion for the existence of the essential spectrum for the elasticity problem in the case the traction-free surface of a finite elastic body has a beak-shaped irregularity (see Corollary 3.5 and Theorem 4.2). This boundary irregularity is angular in two dimensions and cuspidal in one dimension. We obtain further information on the spectral structure in some particular cases, and formulate open questions and hypotheses.     

A criterion for the existence of the essential spectrum for beak-shaped elastic bodies

We establish a criterion for the existence of the essential spectrum for the elasticity problem in the case the traction-free surface of a finite elastic body has a beak-shaped irregularity (see Corollary 3.5 and Theorem 4.2). This boundary irregularity is angular in two dimensions and cuspidal in one dimension. We obtain further information on the spectral structure in some particular cases, and formulate open questions and hypotheses.     

Functional inequalities for transient birth-death processes and their applications

We give a new diagram about uniform decay, empty essential spectrum and various functional inequalities, including Poincaré inequalities, super- and weak-Poincaré inequalities, for transient birth-death processes. This diagram is completely opposite to that in ergodic situation, and substantially points out the difference between transient birth-death processes and recurrent ones. The criterion for the empty essential spectrum is achieved. Some matching sufficient and necessary conditions for weak-Poincaré inequalities and super-Poincaré inequalities are also presented.     

The Gohberg Lemma,compactness, and essential spectrum of operators on compact Lie groups

We prove a version of the Gohberg Lemma on compact Lie groups giving an estimate from below for the distance from a given operator to the set of compact operators. As a consequence, we obtain several results on bounds for the essential spectrum and a criterion for an operator to be compact. The conditions are given in terms of the matrix-valued symbols of operators.     

Expression for the number of eigenvalues of a Friedrichs model

We consider the self-adjoint operator of a generalized Friedrichs model whose essential spectrum may contain lacunas. We obtain a formula for the number of eigenvalues lying on an arbitrary interval outside the essential spectrum of this operator. We find a sufficient condition for the discrete spectrum to be finite. Applying the formula for the number of eigenvalues, we show that there exist an infinite number of eigenvalues on the lacuna for a particular Friedrichs model and obtain the asymptotics for the number of eigenvalues.     

A Discreteness Criterion for the Spectrum of the Laplace–Beltrami Operator on Quasimodel Manifolds

We study the spectrum of the Laplace–Beltrami operator on noncompact Riemannian manifolds of a special form, in particular on model manifolds. We obtain a discreteness criterion for the spectrum in terms of the volume and capacity of some domains on a manifold.     

Oscillation of the perturbed Hill equation and the lower spectrum of radially periodic Schrödinger operators in the plane

Generalizing the classical result of Kneser, we show that the
Sturm-Liouville equation with periodic coefficients and an added perturbation term is oscillatory or non-oscillatory (for ) at the infimum of the essential spectrum, depending on whether surpasses or stays below a critical threshold. An explicit characterization of this threshold value is given. Then this oscillation criterion is applied to the spectral analysis of two-dimensional rotation symmetric Schrödinger operators with radially periodic potentials, revealing the surprising fact that (except in the trivial case of a constant potential) these operators always have infinitely many eigenvalues below the essential spectrum.

     

The essential spectrum of the Laplacian on rapidly branching tessellations

In this paper, we characterize absence of the essential spectrum of the Laplacian under a hyperbolicity assumption for general graphs. Moreover, we present a characterization for absence of the essential spectrum for planar tessellations in terms of curvature.     

算子的拟相似性与本质谱的性质

设算子A和B拟相似,本文通过算子谱的精密结构的分析,给出了算子A的Wolf本质谱、Kato本质谱、Weyl本质谱以及右本质谱的连通分支与算子B的Wolf本质谱的某些子集的相交关系,并肯定地回答了L．A．Fialkow在文献[3]中提出的一个问题．     

On the Finiteness of the Discrete Spectrum of a Four-Particle Lattice Schrödinger Operator

A Hamiltonian describing four bosons that move on a lattice and interact by means of pair zero-range attractive potentials is considered. A stronger version of the Hunziker–Van Vinter–Zhislin theorem on the essential spectrum is established. It is proved that the set of eigenvalues lying to the left of the essential spectrum is finite for any interaction energy of two bosons and is empty if this energy is sufficiently small.     

