Related papers: Zero measure Cantor spectra for continuum one-dime…

The formulation of a new analysis on a zero measure Cantor set $C (\subset I=[0,1])$ is presented. A non-archimedean absolute value is introduced in $C$ exploiting the concept of {\em relative} infinitesimals and a scale invariant…

We numerically study energy spectra and localization properties of the double exchange model at irrational filling factor. To obtain variational ground state, we use a mumerical technique in momentum space by ``embedded'' boundary condition…

It is shown that the complete localization of eigenvectors for the almost Mathieu operator entails the absence of Cantor spectrum for this operator.

We show that, for one-dimensional discrete Schr\"odinger operators, stability of Anderson localization under a class of rank one perturbations implies absence of intervals in spectra. The argument is based on well-known result of Gordon and…

We present a result of absence of absolutely continuous spectrum in an interval of $\R$, for a matrix-valued random Schr\"odinger operator, acting on $L^2(\R)\otimes \R^N$ for an arbitrary $N\geq 1$, and whose interaction potential is…

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…

Each Cantor measure (\mu) with scaling factor 1/(2n) has at least one associated orthonormal basis of exponential functions (ONB) for L^2(\mu). In the particular case where the scaling constant for the Cantor measure is 1/4 and two specific…

We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux $\Phi/2\pi$ through a honeycomb is…

We consider the Schr\"odinger operator with a periodic potential on quasi-1D models of armchair single-wall nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite…

We consider a Schr\"odinger operator $H=-\Delta+V(\vec x)$ in dimension two with a quasi-periodic potential $V(\vec x)$. We prove that the absolutely continuous spectrum of $H$ contains a semiaxis and there is a family of generalized…

We define a class of pseudo-ergodic non-self-adjoint Schr\"odinger operators acting in spaces $l^2(X)$ and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a non-self-adjoint Anderson…

Given a self-adjoint operator H, a self-adjoint trace class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and co-kernel, using limiting absorption principle an explicit set of full Lebesgue measure is defined such…

We study in dimension $d\geq2$ low-energy spectral and scattering asymptotics for two-body $d$-dimensional Schr\"odinger operators with a radially symmetric potential falling off like $-\gamma r^{-2},\;\gamma>0$. We consider angular…

For a large family of real-valued Radon measures m on R^d, including the Kato class, the operators -\Delta + C^2 \Delta^2 + m tend to the Schrodinger operator -\Delta +m in the norm resolvent sense as C tends to zero. If the measure is…

We show that for almost every frequency alpha \in \R \setminus \Q, for every C^omega potential v:\R/\Z \to R, and for almost every energy E the corresponding quasiperiodic Schrodinger cocycle is either reducible or nonuniformly hyperbolic.…

We introduce a unitary almost-Mathieu operator, which is obtained from a two-dimensional quantum walk in a uniform magnetic field. We exhibit a version of Aubry--Andr\'{e} duality for this model, which partitions the parameter space into…

We prove that the quasi-periodic Schr\"{o}dinger operator with a finitely differentiable potential has purely absolutely continuous spectrum for all phases if the frequency is Diophantine and the potential is sufficiently small in the…

We study the spectrum of operators in the Schwartz space of rapidly decreasing functions which associate each function with its composition with a polynomial. In the case where this operator is mean ergodic we prove that its spectrum…

This paper is a survey devoted to the study of probability and infinite ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We focus mostly on the cases when a homeomorphism has either a unique ergodic invariant measure…

Fermi surfaces are basic objects in solid state physics and in the spectral theory of periodic operators. We define several measures connected to Fermi surfaces and study their measure theoretic properties. From this we get absence of…