Convergence of perturbation series for unbounded monotone quasiperiodic operators

Abstract

We consider a class of unbounded quasiperiodic Schrödinger-type operators on ℓ2(Zd) with monotone potentials (akin to the Maryland model) and show that the Rayleigh–Schrödinger 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 frequencies.