## Differential and Integral Equations

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

### A finite range operator with a quasi-periodic potential

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 7 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367969998

**Mathematical Reviews number (MathSciNet)**

MR1364044

**Zentralblatt MATH identifier**

0838.47018

**Subjects**

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

#### Citation

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