Essential Spectrum Research Articles

Perturbations of gaps in the essential spectra of self-adjoint relations

This paper is concerned with the stability of gaps in the essential spectra of self-adjoint relations under non-negative relatively compact perturbation in Hilbert spaces. A stability result about semi-boundedness of self-adjoint relations under relatively bounded perturbation is given. It is shown that the gaps in the essential spectra of self-adjoint relations are invariant under non-negative relatively compact perturbation. The results obtained in the present paper generalize the corresponding results for linear operators to linear relations.

Characterization of essential spectra by quasi-compact perturbations

Characterization of essential spectra by quasi-compact perturbations

Spectral mapping theorems for essential spectra and regularized functional calculi

Gramsch and Lay [8] gave spectral mapping theorems for the Dunford-Taylor calculus of a closed linear operator $T$ , \[ \widetilde{\sigma}_i(f(T)) = f(\widetilde{\sigma}_i(T)), \] for several extended essential spectra $\widetilde {\sigma }_i$ . In this work, we extend such theorems for the regularized functional calculus introduced by Haase [10, 11] assuming suitable conditions on $f$ . At the same time, we answer in the positive a question made by Haase [11, Remark 5.4] regarding the conditions on $f$ which are sufficient to obtain the spectral mapping theorem for the usual extended spectrum $\widetilde \sigma$ . We use the model case of bisectorial-like operators, although the proofs presented here are generic, and are valid for similar functional calculi.

Existence condition of an eigenvalue of the three particle Schr¨odinger operator on a lattice

We consider the three-particle discrete Schrodinger operator Hµ,γ(К), К ϵТ3 associated to a system of three particles (two particle are fermions with mass 1 and third one is an another particle with mass m = 1/y < 1) interacting through zero range pairwise potential µ> 0 on the three-dimensional lattice Z3. It is proved that for γϵ(1, γ0) (γ0≈4,7655) the operator Hµ,γ(π), π=(π,π,π), has no eigenvalue and has only unique eigenvalue with multiplicity three for γ>γ0 lying right of the essential spectrum for sufficiently large µ.

Tamed stability of finite difference schemes for the transport equation on the half-line

In this paper, we prove that, under precise spectral assumptions, some finite difference approximations of scalar leftgoing transport equations on the positive half-line with numerical boundary conditions are ℓ 1 \ell ^1 -stable but ℓ q \ell ^q -unstable for any q > 1 q>1 . The proof relies on the accurate description of the Green’s function for a particular family of finite rank perturbations of Toeplitz operators whose essential spectrum belongs to the closed unit disk and with a simple eigenvalue of modulus 1 1 embedded into the essential spectrum.

Convergent spectral inclusion sets for banded matrices

AbstractWe obtain sequences of inclusion sets for the spectrum, essential spectrum, and pseudospectrum of banded, in general non‐normal, matrices of finite or infinite size. Each inclusion set is the union of the pseudospectra of certain submatrices of a chosen size n. Via the choice of n, one can balance accuracy of approximation against computational cost, and we show, in the case of infinite matrices, convergence as of the respective inclusion set to the corresponding spectral set.

New Characterizations of the Jeribi Essential Spectrum

In this paper, we give several characterizations of the Jeribi essential spectrum of a bounded linear operator defined on a Banach space by using the notion of almost weakly compact operators. As a consequence, we prove the stability of the Jeribi essential spectrum under compact perturbations. Furthermore, some characterizations of the Jeribi essential spectra of 3×3 upper triangular block operator matrix are also given.

On the asymptotic behavior of a size-structured model arising in population dynamics

We study the asymptotic behavior of a semilinear size-structured population model withdelay when the nonlinearity is small in some sense. The novelty in this work is that theoperator governing the linear part of the equation does not generate a compact semigroupunlike in the results present in literature. In such a case the spectrum does not consist whollyof eigenvalues but also has a non-trivial component called Browder’s essential spectrum. Toovercome the lack of compactness, we give a localization of Browder’s essential spectrum of theoperator governing the linear part and we use the Perron-Frobenius spectral analysis adaptedto semigroups of positive operators in Banach lattices to investigate the long time behavior ofthe system.Keywords: Perron-Frobenius, positive operators, structured population models, Browder’sessential spectrum, asymptotic behavior, semigroups of operatorsAMS Subject Classifications: 35B40, 35R10, 47D06.

