Essential Spectrum Of Operator Research Articles

THE DISCRETE SPECTRUM OF TWO PARTICLE HAMILTONIAN ON TWO-DIMENSIONAL LATTICE

We consider a system of two arbitrary quantum particles on a two-dimensional lattice with special dispersion functions (describing site-to-site particle transport), where the particles interact by a chosen attraction potential. We study how the number of eigenvalues of operator family ℎ(?) depends on the particle interaction energy and the total quasi-momentum ? ∈ ? 2 (where ? 2 is a two-dimensional torus). Subject to the particle inter-action energy, we obtain conditions for existence of multiple eigenvalues below the essential spectrum of operator ℎ(?).

Bifurcations of thresholds in essential spectra of elliptic operators under localized non‐Hermitian perturbations

AbstractWe consider the operator urn:x-wiley:00222526:media:sapm12367:sapm12367-math-0001subject to the Dirichlet or Robin condition, where a domain is bounded or unbounded. The symbol stands for a second‐order self‐adjoint differential operator on such that the spectrum of the operator contains several discrete eigenvalues , . These eigenvalues are thresholds in the essential spectrum of the operator . We study how these thresholds bifurcate once we add a small localized perturbation to the operator , where is a small positive parameter and is an abstract, not necessarily symmetric operator. We show that these thresholds bifurcate into eigenvalues and resonances of the operator in the vicinity of for sufficiently small . We prove effective simple conditions determining the existence of these resonances and eigenvalues and find the leading terms of their asymptotic expansions. Our analysis applies to generic nonself‐adjoint perturbations and, in particular, to perturbations characterized by the parity‐time () symmetry. Potential applications of our result embrace a broad class of physical systems governed by dispersive or diffractive effects. As a case example, we employ our findings to develop a scheme for a controllable generation of non‐Hermitian optical states with normalizable power and real part of the complex‐valued propagation constant lying in the continuum. The corresponding eigenfunctions can be interpreted as an optical generalization of bound states embedded in the continuum. For a particular example, the persistence of asymptotic expansions is confirmed with direct numerical evaluation of the perturbed spectrum .

On finitely many resonances emerging under distant perturbations in multi-dimensional cylinders

We consider a general elliptic operator in an infinite multi-dimensional cylinder with several distant perturbations; this operator is obtained by “gluing” several single perturbation operators H(k), k=1,…,n, at large distances. The coefficients of each operator H(k) are periodic in the outlets of the cylinder; the structure of these periodic parts at different outlets can be different. We consider a point λ0∈R in the essential spectrum of the operator with several distant perturbations and assume that this point is not in the essential spectra of middle operators H(k), k=2,…,n−1, but is an eigenvalue of at least one of H(k), k=1,…,n. Under such assumption we show that the operator with several distant perturbations possesses finitely many resonances in the vicinity of λ0. We find the leading terms in asymptotics for these resonances, which turn out to be exponentially small. We also conjecture that the made assumption selects the only case, when the distant perturbations produce finitely many resonances in the vicinity of λ0. Namely, as λ0 is in the essential spectrum of at least one of operators H(k), k=2,…,n−1, we do expect that infinitely many resonances emerge in the vicinity of λ0.

Essential Spectrum of Schrödinger Operators on Periodic Graphs

We give a description of the essential spectra of unbounded operators ℋ q on L2(Γ) determined by the Schrodinger operators −d2/dx2 + q(x) on the edges of Γ and general vertex conditions. We introduce a set of limit operators of ℋ q such that the essential spectrum of ℋ q is the union of the spectra of limit operators. We apply this result to describe the essential spectra of the operators ℋ q with periodic potentials perturbed by terms slowly oscillating at infinity.

Erratum to “The Gohberg Lemma, compactness, and essential spectrum of operators on compact Lie groups”

Erratum to “The Gohberg Lemma, compactness, and essential spectrum of operators on compact Lie groups”

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

Abstract 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.

Study of the essential spectrum of a matrix operator

We consider a matrix operator H corresponding to a system with a nonconserved finite number of particles on a lattice. We describe the structure of the essential spectrum of the operator H and prove that the essential spectrum is a union of at most four intervals.

The essential and discrete spectrum of a model operator associated to a system of three identical quantum particles

A model operator H associated to a system of three identical quantum particles on the three-dimensional lattice ℤ 3 is considered. The existence of eigenvalues lying below the essential spectrum of a family of Friedrichs models under rank-one perturbations h μα (p) , p ∈ T 3 , α = 1, 2, is established. The essential spectrum of the operator H is described by the spectrum of the family of the Friedrichs models h μα (p) , p ∈ T 3 , α = 1, 2. The following results are proven: The operator H has a finite number of eigenvalues lying below zero, if at least one of the Friedrichs models hμα (0), α = 1, 2, has a zero energy resonance. The operator H has infinitely many eigenvalues lying below zero and accumulating at zero, if both operators hμα (0), α = 1,2, have zero energy resonances.

Linear maps preserving essential spectral functions and closeness of operator ranges

Let H be an infinite-dimensional complex Hilbert space and ℬ(H) the algebra of all bounded linear operators on H. Some characterizations are obtained of linear maps Φ on ℬ(H) preserving essential spectral functions such as the intersection of left essential spectrum and right essential spectrum of operators, preserving Fredholm operators, and preserving semi-Fredholm operators, and preserving closeness of operator ranges. A result of Mbekhta, Rodman and Šemrl (Integral Equations Operator Theory 55 (2006) 93-109) on generalized invertibility preserving linear maps is improved.

