JournalsjstVol. 6, No. 2pp. 415–427

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

We consider the quasi-periodic Schrödinger operator

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

in L2(R)L^2(\mathbb R), where the potential is given by

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

with a Diophantine frequency vector ω=(ω1,,ων)Rν\omega = (\omega_1, \dots, \omega_\nu) \in \mathbb R^\nu and exponentially decaying Fourier coefficients c(m)εexp(κ0m)|c(m)| \le \varepsilon \mathrm {exp} (-\kappa_0|m|). In the regime of small ε>0\varepsilon > 0 we show that the spectrum of the operator HH 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