Related papers: Absolutely Continuous Spectrum For Schr\"odinger O…

The subject of the paper are Schr\"odinger operators on tree graphs which are radial having the branching number $b_n$ at all the vertices at the distance $t_n$ from the root. We consider a family of coupling conditions at the vertices…

In this paper we consider two classes of random Hamiltonians on $L^2(\RR^d)$ one that imitates the lattice case and the other a Schr\"odinger operator with non-decaying, non-sparse potential both of which exhibit a.c. spectrum. In the…

We show that a generic quasi-periodic Schr\"odinger operator in $L^2(\mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling…

We investigate spectral properties of limit-periodic Schr\"odinger operators in $\ell^2(\Z)$. Our goal is to exhibit as rich a spectral picture as possible. We regard limit-periodic potentials as generated by continuous sampling along the…

We prove new criteria of stability of the absolutely continuous spectrum of one-dimensional Schr\"odinger operators under slowly decaying perturbations. As applications, we show that the absolutely continuous spectrum of the free and…

I prove that quasi-periodic Schr\"odinger operators in arbitrary dimension have some absolutely continuous spectrum.

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…

It is known that the spectrum of Schr\"odinger operators with sparse potentials consists of singular continuous spectrum. We give a sufficient condition so that the edge of the singular continuous spectrum is not an eigenvalue and construct…

We consider normalized Laplacians and their perturbations by periodic potentials (Schr\"odinger operators) on periodic discrete graphs. The spectrum of the operators consists of an absolutely continuous part (a union of a finite number of…

We consider Schr\"odinger operators $H=-\Delta+V({\mathbf x})$ in ${\mathbb R}^d$, $d\geq2$, with quasi-periodic potentials $V({\mathbf x})$. We prove that the absolutely continuous spectrum of a generic $H$ contains a semi-axis…

This paper deals with general structural properties of one-dimensional Schr"odinger operators with some absolutely continuous spectrum. The basic result says that the omega limit points of the potential under the shift map are…

We consider discrete Schr\"odinger operators with pattern Sturmian potentials. This class of potentials strictly contains the class of Sturmian potentials, for which the spectral properties of the associated Schr\"odinger operators are well…

The Bethe Strip of width $m$ is the cartesian product $\B\times\{1,...,m\}$, where $\B$ is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have "extended states" for small disorder. More precisely, we…

We consider random Schr\"odinger operators on tree graphs and prove absolutely continuous spectrum at small disorder for two models. The first model is the usual binary tree with certain strongly correlated random potentials. These…

Here we show that for Schr\"{o}dinger operator with decaying random potential with fat tail single site distribution, the negative spectrum shows a transition from essential spectrum to discrete spectrum. We study the Schr\"{o}dinger…

We prove that 3-dimensional Schrodinger operator with slowly decaying potential has an absolutely continuous spectrum that fills the positive half-line. The asymptotics of Green's function is obtained as well.

We investigate the spectral properties of Schr\"odinger operators in l^2(Z) with limit-periodic potentials. The perspective we take was recently proposed by Avila and is based on regarding such potentials as generated by continuous sampling…

We consider the discrete Schr\"odinger operator $H=-\Delta+V$ with a sparse potential $V$ and find conditions guaranteeing either existence of wave operators for the pair $H$ and $H_0=-\Delta$, or presence of dense purely point spectrum of…

We consider Schr\"odinger operators in $\ell^2(\mathbb{Z})$ whose potentials are given by the sum of an ergodic term and a random term of Anderson type. Under the assumption that the ergodic term is generated by a homeomorphism of a…

We review a geometric approach to proving absolutely continuous (ac) spectrum for random and deterministic Schr\"odinger operators developed in \cite{FHS1,FHS2,FHS3,FHS4}. We study decaying potentials in one dimension and present a…