Related papers: Singular continuous spectrum is generic
In this paper, we develop spectral analysis of a discrete non-Hermitian quantum system that is a discrete counterpart of some continuous quantum systems on a complex contour. In particular, simple conditions for discreteness of the spectrum…
We study the spectral properties of bounded and unbounded Jacobi matrices whose entries are bounded operators on a complex Hilbert space. In particular, we formulate conditions assuring that the spectrum of the studied operators is…
We study the spectral types of the families of discrete one-dimensional Schr\"odinger operators $\{H_\omega\}_{\omega\in\Omega}$, where the potential of each $H_\omega$ is given by $V_\omega(n)=f(T^n\omega)$ for $n\in\mathbb{Z}$, $T$ is an…
In quantum theory, observables with a continuous spectrum are known to be fundamentally different from those with a discrete and finite spectrum. While some fundamental tests and applications of quantum mechanics originally formulated for…
In this survey paper we review classical results and recent progress about a certain topic in the spectral theory of two-dimensional canonical systems. Namely, we consider the questions whether the spectrum $\sigma$ is discrete, and if it…
We analyze the spectrum of the operator $\Delta^{-1} [\nabla \cdot (K\nabla u)]$, where $\Delta$ denotes the Laplacian and $K=K(x,y)$ is a symmetric tensor. Our main result shows that this spectrum can be derived from the spectral…
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…
Continuous spectrum operators (CSOs), characterized by spectra comprising continuous intervals rather than discrete eigenvalues, are pivotal in quantum mechanics, wave propagation, and systems governed by partial differential equations.…
The spectrum of triangular band matrices defined on the sequence spaces where the entries of each band is a constant or convergent sequence is well studied. In this article, the spectrum and fine spectrum of a new generalised difference…
Consider the Dirichlet Laplacian operator $-\Delta^D$ in a periodic waveguide $\Omega$. On the condition that $\Omega$ is sufficiently thin, we show that its spectrum $\sigma(-\Delta^D)$ is absolutely continuous (in each finite region). In…
This article presents a new proof of a theorem concerning bounds of the spectrum of the product of unitary operators and a generalization for differentiable curves of this theorem. The proofs involve metric geometric arguments in the group…
First we study the spectral singularity at infinity and investigate the connections of the spectral singularities and the spectrality of the Hill operator. Then we consider the spectral expansion when there is not the spectral singularity…
We prove $-\Delta +V$ has purely discrete spectrum if $V\geq 0$ and, for all $M$, $|\{x\mid V(x)<M\}|<\infty$ and various extensions.
We review recent developments in the spectral theory of continuum one-dimensional quasicystals, yielding purely singular continuous spectrum for these Schr\"odinger operators. Allowing measures as potentials we can generalize some results…
The problem of approximating the discrete spectra of families of self-adjoint operators that are merely strongly continuous is addressed. It is well-known that the spectrum need not vary continuously (as a set) under strong perturbations.…
We consider CMV matrices with dynamically defined Verblunsky coefficients. These coefficients are obtained by continuous sampling along the orbits of an ergodic transformation. We investigate whether certain spectral phenomena are generic…
Regular sequences are natural generalisations of fixed points of constant-length substitutions on finite alphabets, that is, of automatic sequences. Using the harmonic analysis of measures associated with substitutions as motivation, we…
The purpose of this note is to describe a space that is regular but not completely regular, but only barely so: all closed sets are $G_\delta$-sets and every singleton is a zero-set.
In this paper we prove that generic metric spaces are everywhere dense in the proper class of all metric spaces endowed with the Gromov-Hausdorff distance.
Let $\mathcal{H}$ be a right quaternionic Hilbert space and let $T$ be a bounded normal right quaternionic linear operator on $\mathcal{H}$. In this paper, we prove that there exists a unique spectral measure $E$ in $\mathcal{H}$ such that…