Weakly resonant tunneling interactions for adiabatic quasi-periodic Schrodinger operators

TL;DR

This paper investigates how weakly resonant adiabatic quasi-periodic perturbations influence the spectral properties of one-dimensional periodic Schrödinger operators, revealing complex resonance effects that alter the spectrum's structure.

Contribution

It introduces a detailed analysis of resonance effects in adiabatic quasi-periodic Schrödinger operators, showing their impact on the spectrum's geometry and nature, including the emergence of absolutely continuous spectrum islands.

Findings

Resonance effects cause intertwining of spectral intervals.

Spectral regions can contain both absolutely continuous and singular spectra.

Exponential smallness of absolutely continuous spectrum islands in singular regions.

Abstract

In this paper, we study spectral properties of the one dimensional periodic Schrodinger operator with an adiabatic quasi-periodic perturbation. We show that in certain energy regions the perturbation leads to resonance effects related to the ones observed in the problem of two resonating quantum wells. These effects affect both the geometry and the nature of the spectrum. In particular, they can lead to the intertwining of sequences of intervals containing absolutely continuous spectrum and intervals containing singular spectrum. Moreover, in regions where all of the spectrum is expected to be singular, these effects typically give rise to exponentially small "islands" of absolutely continuous spectrum.