Essential Spectrum Research Articles

On Perturbation of Thresholds in Essential Spectrum under Coexistence of Virtual Level and Spectral Singularity

On Perturbation of Thresholds in Essential Spectrum under Coexistence of Virtual Level and Spectral Singularity

On creating new essential spectrum by self-adjoint extension of gapped operators

On creating new essential spectrum by self-adjoint extension of gapped operators

On the Number of Components of the Essential Spectrum of One 2 × 2 Operator Matrix

On the Number of Components of the Essential Spectrum of One 2 × 2 Operator Matrix

Generalized essential spectra involving the class of g-g-Riesz operators

Abstract In this paper, we explore the spectral properties of unbounded generalized Fredholm operators acting on a non-reflexive Banach space X. The results are formulated in terms of some topological conditions made on X or on its dual X * {X^{*}} . In addition, we introduce the concept of the so-called g-g-Riesz linear operators as an extension of Riesz operators. The obtained results are used to discuss the incidence of the behavior of generalized essential spectra. Furthermore, a relation between the generalized essential spectrum and the left (resp. the right) essential spectrum by means of g-Riesz perturbation is provided.

Bound states of weakly deformed soft waveguides

In this paper we consider the two-dimensional Schrödinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region, whose thickness is varying and is determined by the function R ∋ x ↦ d + ε f ( x ), where d > 0 is a constant, ε > 0 is a small parameter, and f is a compactly supported continuous function. We prove that if ∫ R f d x > 0, then the respective Schrödinger operator has a unique simple eigenvalue below the threshold of the essential spectrum for all sufficiently small ε > 0 and we obtain the asymptotic expansion of this eigenvalue in the regime ε → 0. An asymptotic expansion of the respective eigenfunction as ε → 0 is also obtained. In the case that ∫ R f d x < 0 we prove that the discrete spectrum is empty for all sufficiently small ε > 0. In the critical case ∫ R f d x = 0, we derive a sufficient condition for the existence of a unique bound state for all sufficiently small ε > 0.

Recurrent Subspaces in Banach Spaces

Abstract We study the spaceability of the set of recurrent vectors $\text{Rec}(T)$ for an operator $T:X\longrightarrow X$ on a Banach space $X$. In particular, we find sufficient conditions for a quasi-rigid operator to have a recurrent subspace; when $X$ is a complex Banach space, we show that having a recurrent subspace is equivalent to the fact that the essential spectrum of the operator intersects the closed unit disk, and we extend the previous result to the real case. As a consequence, we obtain that a weakly-mixing operator on a real or complex separable Banach space has a hypercyclic subspace if and only if it has a recurrent subspace. The results exposed exhibit a symmetry between the hypercyclic and recurrent spaceability theories showing that, at least for the spaceable property, hypercyclicity and recurrence can be treated as equals.

The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous obstacle with Lipschitz boundary and regularization

The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous obstacle with Lipschitz boundary and regularization

A resolvent approach for the stability of the critical or essential spectra to perturbed C0-semigroups on Banach spaces with application in biological transport

We give sufficient conditions in terms of the resolvents assuring the stability of the critical or essential spectra to perturbed C0-semigroups on an arbitrary given Banach space. Our theoretical result applies in Lp-biological transport, 1≤p<+∞. Our results improve and complete, significatively, well known earlier ones.

2×2 operator matrix with real parameter and its spectrum

In the present paper we consider a linear bounded self-adjoint 2×2 block operator matrix Aμ (so called generalized Friedrichs model) with real parameter μ ∈ R. It is associated with the Hamiltonian of a system consisting of at most two particles on a d -dimensional lattice Zd, interacting via creation and annihilation operators. Aμ is linear bounded self-adjoint operator acting in the two-particle cut subspace of the Fock space, that is, in the direct sum of zero-particle and one-particle subspaces of a Fock space. We find the essential and discrete spectra of the block operator matrix Aμ. The Fredholm determinant and resolvent operator associated to Aμ are constructed. The spectrum of Aμ plays an important role in the study of the spectral properties of the Hamiltonians associated with the energy operator of a lattice system describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles on a lattice.

Numerical spectral analysis of standing waves in quantum hydrodynamics with viscosity

&lt;abstract&gt;&lt;p&gt;We study the spectrum of the linearization around standing wave profiles for two quantum hydrodynamics systems with linear and nonlinear viscosity. The essential spectrum for such profiles is stable; we investigate the point spectrum using an Evans function technique. For both systems we show numerically that there exists a real unstable eigenvalue, thus providing numerical evidence for spectral instability.&lt;/p&gt;&lt;/abstract&gt;

Relative residual bounds for eigenvalues in gaps of the essential spectrum

Relative residual bounds for eigenvalues in gaps of the essential spectrum

Generalized lower characteristic involving measures of non-strict singularity

Abstract This work establishes a connection between the class of generalized lower characteristic operators and [ ⋅ ] a {\left[\cdot ]}_{a} acting on a Banach space involving measures of non-strict singularity. This study presents findings on the Jeribi essential spectra of the sum of two bounded linear operators and the Jeribi essential spectra of each operator. The results are based on certain requirements related to the measure of non-strict singularity of Fredholm inverses on a Banach space X. A particular application is provided.

