Structure of the spectrum of infinite dimensional Hamiltonian operators

This paper deals with the structure of the spectrum of infinite dimensional Hamiltonian operators.It is shown that the spectrum,the union of the point spectrum and residual spectrum,and the continuous spectrum are all symmetric with respect to the imaginary axis of the complex plane. Moreover,it is proved that the residual spectrum does not contain any pair of points symmetric with respect to the imaginary axis;and a complete characterization of the residual spectrum in terms of the point spectrum is then given.As applications of these structure results,we obtain several necessary and sufficient conditions for the residual spectrum of a class of infinite dimensional Hamiltonian operators to be empty.     

关于赋权树的谱半径的精确上界

The spectrum of weighted graphs are often used to solve the problems in the design of networks and electronic circuits. In this paper, we derive the sharp upper bound of spectral radius of all weighted trees on given order and edge independence number, and obtain all such trees that their spectral radius reach the upper bound.     

The generalized regular points and narrow spectrum points of bounded linear operators on Hilbert spaces

In this paper, we introduce the concepts of generalized regular points and narrow spectrum points of bounded linear operators on Hilbert spaces. The concept of generalized regular points is an extension of the concept regular points, and so, the set of all spectrum points is reduced to the narrow spectrum. We present not only the same and different properties of spectrum and of narrow spectrum but also show the relationship between them. Finally, the well known problem about the invariant subspaces of bounded linear operators on separable Hilbert spaces is simplified to the problem of the operator with narrow spectrum only.     

Weyl Spectrum of Upper Triangular Operator Matrices

This paper is concerned with general n × n upper-triangular operator matrices with given diagonal entries. The characterizations of perturbations of their left(resp. right) Weyl spectrum and Weyl spectrum are given, based on the space decomposition technique. Moreover, some sufficient and necessary conditions are given under which the left(resp. right) Weyl spectrum and the Weyl spectrum of such operator matrix, respectively, coincide with the union of the left(resp. right) Weyl spectrum and the Weyl spectrum of its diagonal entries.     

Hermitian Adjacency Spectrum of Cayley Digraphs over Dihedral Group

We first study the spectrum of Hermitian adjacency matrix(H-spectrum)of Cayley digraphs X(D 2n,S)on dihedral group D2n with|S|=3.Then we show that all Cayley digraphs X(D2P,S)with|S|=3 and p odd prime are Cay-DS,namely,for any Cayley digraph X(D2P,T),X(D2P,T)and X(D2P,S)having the same H-spectrum implies that they are isomorphic.     

NOTES ON THE SPECTRAL PROPERTIES OF THE WEIGHTED MEAN DIFFERENCE OPERATOR G(u,v;?) OVER THE SEQUENCE SPACE ?_1

In the study by Baliarsingh and Dutta [Internat. J.Anal., Vol.2014(2014), Article ID 786437], the authors computed the spectrum and the fine spectrum of the product operator G(u, v; ?) over the sequence space ?_1. The product operator G(u, v; ?) over ?_1 is defined by(G(u,v; ?) x)_k=k∑i=1u_kv_i(x_i-x_(i-1)) with x_k = 0 for all k 0, where x =(x_k) ∈ ?_1,and u and v are either constant or strictly decreasing sequences of positive real numbers satisfying certain conditions. In this article we give some improvements of the computation of the spectrum of the operator G(u v; ?) on the sequence space ?_1.     

Spectrum of a class of fourth order left-definite differential operators

The spectrum of a class of fourth order left-definite differential operators is studied. By using the theory of indefinite differential operators in Krein space and the relationship between left-definite and right-definite operators, the following conclusions are obtained： if a fourth order differential operator with a self-adjoint boundary condition that is left-definite and right-indefinite, then all its eigenvalues are real, and there exist countably infinitely many positive and negative eigenvalues which are unbounded from below and above, have no finite cluster point and can be indexed to satisfy the inequality …≤λ-2≤λ-1≤λ-0〈0〈λ0≤λ1≤λ2≤…     

THE SPECTRUM OF CONTRACTIVE OPERATORS ON π_k

In this paper, it is shown that, for a contraction on π_k, the intersection of its spectrum with the exterior of the unit disk is a finite set of isolated eigenvalues, each of which has finite multplicity. Futhermore some relations between its spectrum and the spectrum of its minimal unitary dilation are established.     

