Related papers: Gap Labelling for Discrete One-Dimensional Ergodic…
We consider Schr\"odinger operators $H^h = (ih d+{\bf A})^* (ih d+{\bf A})$ with the periodic magnetic field ${\bf B}=d{\bf A}$ on covering spaces of compact manifolds. Under some assumptions on $\bf B$, we prove that there are arbitrarily…
This work studies geometrical characterizations of the essential spectrum $\sigma_{\text ess}$ of Schr\"odinger operators on graphs. Especially we focus on generalizing characterizations which are given in terms of the concept of right…
A new characterization of the singular packing subspaces of general bounded self-adjoint operators is presented, which is used to show that the set of operators whose spectral measures have upper packing dimension equal to one is a…
We construct multidimensional Schr\"odinger operators with a spectrum that has no gaps at high energies and that is nowhere dense at low energies. This gives the first example for which this widely expected topological structure of the…
We exhibit d-dimensional limit-periodic Schrodinger operators that are uniformly localized in the strongest sense possible. That is, for each of these operators, there is a uniform exponential decay rate such that every element of the hull…
We consider ergodic random magnetic Schr\"odinger operators on the metric graph $\mathbb{Z}^d$ with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting…
In this paper, we prove a discrete version of the Bethe-Sommerfeld conjecture. Namely, we show that the spectra of multi-dimensional discrete periodic Schr\"odinger operators on $\mathbb{Z}^d$ lattice with sufficiently small potentials…
Quasiperiodic systems show a universal gap structure due to quasiperiodicity which is analogous to gap openings at the Brillouin zone boundary in periodic systems. The integrated density of states (IDoS) below those energy gaps are…
We prove sharp lower bounds on the spectral gap of 1-dimensional Schr\"odinger operators with Robin boundary conditions for each value of the Robin parameter. In particular, our lower bounds apply to single-well potentials with a centered…
We compute the Schwartzman group associated with an ergodic affine automorphism of a compact connected abelian group given by the composition of an automorphism of the group and a translation by an element in the path component of the…
We study discrete Schr\"odinger operators on the graphs corresponding to the triangular lattice, the hexagonal lattice, and the square lattice with next-nearest neighbor interactions. For each of these lattice geometries, we analyze the…
Commutator relations are used to investigate the spectra of Schr\"odinger Hamiltonians, $H = -\Delta + V({x}),$ acting on functions of a smooth, compact $d$-dimensional manifold $M$ immersed in $\bbr^{\nu}, \nu \geq d+1$. Here $\Delta$…
We demonstrate criteria, purely based on finite subwords of the potential, to guarantee spectral inclusion as well as Hausdorff approximation of pseudospectra or even spectra of generalized Schr\"odinger operators on the discrete line or…
We consider discrete one-dimensional Schr\"odinger operators with strictly ergodic, aperiodic potentials taking finitely many values. The well-known tendency of these operators to have purely singular continuous spectrum of zero Lebesgue…
Using an extension of the H\"ormander product of distributions, we obtain an intrinsic formulation of one-dimensional Schr\"odinger operators with singular potentials. This formulation is entirely defined in terms of standard {\it Schwartz}…
We consider discrete Schr{\"o}dinger operators on ${\mathbb{Z}}^d$ for which the perturbation consists of the sum of a long-range type potential and a Wigner-von Neumann type potential. Still working in a framework of weighted Mourre…
I present an example of a discrete Schr"odinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than…
The paper is devoted to the study of the essential spectrum of discrete Schr\"{o}dinger operators on the lattice $\mathbb{Z}^{N}$ by means of the limit operators method. This method has been applied by one of the authors to describe the…
We analyze Schr\"odinger operators whose potential is given by a singular interaction supported on a sub-manifold of the ambient space. Under the assumption that the operator has at least two eigenvalues below its essential spectrum we…
We show that the measure of the spectrum of Schr\"odinger operator with potential defined by non-constant function over any minimal aperiodic finite subshift tends to zero, as the coupling constant tends to infinity. We also obtained a…