Related papers: Multidimensional Almost-Periodic Schr\"odinger Ope…

We show that a large class of limit-periodic Schr\"odinger operators has purely absolutely continuous spectrum in arbitrary dimensions. This result was previously known only in dimension one. The proof proceeds through the non-perturbative…

We establish a new and simple criterion that suffices to generate many spectral gaps for periodic word models. This leads to new examples of ergodic Schr\"odinger operators with Cantor spectra having zero Hausdorff dimension that…

The discrete spectra of certain two-dimensional Schrodinger operators are numerically calculated. These operators have interesting spectral properties, i.e. their kernels are multi-dimensional and the deformations of potentials via the…

We study the spectral properties of ergodic Schr\"{o}dinger operators that are associated to a certain family of non-primitive substitutions on a binary alphabet. The corresponding subshifts provide examples of dynamical systems that go…

We study Schroedinger operators with periodic magnetic field in Euclidean 2-space, in the case of irrational magnetic flux. Positive measure Cantor spectrum is generically expected in the presence of an electric potential. We show that,…

We investigate the kernels of the transformation operators for one-dimensional Schroedinger operators with potentials, which are asymptotically close to Bohr almost periodic infinite-gap potentials.

In this paper we study spectral properties of Schr\"odinger operators with quasi-periodic potentials related to quasi-periodic action minimizing trajectories for analytic twist maps. We prove that the spectrum contains a component of…

We discuss discrete one-dimensional Schr\"odinger operators whose potentials are generated by an invertible ergodic transformation of a compact metric space and a continuous real-valued sampling function. We pay particular attention to the…

We consider continuous $SL(2,\mathbb{R})$-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be…

We continue the investigation of the existence of absolutely continuous (a.c.) spectrum for the discrete Schr\"odinger operator $\Delta+V$ on $\ell^2(\Z^d)$, in dimensions $d\geq 2$, for potentials $V$ satisfying the long range condition…

For the square tridiagonal Fibonacci Hamiltonian, we prove existence of an open set of parameters which yield mixed interval-Cantor spectra (i.e. spectra containing an interval as well as a Cantor set), as well as mixed density of states…

We consider the quasi-periodic Schr\"odinger operator $$ [H \psi](x) = -\psi"(x) + V(x) \psi(x) $$ in $L^2(\mathbb{R})$, where the potential is given by $$ V(x) = \sum_{m \in \mathbb{Z}^\nu \setminus \{ 0 \}} c(m)\exp (2\pi i m \omega x) $$…

We give a spectral description of the semi-classical Schrodinger operator with a piecewise linear, complex valued potential. Moreover, using these results, we show how an arbitrarily small bounded perturbation of a non-self-adjoint operator…

We consider separable 2D discrete Schr\"odinger operators generated by 1D almost Mathieu operators. For fixed Diophantine frequencies we prove that for sufficiently small couplings the spectrum must be an interval. This complements a result…

This paper is concerned with discrete, one-dimensional Schr\"odinger operators with real analytic potentials and one Diophantine frequency. Using localization and duality we show that almost every point in the spectrum admits a…

We prove that limit-periodic Dirac operators generically have spectra of zero Lebesgue measure and that a dense set of them have spectra of zero Hausdorff dimension. The proof combines ideas of Avila from a Schr\"odinger setting with a new…

We consider Sch\"odinger operators on the half-line, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if $\Delta +…

We prove that the spectrum of a limit-periodic Schr\"odinger operator is homogeneous in the sense of Carleson whenever the potential obeys the Pastur--Tkachenko condition. This implies that a dense set of limit-periodic Schr\"odinger…

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 a family of multi-dimensional Schr\"odinger operators $-\Delta+t V$ with a real $t$. The potential $V$ in our model decays at infinity in a special way, so that it satisfies a certain integral condition. We prove that the…