Convergence of perturbation series for unbounded monotone quasiperiodic operators

TL;DR

This paper proves the convergence of perturbation series for a class of unbounded quasiperiodic operators with monotone potentials, leading to a new proof of Anderson localization with explicit eigenvalue and eigenvector expansions.

Contribution

It introduces a novel convergence proof for perturbation series in unbounded quasiperiodic operators, extending localization results to more general potential classes.

Findings

Convergence of Rayleigh--Schr"odinger perturbation series in small kinetic energy regime.

Explicit series expansions for eigenvalues and eigenvectors.

Generalization of Anderson localization to broader operator classes.

Abstract

We consider a class of unbounded quasiperiodic Schr\"odinger-type operators on $\ell^2(\mathbb Z^d)$ with monotone potentials (akin to the Maryland model) and show that the Rayleigh--Schr\"odinger perturbation series for these operators converges in the regime of small kinetic energies, uniformly in the spectrum. As a consequence, we obtain a new proof of Anderson localization in a more general than before class of such operators, with explicit convergent series expansions for eigenvalues and eigenvectors. This result can be restricted to an energy window if the potential is only locally monotone and one-to-one. A modification of this approach also allows the potential to be non-strictly monotone and have a flat segment, under additional restrictions on the frequency.