Differential and Integral Equations

A finite range operator with a quasi-periodic potential

Steve Surace

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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

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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

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