# The spectrum of a Schrödinger operator with small quasi-periodic potential is homogeneous

### David Damanik

Rice University, Houston, United States### Michael Goldstein

University of Toronto, Canada### Milivoje Lukic

Rice University, Houston, USA

## Abstract

We consider the quasi-periodic Schrödinger operator

$[H \psi] (x) = -\psi''(x) + V(x) \psi(x)$

in $L^2(\mathbb R)$, where the potential is given by

$V(x) = \sum_{m \in \Z^\nu \setminus \{ 0 \}} c(m)\exp (2\pi i m \omega x)$

with a Diophantine frequency vector $\omega = (\omega_1, \dots, \omega_\nu) \in \mathbb R^\nu$ and exponentially decaying Fourier coefficients $|c(m)| \le \varepsilon \mathrm {exp} (-\kappa_0|m|)$. In the regime of small $\varepsilon > 0$ we show that the spectrum of the operator $H$ is homogeneous in the sense of Carleson.

## Cite this article

David Damanik, Michael Goldstein, Milivoje Lukic, The spectrum of a Schrödinger operator with small quasi-periodic potential is homogeneous. J. Spectr. Theory 6 (2016), no. 2, pp. 415–427

DOI 10.4171/JST/128