Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schr\"{o}dinger operators

TL;DR

This paper investigates one-dimensional adiabatic quasi-periodic Schrödinger operators, revealing how strong resonant tunneling influences spectral properties and causes level repulsion, which varies with the spectrum's singularity.

Contribution

It demonstrates the emergence of level repulsion linked to the local spectral type, providing new insights into spectral behavior under strong resonant tunneling in quasi-periodic operators.

Findings

Level repulsion phenomenon depends on spectral singularity

Stronger resonant tunneling leads to weaker level repulsion

Spectral type influences the strength of level interactions

Abstract

In this paper, we consider one dimensional adiabatic quasi-periodic Schr\"{o}dinger operators in the regime of strong resonant tunneling. We show the emergence of a level repulsion phenomenon which is seen to be very naturally related to the local spectral type of the operator: the more singular the spectrum, the weaker the repulsion.