Comb domains of Schrödinger operators with small quasiperiodic potentials

TL;DR

This work analyzes Schrödinger operators on $\mathbb{R}$ with small analytic quasiperiodic potentials, establishing a precise link between spectral data and Marchenko--Ostrovskii (MO) maps via comb domains. By combining continuity and stability results for MO maps with periodic approximation and geometric function theory, the authors derive exponential decay relations between gap sizes and comb-domain slit heights, and show Lipschitz and $C^1$ regularity properties for gap edges and the action variables $I_m$. They develop a robust framework for translating spectral data into Dirichlet data on the isospectral torus, enabling a Hamiltonian/AKade framework-like structure (actions and angles) in the quasiperiodic setting with applications to the KdV equation for small initial data. The findings provide a concrete, quantitative bridge between spectral geometry, periodic approximation, and integrable-system techniques in the small-potential regime, with implications for homogeneity of the spectrum and dynamical evolution under KdV.

Abstract

We characterize spectra of Schrödinger operators with small quasiperiodic analytic potentials in terms of their comb domains, and study action variables motivated by the KdV integrable system.

Theorems & Definitions (70)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1: JMerratum
• Lemma 2.2
• proof
• Proposition 2.3
• proof