• Corpus ID: 247315112

Gap Labelling for Discrete One-Dimensional Ergodic Schr\"odinger Operators

  title={Gap Labelling for Discrete One-Dimensional Ergodic Schr\"odinger Operators},
  author={David Damanik and Jake Fillman},
In this survey, we give an introduction to and proof of the gap labelling theorem for discrete one-dimensional ergodic Schrödinger operators via the Schwartzman homomorphism. To keep the paper relatively self-contained, we include background on the integrated density of states, the oscillation theorem for 1D operators, and the construction of the Schwartzman homomorphism. We illustrate the result with some examples. In particular, we show how to use the Schwartzman formalism to recover the… 

The Spectrum of Schrödinger Operators with Randomly Perturbed Ergodic Potentials

We consider Schrödinger operators in $$\ell ^2({\mathbb Z})$$ ℓ 2 ( Z ) whose potentials are given by the sum of an ergodic term and a random term of Anderson type. Under the assumption that the

Johnson-Schwartzman Gap Labelling for Ergodic Jacobi Matrices

We consider two-sided Jacobi matrices whose coefficients are obtained by continuous sampling along the orbits of a homeomorphim of a compact metric space. Given an ergodic probability measure, we

Spectral characteristics of Schrödinger operators generated by product systems

We study ergodic Schr\"odinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational

The Almost Sure Essential Spectrum of the Doubling Map Model is Connected

We consider discrete Schrödinger operators on the half line with potentials generated by the doubling map and continuous sampling functions. We show that the essential spectrum of these operators is

Gap Labels for Zeros of the Partition Function of the 1D Ising Model via the Schwartzman Homomorphism

. Inspired by the 1995 paper of Baake–Grimm–Pisani, we aim to explain the empirical observation that the distribution of Lee–Yang zeros corresponding to a one-dimensional Ising model appears to

The Schwartzman Group of an Affine Transformation

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

Must the Spectrum of a Random Schrödinger Operator Contain an Interval?

We consider Schrödinger operators in ℓ2(Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}

Opening gaps in the spectrum of strictly ergodic

We consider Schr¨ odinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the

Uniform Hyperbolicity for Szeg\H{o} Cocycles and Applications to Random CMV Matrices and the Ising Model

We consider products of the matrices associated with the Szeg\H{o} recursion from the theory of orthogonal polynomials on the unit circle and show that under suitable assumptions, their norms grow

Cantor Spectrum for Schr\"odinger Operators with Potentials arising from Generalized Skew-shifts

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

Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrödinger operators

All the main results in this notes are well known. Yet it’s still interesting to give a concise, detailed and self-contained descriptions of some of the basic relations between the one dimensional


The spectra of Schrodinger operators on the square and cubic lattice are investigated by means of non-commutative topological K-theory. Using a general gap-labelling theorem, it is shown how to


Let be the number of eigenvalues not exceeding for the selfadjoint boundary problem with random potential , and let Our problem is to clarify the conditions under which this function will exist and

Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian

We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the

Almost localization and almost reducibility

We develop a quantitative version of Aubry duality and use it to obtain several sharp estimates for the dynamics of Schr\"odinger cocycles associated to a non-perturbatively small analytic potential


We study one dimensional tight binding hamiltonians with potentials given by automatic sequences. By means of Shubin’s formula, we show how K-theory allows to prove gap labelling theorems for their

Schrödinger operators with dynamically defined potentials

  • D. Damanik
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2016
In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schrödinger operators whose potentials are obtained by real-valued sampling along the orbits of an