HYPERCYCLICITY,SUPERCYCLICITY AND GENERALIZED RAKOEVIC'S PROPERTY

A Banach space operator satisfies generalized Rakoevi's property(gw) if the complement of its upper semi B-Weyl spectrum in its approximate point spectrum is the set of eigenvalues of T which are isolated in the spectrum of T. In this note, we characterize hypecyclic and supercyclic operators satisfying the property(gw).     

THE SET OF THE INVARIANT MEASURES OF BOUNDED SPIN-FLIP PROCESSES WITH POTENTIAL

In this paper, some properties of the invariant measures of bounded spin-flip processes are discussed. It is proved that all invariant measures of a bounded spinflip process with potential are reversible measures if its speed functions have locally finite range interaction. Therefore, all invariant measures of the process are the Gibbs states of the potential. It is also proved that for each given potential, there exists a spin-flip process suth that all of its invariant measures are the Gibbs states of the potential.     

Spectra of first-order formulas with a low quantifier depth and a small number of quantifier alternations

Spectra of first-order formulas are studied. The spectrum of a first-order formula is the set of all positive α such that either this formula is true for the random graph G(n, n ^?α) with an asymptotic probability being neither 0 nor 1 or the limit does not exist. It is well known that there exists a first-order formula with an infinite spectrum. The minimum number of quantifier alternations in such a formula is found.     

Complete Continuity for a Class of Banach Algebras

For quotiens of group algebras, we stady elements such that the multiplication by these elements generates a compact (left or right) multiplication operator. The spectrum of such an algebra is analyzed in the case where all such (left or right) multiplication operators are weakly compact. Bibliography: 33 titles.     

On the spectra of a Cantor measure

We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75 (1998) 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also extends many previous results in the literature. In fact, most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition. We also obtain two new conditions for a labeling tree to generate a spectrum when other digits (digits not necessarily in {0,1,2,3}) are used in the base 4 expansion of integers and when bad branches are allowed in the spectral labeling. These new conditions yield new examples of spectra and in particular lead to a surprizing example which shows that a maximal set of orthogonal exponentials is not necessarily an orthonormal basis.     

Fresh function spectra

In this paper, we investigate the fresh function spectrum of forcing notions, where a new function on an ordinal is called fresh if all its initial segments are in the ground model. We determine the fresh function spectrum of several forcing notions and discuss the difference between fresh functions and fresh subsets. Furthermore, we consider the question which sets are realizable as the fresh function spectrum of a homogeneous forcing. We show that under GCH all sets with a certain closure property are realizable, while consistently there are sets which are not realizable.     

On the whole spectrum of Timoshenko beams. Part I: a theoretical revisitation

The problem of free vibrations of the Timoshenko beam model is here addressed. A careful analysis of the governing equations allows identifying that the vibration spectrum consists of two parts, separated by a transition frequency, which, depending on the applied boundary conditions, might be itself part of the spectrum. For both parts of the spectrum, the values of natural frequencies are computed and the expressions of eigenmodes are provided. This allows to acknowledge that the nature of vibration modes changes when moving across the transition frequency. Among all possible combination of end constraints which can be applied to single-span beams, the case of a simply supported beam is considered. These theoretical results can be used as benchmarks for assessing the correctness of the numerical values provided by several numerical techniques, e.g. traditional Lagrangian-based finite element models or the newly developed isogeometric approach.     

