Related papers: Absolutely Continuous Spectrum for the Anderson Mo…
We prove delocalization for the Anderson model on an infinite regular tree (or Cayley graph or Bethe lattice) at low disorder. This extends earlier results of Klein and Aizenman--Warzel by filling in the previously missing parts of the…
I prove that quasi-periodic Schr\"odinger operators in arbitrary dimension have some absolutely continuous spectrum.
We summarize recent works on the stability under disorder of the absolutely continuous spectra of random operators on tree graphs. The cases covered include: Schr\"odinger operators with random potential, quantum graph operators for trees…
We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely…
Our recently established criterion for the formation of extended states on tree graphs in the presence of disorder is shown to have the surprising implication that for bounded random potentials, as in the Anderson model, there is no…
We give sufficient conditions for the presence of the absolutely continuous spectrum of a Schr\"odinger operator on a regular rooted tree without loops (also called regular Bethe lattice or Cayley tree).
We study the spectrum of random operators on a large class of trees. These trees have finitely many cone types and they can be constructed by a substitution rule. The random operators are perturbations of Laplace type operators either by…
We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures. We use these…
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 prove an averaging formula for the derivative of the absolutely continuous part of the density of states measure for an ergodic family of CMV matrices. As a consequence, we show that the spectral type of such a family is almost surely…
We present general principles for the preservation of a.c. spectrum under weak perturbations. The Schrodinger operators on the strip and on the Caley tree (Bethe lattice) are considered.
We consider perturbations of quasi-periodic Schr\"odinger operators on the integer lattice with analytic sampling functions by decaying potentials and seek decay conditions under which various spectral properties are preserved. In the…
We consider multi-type Galton Watson trees that are close to a tree of finite cone type in distribution. Moreover, we impose that each vertex has at least one forward neighbor. Then, we show that the spectrum of the Laplace operator…
The subject of this work are random Schroedinger operators on regular rooted tree graphs $\T$ with stochastically homogeneous disorder. The operators are of the form $H_\lambda(\omega) = T + U + \lambda V(\omega)$ acting in $\ell^2(\T)$,…
In this work the spectral theory of self-adjoint operator $A$ represented by Jacobi matrix is considered. The approach is based on the continued fraction representation of the resolvent matrix element of $A$. Different criteria of absolute…
We investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely continuous spectra of two model examples of generalized indefinite…
We prove that the spectrum of the discrete Schr\"odinger operator on $\ell^2(Z^2)$, $(\psi_{n,m})\mapsto -(\psi_{n+1,m} +\psi_{n-1,m} +\psi_{n,m+1} +\psi_{n,m-1})+V_n\psi_{n,m}$ is absolutely continuous.
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…
We consider the multi-particle Anderson model in the continuum and show that under some mild assumptions on the inter-particle interaction and the external potential, its lower spectral edge is almost surely constant and is the same with…
We show that the Kronecker sum of d >= 2 copies of a random one-dimensional sparse model displays a spectral transition of the type predicted by Anderson, from absolutely continuous around the center of the band to pure point around the…