AMD-Numbers, Compactness, Strict Singularity and the Essential Spectrum of Operators

Abstract For an operator 𝑇 acting from an infinite-dimensional Hilbert space 𝐻 to a normed space 𝑌 we define the upper AMD-number and the lower AMD-number as the upper and the lower limit of the net (δ(𝑇|𝐸))𝐸∈𝐹𝐷(𝐻), with respect to the family 𝐹𝐷(𝐻) of all finite-dimensional subspaces of 𝐻. When these numbers are equal, the operator is called AMD-regular. It is shown that if an operator 𝑇 is compact, then and, conversely, this property implies the compactness of 𝑇 provided 𝑌 is of cotype 2, but without this requirement may not imply this. Moreover, it is shown that an operator 𝑇 has the property if and only if it is superstrictly singular. As a consequence, it is established that any superstrictly singular operator from a Hilbert space to a cotype 2 Banach space is compact. For an operator 𝑇, acting between Hilbert spaces, it is shown that and are respectively the maximal and the minimal elements of the essential spectrum of , and that 𝑇 is AMD-regular if and only if the essential spectrum of |𝑇| consists of a single point.

Bishop’s property ($\beta $) and essential spectra of quasisimilar operators

We analyze the notion of Bishop’s property (β) to obtain some new concepts. We describe some conditions in terms of these concepts for an operator to have its essential spectrum (spectrum) contained in the essential spectrum (spectrum) of every operator quasisimilar to it. A subfamily of such operators is proved to be dense in L(H). 1. Preliminaries and notations The concept of quasisimilarity of linear operators was introduced by B. Sz-Nagy and C. Foias [1] in 1967. Quasisimilarity is an equivalent relation weaker than similarity. Similarity preserves the spectrum and essential spectrum of an operator, but this fails to be true for quasisimilarity. Suppose that S q ∼ T . What condition should be imposed on S and T to insure the equality relation σe(S) = σe(T ) (σ(S) = σ(T ))? A list of results have been announced along this line. We would like to recall that Yang [2] proved that two quasisimilar M -hyponormal operators have equal essential spectra and M. Putinar [3] proved that two densely similar tuples of operators having Bishop’s property (β) ([4]) have equal essential spectra. However, the foregoing works paid more attention to the equality of essential spectra and spectra than the inclusion relations among them, and the methods applied formerly to different families of operators were varied. We are now going to seek some general conditions for a bounded linear operator S ∈ L(H) to have its essential spectrum (spectrum) contained in that of every operator quasisimilar to it. We call such an operator S a (Q) ((P)) operator, denoted as S ∈ (Q) (S ∈ (P )). Of course, if both S and T ∈ (Q) ((P )) and S q ∼ T , then S and T have equal essential spectra (spectra). The results in [2], [3] motivate us to analyze Bishop’s property (β). It is well known that the subdecomposability of an operator T is equivalent to having Bishop’s property (β) [5]. In section 2, we “localize” the property (β) of an operator S to obtain the concepts A(S), E1(S), E2(S), C1(S), C2(S) as defined below and discuss their mutual relations and the relations between them and the spectral structure of S. In section 3, we establish certain sufficient conditions for (Q) ((P), etc.) operators (Theorems 1,2). We then give a number of corollaries and Received by the editors March 27, 1998. 1991 Mathematics Subject Classification. Primary 47B40, 47A10.

Toeplitz operators with PC symbols on general Carleson Jordan curves with arbitrary Muckenhoupt weights

We describe the spectra and essential spectra of Toeplitz operators with piecewise continuous symbols on the Hardy space H p ( Γ , ω ) H^p(\Gamma ,\omega ) in case 1 > p > ∞ 1>p>\infty , Γ \Gamma is a Carleson Jordan curve and ω \omega is a Muckenhoupt weight in A p ( Γ ) A_p(\Gamma ) . Classical results tell us that the essential spectrum of the operator is obtained from the essential range of the symbol by filling in line segments or circular arcs between the endpoints of the jumps if both the curve Γ \Gamma and the weight are sufficiently nice. Only recently it was discovered by Spitkovsky that these line segments or circular arcs metamorphose into horns if the curve Γ \Gamma is nice and ω \omega is an arbitrary Muckenhoupt weight, while the authors observed that certain special so-called logarithmic leaves emerge in the case of arbitrary Carleson curves with nice weights. In this paper we show that for general Carleson curves and general Muckenhoupt weights the sets in question are logarithmic leaves with a halo, and we present final results concerning the shape of the halo.

Essential Spectra of Operators in the Class B n (Ω)

Essential Spectra of Operators in the Class B n (Ω)

Essential spectra of operators in the class ℬ_{𝓃}(Ω)

For a connected open subset Ω \Omega of the plane and n n a positive integer, let B n ( Ω ) {\mathcal {B}_n}(\Omega ) be the space introduced by Cowen and Douglas in their paper Complex Geometry and Operator Theory. Our paper deals with characterizing the essential spectrum of an operator T T in B n ( Ω ) {\mathcal {B}_n}(\Omega ) for which σ ( T ) = Ω ¯ \sigma (T) = \bar \Omega and the point spectrum of T ∗ {T^ * } is empty. This class of operators forms an important part of B n ( Ω ) {\mathcal {B}_n}(\Omega ) denoted by B n ′ ( Ω ) {\mathcal {B}’_n}(\Omega ) . We use this characterization to give another proof of the result of Axler, Conway and McDonald on determining the essential spectrum of the Bergman operator. Let A n ( G ) = { S : T = S ∗ is in B ′ n ( G ∗ ) } {A_n}(G) = \left \{ {S:T = {S^ * }{\text {is}}\;{\text {in}}{{\mathcal {B}’}_n}({G^ * })} \right \} . We also characterize the weighted shifts in A 1 ( G ) {A_1}(G) .