arXiv subjects

Siegfried Beckus

Publications and source records attributed to Siegfried Beckus.

24 records · Page 2Linked to original sources

arXiv2017

Shnol-type theorem for the Agmon ground state

Let $H$ be a Schrödinger operator defined on a noncompact Riemannian manifold $Ω$, and let $W\in L^\infty(Ω;\mathbb{R})$. Suppose that the operator $H+W$ is critical in $Ω$, and let $φ$ be the corresponding Agmon ground state. We prove that if $u$ is a generalized eigenfunction of $H$ satisfying $|u|\leq φ$ in $Ω$, then the corresponding eigenvalue is in the spectrum of $H$. The conclusion also holds true if for some $K\Subset Ω$ the operator $H$ admits a positive solution in $\tildeΩ=Ω\setminus K$, and $|u|\leq ψ$ in $\tildeΩ$, where $ψ$ is a positive solution of minimal growth in a neighborhood of infinity in $Ω$. Under natural assumptions, this result holds true also in the context of infinite graphs, and Dirichlet forms.

math.SP↗

arXiv2016

On the spectrum of operator families on discrete groups over minimal dynamical systems

It is well known that, given an equivariant and continuous (in a suitable sense) family of selfadjoint operators in a Hilbert space over a minimal dynamical system, the spectrum of all operators from that family coincides. As shown recently similar results also hold for suitable families of non-selfadjoint operators in $\ell^p (\ZM)$. Here, we generalize this to a large class of bounded linear operator families on Banach-space valued $\ell^p$-spaces over countable discrete groups. We also provide equality of the pseudospectra for operators in such a family. A main tool for our analysis are techniques from limit operator theory.

math.SP↗

arXiv2016

Spectral approximation of aperiodic Schrödinger operators

We study the (Hölder-)continuous behavior of the spectra belonging to a family of linear bounded operators $(A_t)_{t\in T}$ indexed by a topological space $T$. For the cases of self-adjoint, unitary and normal operators, a characterization of the continuity of $Σ:T\to \mathcal{K}(\mathbb{R}), t\mapsto σ(A_t),$ is proven while the distance of the spectra is measured by the Hausorff metric. If $T$ is a metric space, the Hölder-continuous behavior of $Σ$ is characterized for self-adjoint and unitary operators. Here we observe interesting effects, namely the rate of convergence is bisect whenever spectral gaps closes. Based on this, we provide a tool to prove the continuity of the spectra for large classes of operators. In particular, we apply this theory to generalized Schrödinger operators and show that the continuity of the spectra is characterized by the continuous variation of the underlying dynamical systems. Finally, we analyze the existence of periodic dynamical systems approximating a given dynamical system. This leads to periodic approximations of the corresponding Schrödinger operators by the previously developed theory. We prove that local symmetries of the patterns and the presence of a substitution is a sufficient criteria for periodic approximations of subshifts in $\mathbb{Z}^d$. For $d=1$, a characterization is proven for the existence of periodic approximations. For these approaches, the notion of a dictionary is further developed and defined independently of a given configuration. We prove that the set of dictionaries equipped with the local pattern topology is homeomorphic to the space of subshifts. This yields a useful tool to analyze these systems. Furthermore, it delivers the connection of the existence of periodic orbits in a subshift of finite type and the existence of periodic approximations for subshifts.

math.SP↗

arXiv2016

Continuity of the spectrum of a field of self-adjoint operators

Given a family of self-adjoint operators $(A_t)_{t\in T}$ indexed by a parameter $t$ in some topological space $T$, necessary and sufficient conditions are given for the spectrum $σ(A_t)$ to be Vietoris continuous with respect to $t$. Equivalently the boundaries and the gap edges are continuous in $t$. If $(T,d)$ is a complete metric space with metric $d$, these conditions are extended to guarantee Hölder continuity of the spectral boundaries and of the spectral gap edges. As a corollary, an upper bound is provided for the size of closing gaps.

math.SP↗

arXiv2014

Note on spectra of non-selfadjoint operators over dynamical systems

We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.

math.SP↗

arXiv2013

Spectrum of Lebesgue measure zero for Jacobi matrices of quasicrystals

We study one-dimensional random Jacobi operators corresponding to strictly ergodic dynamical systems. In this context, we characterize the spectrum of these operators by non-uniformity of the transfer matrices and the set where the Lyapunov exponent vanishes. Adapting this result to subshifts satisfying the so-called Boshernitzan condition, it turns out that the spectrum is supported on a Cantor set with Lebesgue measure zero. This generalizes earlier results for Schrödinger operators.

math-ph↗