Distributions of resonances of supercritical quasi-periodic operators

TL;DR

This paper investigates how the distribution of resonances influences the spectral properties of supercritical quasi-periodic Schrödinger operators, providing counterexamples to a longstanding conjecture about spectral transitions.

Contribution

It reveals the significance of resonance distributions in spectral classification and disproves Jitomirskaya's second spectral transition line conjecture from the 1990s.

Findings

Resonance distribution affects spectral type

Disproves Jitomirskaya's conjecture

Provides new insights into spectral transitions

Abstract

We discover that the distribution of (frequency and phase) resonances plays a role in determining the spectral type of supercritical quasi-periodic Schr\"odinger operators. In particular, we disprove the second spectral transition line conjecture of Jitomirskaya in the early 1990s.