Differential and Integral Equations

A finite range operator with a quasi-periodic potential

Steve Surace

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider the operator $ H=\varepsilon h+\cos(\alpha\cdot j+\vartheta) $ for $j\in\mathbb{Z}^\nu, $ where $h$ is self-adjoint, translation invariant and finite range. The vector $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_\nu)$ is assumed to have the diophantine property $ |j\cdot\alpha\,\text{mod}\,2\pi|\ge C/|j|^2, $ where $j\ne0$ and $C$ is some constant. For $\varepsilon$ sufficiently small we prove that $H$ has pure point spectrum for almost every $\vartheta$. Moreover, every polynomially bounded eigenfunction of $H$ decays exponentially fast. Finally, we will show how this operator comes up in the study of electrons in a transverse magnetic field subject to a two dimensional periodic potential.

Article information

Differential Integral Equations, Volume 9, Number 1 (1996), 213-237.

First available in Project Euclid: 7 May 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B39: Difference operators [See also 39A70]
Secondary: 39A70: Difference operators [See also 47B39] 47N50: Applications in the physical sciences 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis


Surace, Steve. A finite range operator with a quasi-periodic potential. Differential Integral Equations 9 (1996), no. 1, 213--237. https://projecteuclid.org/euclid.die/1367969998

Export citation