Related papers: Schauder Bases and Operator Theory III: Schauder S…
The spectrum of the Schr\"odinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also if the waveguide is bent…
Affiliated and normal operators in octonion Hilbert spaces are studied. Theorems about their properties and of related algebras are demonstrated. Spectra of unbounded normal operators are investigated.
We consider discrete Schr\"odinger operators with periodic potentials on periodic graphs perturbed by guided non-positive potentials, which are periodic in some directions and finitely supported in other ones. The spectrum of the…
The primary purpose of this paper is to show the existence of normal square and nth roots of some classes of bounded operators on Hilbert spaces. Two interesting simple results hold. Namely, under simple conditions we show that if any…
We give a spectral description of the semi-classical Schrodinger operator with a piecewise linear, complex valued potential. Moreover, using these results, we show how an arbitrarily small bounded perturbation of a non-self-adjoint operator…
Many novel and unique physical phenomena of incommensurate systems can be illustrated and predicted using the spectra of the associated Schr\"odinger operators. However, the absence of periodicity in these systems poses significant…
We consider Sch\"odinger operators on the half-line, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if $\Delta +…
We prove that for every bounded linear operator $T:X\to X$, where $X$ is a non-reflexive quotient of a von Neumann algebra, the point spectrum of $T^*$ is non-empty (i.e. for some $\lambda\in\mathbb C$ the operator $\lambda I-T$ fails to…
We study spectral properties of one-dimensional singular perturbations of an unbounded selfadjoint operator and give criteria for the possibility to remove the whole spectrum by a perturbation of this type. A counterpart of our results for…
In this paper, we study the following magnetic Schr\"odinger operator in $\mathbb{R}^3$: \[ H=(i \nabla +A)^2- \tilde{V}, \] where $\tilde{V}$ is non-negative potential supported over the tube built along a curve which is a local…
Motivated by the well known theorem of Schauder, we study the relationship between various s-numbers of an operator T and its adjoint T* between Banach spaces.
In this paper we investigate the spectral and scattering theory for operators acting on topological crystals and on their perturbations. A special attention is paid to perturbations obtained by the addition of an infinite number of edges,…
We show that any $L^2$-bounded rational function in free semicircular random variables is a bounded operator, which implies the coincidence of the usual spectrum and $L^2$-spectrum for rational functions. Based on this observation, we also…
We survey results concerning the spectral properties of limit-periodic operators. The main focus is on discrete one-dimensional Schr\"odinger operators, but other classes of operators, such as Jacobi and CMV matrices, continuum…
The research on spectral inequalities for discrete Schrodinger Operators has proved fruitful in the last decade. Indeed, several authors analysed the operator's canonical relation to a tridiagonal Jacobi matrix operator. In this paper, we…
In this manuscript, we investigate the properties of systems formed by translations of an operator in the Schatten $p$-classes $\mathcal{T}^p$. We establish the existence of Schauder frames of integer translates in $\mathcal{T}^p$ for…
We consider a periodic magnetic Schr\"odinger operator on a noncompact Riemannian manifold $M$ such that $H^1(M, \RR)=0$ endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no…
We consider a random family of Schr\"odinger operators on a cover $X$ of a compact Riemannian manifold $M = X/\Gamma$. We present several results on their spectral theory, in particular almost sure constancy of the spectral components and…
In this paper the spectrum of composition operators on the space of real analytic functions is investigated. In some cases it is completely determined while in some other cases it is only estimated.
We prove that one-dimensional reflectionless Schr\"odinger operators with spectrum a homogeneous set in the sense of Carleson, belonging to the class introduced by Sodin and Yuditskii, have purely absolutely continuous spectra. This class…