Asymptotic behavior of spectral gaps in a regularly perturbed periodic waveguide

Since the spectrum of a periodic waveguide is the union of a countable family of closed bounded segments (spectral bands), it can contain opened spectral gaps, i.e., intervals in the real positive semi-axis that are free of the spectrum but have both endpoints in it. A cylindrical waveguide has an intact spectrum that is a closed ray. We consider a small periodic perturbation of the waveguide wall, and, by means of an asymptotic analysis of the eigenvalues in the model problem on the periodicity cell, we show how a spectral gap opens when the cylindrical waveguide converts into a periodic one. Indeed, a cylindrical waveguide can be interpreted as a periodic one with an arbitrary period, but all its spectral bands touch each other. A periodic perturbation of the waveguide wall provides the splitting of the band edges. This effect is known in the physical literature for waveguides of different shapes, and, in this paper, we provide a rigorous mathematical proof of the effect. Several variants of the edge splitting (alone and coupled, simple and multiple knots) are examined. Explicit formulas are obtained for a plane waveguide.     

Spectral properties of hamiltonians with a magnetic field at a fixed pseudomomentum. I

It is shown that the energy operator of an n-particle neutral system with a fixed pseudomomentum in a homogeneous magnetic field can be written as an operator in the space of the relative motion. The Hunziker-Van Winter-Zhislin theorem on localization of the essential spectrum is proved for this operator with regard to the permutation symmetry for all n≥2. Conditions for the finiteness and infiniteness of the discrete spectrum and spectral asymptotic formulas with estimates for the remainder are established for the case n=2. In particular, these results can be applied to the Hamiltonian of the hydrogen atom in a homogeneous magnetic field. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 3, pp. 413–431, December, 1997. This work was supported by RFBR (Grant 9601-00478) and by INTAS (Grant 95-0414).     

The spectral characterization of graphs of index less than 2 with no path as a component

A graph is said to be determined by the adjacency and Laplacian spectrum (or to be a DS graph, for short) if there is no other non-isomorphic graph with the same adjacency and Laplacian spectrum, respectively. It is known that connected graphs of index less than 2 are determined by their adjacency spectrum. In this paper, we focus on the problem of characterization of DS graphs of index less than 2. First, we give various infinite families of cospectral graphs with respect to the adjacency matrix. Subsequently, the results will be used to characterize all DS graphs (with respect to the adjacency matrix) of index less than 2 with no path as a component. Moreover, we show that most of these graphs are DS with respect to the Laplacian matrix.     

Dynamic behavior of a heat equation with memory

This paper addresses the spectrum‐determined growth condition for a heat equation with exponential polynomial kernel memory. By introducing some new variables, the time‐variant system is transformed into a time‐invariant one. The detailed spectral analysis is presented. It is shown that the system demonstrates the property of hyperbolic equation that all eigenvalues approach a line that is parallel to the imaginary axis. The residual spectral set is shown to be empty and the set of continuous spectrum is exactly characterized. The main result is the spectrum‐determined growth condition that is one of the most difficult problems for infinite‐dimensional systems. Consequently, a strong exponential stability result is concluded. Copyright © 2008 John Wiley & Sons, Ltd.     

Interplay between Riccati,Ermakov, and Schrödinger equations to produce complex‐valued potentials with real energy spectrum

Nonlinear Riccati and Ermakov equations are combined to pair the energy spectrum of 2 different quantum systems via the Darboux method. One of the systems is assumed Hermitian, exactly solvable, with discrete energies in its spectrum. The other system is characterized by a complex‐valued potential that inherits all the energies of the former one and includes an additional real eigenvalue in its discrete spectrum. If such eigenvalue coincides with any discrete energy (or it is located between 2 discrete energies) of the initial system, its presence produces no singularities in the complex‐valued potential. Non‐Hermitian systems with spectrum that includes all the energies of either Morse or trigonometric Pöschl‐Teller potentials are introduced as concrete examples.