Spectrum of One-Dimensional Potential Perturbed by a Small Convolution Operator: General Structure

We consider an operator of multiplication by a complex-valued potential in L2(R), to which we add a convolution operator multiplied by a small parameter. The convolution kernel is supposed to be an element of L1(R), while the potential is a Fourier image of some function from the same space. The considered operator is not supposed to be self-adjoint. We find the essential spectrum of such an operator in an explicit form. We show that the entire spectrum is located in a thin neighbourhood of the spectrum of the multiplication operator. Our main result states that in some fixed neighbourhood of a typical part of the spectrum of the non-perturbed operator, there are no eigenvalues and no points of the residual spectrum of the perturbed one. As a consequence, we conclude that the point and residual spectrum can emerge only in vicinities of certain thresholds in the spectrum of the non-perturbed operator. We also provide simple sufficient conditions ensuring that the considered operator has no residual spectrum at all.

Spectral minimal partitions of unbounded metric graphs

We investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schrödinger operator of the form -\Delta + V with suitable (electric) potential V , which is taken as a fixed, underlying function on the whole graph. We show that there is a strong link between spectral minimal partitions and infimal partition energies on the one hand, and the infimum \Sigma of the essential spectrum of the corresponding Schrödinger operator on the whole graph on the other. Namely, we show that for any k\in\mathbb{N} , the infimal energy among all admissible k -partitions is bounded from above by \Sigma , and if it is strictly below \Sigma , then a spectral minimal k -partition exists. We illustrate our results with several examples of existence and non-existence of minimal partitions of unbounded and infinite graphs, with and without potentials. The nature of the proofs, a key ingredient of which is a version of the characterization of the infimum of the essential spectrum known as Persson’s theorem for quantum graphs, strongly suggests that corresponding results should hold for Schrödinger operator-based partitions of unbounded domains in Euclidean space.

Keller and Lieb–Thirring estimates of the eigenvalues in the gap of Dirac operators

We estimate the lowest eigenvalue in the gap of the essential spectrum of a Dirac operator with mass in terms of a Lebesgue norm of the potential. Such a bound is the counterpart for Dirac operators of the Keller estimates for the Schrödinger operator, which are equivalent to some Gagliardo–Nirenberg–Sobolev interpolation inequalities. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach the bottom of the gap in the essential spectrum. The Keller estimate is then extended to a Lieb–Thirring inequality for the eigenvalues in the gap. Most of our result are established in the Birman–Schwinger reformulation.

THE NUMBER OF EIGENVALUES OF THE DISCRETE SCHRÖDINGER OPERATOR

In this paper we consider the family of Schrdinger operators, associated to the Bose-Hubbard Hamiltonian of a single boson on the one dimensional lattice . We find suffiently conditions for the parameters , such that the operator has no one eigenvalues below the essential spectrum and three eigenvalues above the essential spectrum.

A comparative study for the spectral properties of Toeplitz and Hankel operators

In this introductory review, we study Hankel and Toeplitz operators considering them as acting on certain spaces of analytic functions, namely Hardy spaces and compare their spectral properties such as their compactness criteria. In contrast to Toeplitz operators, the symbol of a Hankel operator is not uniquely determined by the operator. We also connect Toeplitz operators with Fredholm operators and give some of the most beautiful properties of Toeplitz operators such as the essential spectrum of Toeplitz operator with continuous symbol and the index of Toeplitz operator introducing Fredholm operators firstly.

Volume integral equations for electromagnetic scattering by an orthotropic infinite cylinder

We investigate time-harmonic electromagnetic wave scattering by an inhomogeneous anisotropic medium. For the case of an orthotropic cylindrical obstacle and fields invariant along the axis of the cylinder, the problem can be reduced to a two-dimensional one. This 2D problem can be written equivalently in terms of two different strongly singular volume integral equations (VIEs), one is vector-valued and the other is scalar-valued. The aim of this study is to analyze the essential spectrum of the integral operators that describe these VIEs in the case where the physical parameters are piecewise valued matrix functions with some bounds on their eigenvalues. For Lipschitz interfaces, we show that the spectrum is contained in some subsets which depend on the spectral properties of the scattering parameters. The results on the spectrum will then be used to derive sufficient conditions which ensure the uniqueness of solutions to the diffraction problem.