Spectral sets of generalized Hausdorff matrices on spaces of holomorphic functions on [formula omitted

Here, we study a family of bounded operators H, acting on Banach spaces of holomorphic functions X↪O(D), which are subordinated in terms of a C0-semigroup of weighted composition operators (vtCϕt), i.e., H=∫0∞vtCϕtdν(t) in the strong sense for some Borel measure ν. This family of operators extends the so-called generalized Hausdorff operators. We obtain the spectrum, point spectrum and essential spectrum of H under mild assumptions on (vtCϕt),ν and X. In particular, we obtain these spectral sets for a wide family of generalized Hausdorff operators acting on Hardy spaces, weighted Bergman spaces, weighted Dirichlet spaces and little Korenblum classes. The description for the spectra of the infinitesimal generator of (vtCϕt) is the key for our findings.

Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems

Koopman operators linearize nonlinear dynamical systems, making their spectral information of crucial interest. Numerous algorithms have been developed to approximate these spectral properties, and dynamic mode decomposition (DMD) stands out as the poster child of projection-based methods. Although the Koopman operator itself is linear, the fact that it acts in an infinite-dimensional space of observables poses challenges. These include spurious modes, essential spectra, and the verification of Koopman mode decompositions. While recent work has addressed these challenges for deterministic systems, there remains a notable gap in verified DMD methods for stochastic systems, where the Koopman operator measures the expectation of observables. We show that it is necessary to go beyond expectations to address these issues. By incorporating variance into the Koopman framework, we address these challenges. Through an additional DMD-type matrix, we approximate the sum of a squared residual and a variance term, each of which can be approximated individually using batched snapshot data. This allows verified computation of the spectral properties of stochastic Koopman operators, controlling the projection error. We also introduce the concept of variance-pseudospectra to gauge statistical coherency. Finally, we present a suite of convergence results for the spectral information of stochastic Koopman operators. Our study concludes with practical applications using both simulated and experimental data. In neural recordings from awake mice, we demonstrate how variance-pseudospectra can reveal physiologically significant information unavailable to standard expectation-based dynamical models.

Superpixels meet essential spectra for fast Raman hyperspectral microimaging.

In the context of spectral unmixing, essential information corresponds to the most linearly dissimilar rows and/or columns of a two-way data matrix which are indispensable to reproduce the full data matrix in a convex linear way. Essential information has recently been shown accessible on-the-fly via a decomposition of the measured spectra in the Fourier domain and has opened new perspectives for fast Raman hyperspectral microimaging. In addition, when some spatial prior is available about the sample, such as the existence of homogeneous objects in the image, further acceleration for the data acquisition procedure can be achieved by using superpixels. The expected gain in acquisition time is shown to be around three order of magnitude on simulated and real data with very limited distortions of the estimated spectrum of each object composing the images.

Asymptotics for Christoffel functions associated to continuum Schrödinger operators

We prove asymptotics of the Christoffel function, λL(ξ), of a continuum Schrödinger operator for points in the interior of the essential spectrum under some mild conditions on the spectral measure. It is shown that LλL(ξ) has a limit and that this limit is given by the Radon–Nikodym derivative of the spectral measure with respect to the Martin measure. Combining this with a recently developed local criterion for universality limits at scale λL(ξ), we compute universality limits for continuum Schrödinger operators at scale L and obtain clock spacing of the eigenvalues of the finite range truncations.

Spectral analysis and hydrodynamic manifolds for the linearized Shakhov model

We perform a complete spectral analysis of the linearized Shakhov model involving two relaxation times τfast and τslow. Our results are based on spectral functions derived from the theory of finite-rank perturbations, which allows us to derive an explicit expression for the critical wave number kcrit limiting the number of discrete eigenvalues above the essential spectrum together with the existence of a finite-dimensional slow manifold defining non-local hydrodynamics. We discuss the merging of hydrodynamic modes as well as the existence of second sound and the appearance of ghost modes beneath the essential spectrum in dependence of the Prandtl number.

The Schwartzman group of an affine transformation

We compute the Schwartzman group associated with an ergodic affine automorphism of a compact connected abelian group given by the composition of an automorphism of the group and a translation by an element in the path component of the identity. We show that the Schwartzman group can be characterized by evaluating the invariant characters of the automorphism at the group element by which one translates. As a byproduct, we show that the set of labels associated with the doubling map on the dyadic solenoid is trivial, which in turn allows us to show that any ergodic family of Jacobi matrices defined over the doubling map has connected almost-sure essential spectrum.

The Structure of the Essential Spectrum and the Discrete Spectrum of the Energy Operator for Six-Electron Systems in the Hubbard Model. The Second Singlet State

The Structure of the Essential Spectrum and the Discrete Spectrum of the Energy Operator for Six-Electron Systems in the Hubbard Model. The Second Singlet State

Absence of excited eigenvalues for Fröhlich type polaron models at weak coupling

We consider a class of polaron models, including the Fröhlich model, at zero total momentum, and show that at sufficiently weak coupling there are no excited eigenvalues below the essential spectrum.