BROWDER AND SEMI-BROWDER OPERATORS＊

In this article,we study characterization,stability,and spectral mapping theorem for Browder's essential spectrum,Browder's essential defect spectrum and Browder's essential approximate point spectrum ...     

Spectrum of the energy operator of a two-magnon system in the three-dimensional isotropic Heisenberg ferromagnet model with impurity

We consider the energy operator of two-magnon systems in the three-dimensional isotropic Heisenberg ferromagnet model with impurity and with the nearest-neighbor interaction. We investigate the structure of the essential spectrum and discrete spectrum of the system on a three-dimensional lattice. We show that the essential spectrum consists of the union of at most four segments and that the discrete spectrum is finite at the edge of the essential spectrum.     

SPECTRAL MAPPING THEOREM FOR ASCENT,ESSENTIAL ASCENT,DESCENT AND ESSENTIAL DESCENT SPECTRUM OF LINEAR RELATIONS

In [7], Cross showed that the spectrum of a linear relation T on a normed space satisfies the spectral mapping theorem. In this paper, we extend the notion of essential ascent and descent for an operator acting on a vector space to linear relations acting on Banach spaces. We focus to define and study the descent, essential descent, ascent and essential ascent spectrum of a linear relation everywhere defined on a Banach space X. In particular, we show that the corresponding spectrum satisfy the polynomial version of the spectral mapping theorem.     

Essential spectrum of a system of singular differential operators and the asymptotic Hain-Lüst operator

We consider a matrix differential operator with singular entries which arises in magnetohydrodynamics. By means of the asymptotic Hain-Lüst operator and some pseudo-differential operator techniques, we determine the essential spectrum of this operator. Whereas in the regular case, the essential spectrum consists of two intervals, it turns out that in the singular case two additional intervals due to the singularity may arise. In addition, we establish criteria for the essential spectrum to lie in the left half-plane.

     

Perturbations of embedded eigenvalues for the planar bilaplacian

Operators on unbounded domains may acquire eigenvalues that are embedded in the essential spectrum. Determining the fate of these embedded eigenvalues under small perturbations of the underlying operator is a challenging task, and the persistence properties of such eigenvalues are linked intimately to the multiplicity of the essential spectrum. In this paper, we consider the planar bilaplacian with potential and show that the set of potentials for which an embedded eigenvalue persists is locally an infinite-dimensional manifold with infinite codimension in an appropriate space of potentials.     

Spectral mapping theorem for ascent,essential ascent,descent and essential descent spectrum of linear relations

In [7], Cross showed that the spectrum of a linear relation T on a normed space satisfies the spectral mapping theorem. In this paper, we extend the notion of essential ascent and descent for an operator acting on a vector space to linear relations acting on Banach spaces. We focus to define and study the descent, essential descent, ascent and essential ascent spectrum of a linear relation everywhere defined on a Banach space X. In particular, we show that the corresponding spectrum satisfy the polynomial version of the spectral mapping theorem.     

Spectral approximation by L 2 discretization of the Laplace operator on manifolds of bounded geometry

The author considers a discretization of the p-form Laplacian on open complete Riemannian manifolds of bounded geometry. Following Dodziuk and Patodi [8], the eigenvalues below the essential spectrum together with their eigenforms are approximated by eigenvalues and eigencochains of a semicombinatorical Laplacian acting on L ₂-cochains. We obtain a similar result for a ray which is contained in the essential spectrum. An example of a manifold of bounded geometry which admits eigenvalues below the essential spectrum is constructed.