Essential positivity

We define essentially positive operators on Hilbert space as a class of self-adjoint operators whose essential spectra is contained in the non-negative real numbers and describe their basic properties. Using Toeplitz operators and the Berezin transform, we further illustrate the notion of essential positivity in the Hardy space and the Bergman space.

Reconstruction of the Schrödinger Operator with a Singular Potential on Half-Line by Its Prescribed Essential Spectrum

Reconstruction of the Schrödinger Operator with a Singular Potential on Half-Line by Its Prescribed Essential Spectrum

Coupling resonances and spectral properties of the product of resolvent and perturbation

Given a self-adjoint operator H0 and a relatively compact self-adjoint perturbation V, we study in some detail the spectral properties of the product (H0−z)−1V, z∈C. The eigenvalues of this product are related simply to those of the corresponding product in the situation when the operator H0 is perturbed to H0+sV for s∈C. For some numbers rz, the eigenvalues of (H0+sV−z)−1V, s∈C, are (s−rz)−1. We study the root spaces of the eigenvalues (s−rz)−1 and complex-analytic properties of the functions rz such as branching points. In particular, for a generic case, we give a variety of necessary and sufficient conditions for branching. The functions rz, called coupling resonances, are important in the spectral analysis of H0+rV for any real number r∈R. For instance, they afford a description of the spectral shift function (SSF) of the pair H0 and V, as well as the absolutely continuous and singular parts of the SSF. A thorough study of real-valued coupling resonances rλ for real λ outside of the essential spectrum was carried out in a recent work by the first author. Here we extend this study to the complex domain, motivated by the fact, which is well known in the case of a rank-one perturbation, that the behaviour of coupling resonances rz near the essential spectrum provides valuable information about the latter.

Properties of Fredholm, Weyl and Jeribi essential S-spectra in a right quaternionic Hilbert space

The paper aims to extend the concept of Fredholm, Weyl and Jeribi essential spectra in the quaternionic setting. Furthermore, some properties and stability of the corresponding spectra of Fredholm and Weyl operators have been investigated in this setting. To achieve the goal, a characterization of the sum of two invariant bounded linear operators has been obtained in order to explore various properties of the Fredholm operator and Weyl operator under some assumptions in quaternionic setting. Also, various sequential properties of the pseudo-resolvent operator, right quaternionic linear operator, Weyl operator, Weyl S-spectrum, Jeribi essential S-spectrum and some properties of [Formula: see text] block operator matrices have been discussed. The spectral mapping theorem of essential S-spectrum, Weyl S-spectrum and Jeribi essential S-spectrum for self-adjoint operators has been established. A characterization of the essential S-spectrum and Weyl S-spectrum of the sum of two bounded linear operators concludes this investigation.

Two-fermion lattice Hamiltonian with first and second nearest-neighboring-site interactions

We study the Schrödinger operators , with the fixed quasimomentum of the particles pair, associated with a system of two identical fermions on the two-dimensional lattice with first and second nearest-neighboring-site interactions of magnitudes and , respectively. We establish a partition of the plane so that in each its connected component, the Schrödinger operator has a definite (fixed) number of eigenvalues, which are situated below the bottom of the essential spectrum and above its top. Moreover, we establish a sharp lower bound for the number of isolated eigenvalues of in each connected component.

Negative eigenvalues of non-local Schrödinger operators with sign-changing potentials

Simon’s results on the negative spectrum of recurrent Schrödinger operators ( d = 1 , 2 d=1,2 ) are extended to a wider class of potentials and to non-local operators. An example of L 1 − L^1- potental is constructed for which the essential spectrum of two dimensional Schrödinger operator covers the whole axis. Some counterexamples are provided for transient operators ( d ≥ 3 d\geq 3 ) showing that the assumptions on the potential for the validity of the Cwikel-Lieb-Rozenblum estimate can’t be improved significantly.