Homogeneity of the spectrum for quasi-periodic Schrödinger operators

  • David Damanik

    Rice University, Houston, USA
  • Michael Goldstein

    University of Toronto, Canada
  • Wilhelm Schlag

    University of Chicago, USA
  • Mircea Voda

    University of Chicago, USA


We consider the one-dimensional discrete Schrödinger operator

, with real-analytic potential . Assume for all . Let be the spectrum of . For all obeying the Diophantine condition , we show the following: if , then is homogeneous in the sense of Carleson (see [Car83]). Furthermore, we prove, that if , are two gaps with , then , . Moreover, the same estimates hold for the gaps in the spectrum on a finite interval, that is, for , , where is the Schrödinger operator restricted to the interval with Dirichlet boundary conditions. In particular, all these results hold for the almost Mathieu operator with . For the supercritical almost Mathieu operator, we combine the methods of [GolSch08] with Jitomirskaya's approach from [Jit99] to establish most of the results from [GolSch08] with obeying a strong Diophantine condition.

Cite this article

David Damanik, Michael Goldstein, Wilhelm Schlag, Mircea Voda, Homogeneity of the spectrum for quasi-periodic Schrödinger operators. J. Eur. Math. Soc. 20 (2018), no. 12, pp. 3073–3111

DOI 10.4171/JEMS/829