Related papers: Absolutely continuous spectrum of Stark operators
In this paper, we study the scattering theory of a class of continuum Schr\"{o}dinger operators with random sparse potentials. The existence and completeness of wave operators are proven by establishing the uniform boundedness of modified…
We consider a class of fully nonlinear integro-differential operators where the nonlocal integral has two components: the non-degenerate one corresponds to the $\alpha$-stable operator and the second one (possibly degenerate) corresponds to…
We establish a quantitative criterion for an operator defined on a Galton-Watson random tree for having an absolutely continuous spectrum. For the adjacency operator, this criterion requires that the offspring distribution has a relative…
Given a noncompact spin manifold $M$ with a fixed topological spin structure and two complete Riemannian metrics $g$ and $h$ on $M$ with bounded sectional curvatures, we prove a criterion for the existence and completeness of the wave…
A family $A_\alpha$ of differential operators depending on a real parameter $\alpha\ge 0$ is considered. This family was suggested by Smilansky as a model of an irreversible quantum system. We find the absolutely continuous spectrum…
On the half line $[0,\infty)$ we study first order differential operators of the form $B 1/i d/(dx) + Q(x)$, where $B:=\mat{B_1}{0}{0}{-B_2}$, $B_1,B_2\in M(n,\C)$ are self--adjoint positive definite matrices and $Q:\R_+\to M(2n,\C)$,…
We consider the Schr\"odinger operator $H = -\Delta + V$ in a layer or in a $d$-dimensional cylinder. The potential $V$ is assumed to be periodic with respect to some lattice. We establish the absolute continuity of $H$, assuming $V \in…
We present a spectral analysis for matrix scaling and operator scaling. We prove that if the input matrix or operator has a spectral gap, then a natural gradient flow has linear convergence. This implies that a simple gradient descent…
Avila and Jitomirskaya prove that the quasi-periodic Schr\"{o}dinger operator $H_{\lambda v,\alpha,\theta}$ has purely absolutely continuous spectrum for $\alpha $ in sub-exponential regime (i.e., $\beta(\alpha)=0$) with small $\lambda$, if…
We consider standard and extended CMV matrices with small quasi-periodic Verblunsky coefficients and show that on their essential spectrum, all spectral measures are purely absolutely continuous. This answers a question of Barry Simon from…
Let $A(k)u(k)=f(k) (1)$ be an operator equation, $X$ and $Y$ are Banach spaces, $k\in\Delta\subset\C$ is a parameter, $A(k):X\to Y$ is a map, possibly nonlinear. Sufficient conditions are given for continuity of $u(k)$ with respect to $k$.…
We study the problem of whether all bipartite quantum states having a prescribed spectrum remain positive under the reduction map applied to one subsystem. We provide necessary and sufficient conditions, in the form of a family of linear…
An operator $T\in B(H)$ is said to satisfy property ($UW${\scriptsize \it{E}}) if the complement in the approximate point spectrum of the essential approximate point spectrum coincides with the isolated eigenvalues of the spectrum. Via the…
We introduce compactness classes of Hilbert space operators by grouping together all operators for which the associated singular values decay at a certain speed and establish upper bounds for the norm of the resolvent of operators belonging…
A parametric constrained convex optimal control problem, where the initial state is perturbed and the linear state equation contains a noise, is considered in this paper. Formulas for computing the subdifferential and the singular…
For a discrete dynamics defined by a sequence of bounded and not necessarily invertible linear operators, we give a complete characterization of exponential stability in terms of invertibility of a certain operator acting on suitable Banach…
A class of linear hyperbolic partial differential equations, sometimes called networks of waves, is considered. For this class of systems, necessary and sufficient conditions are formulated on the system matrices for the operator dynamics…
In this work we introduce the notion of an angular spectrum for a linear discrete time nonautonomous dynamical system. The angular spectrum comprises all accumulation points of longtime averages formed by maximal principal angles between…
Performance of optimization on quadratic problems sensitively depends on the low-lying part of the spectrum. For large (effectively infinite-dimensional) problems, this part of the spectrum can often be naturally represented or approximated…
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…