Perturbation theory for almost-periodic potentials I. One-dimensional case

TL;DR

This paper analyzes the asymptotic behavior of the integrated density of states for one-dimensional Schrödinger operators with almost-periodic potentials, revealing detailed expansions and exceptional sets for various potential types.

Contribution

It provides the first complete asymptotic expansions of the IDS for quasi-periodic potentials and characterizes the super-resonance set for non-quasi-periodic cases.

Findings

Asymptotic expansions of IDS are complete for quasi-periodic potentials.

Existence of an uncountable measure-zero super-resonance set for non-quasi-periodic potentials.

Spectral gap lengths have asymptotic expansions in powers of epsilon.

Abstract

We consider the family of operators $H^{(\epsilon)}:=-\frac{d^2}{dx^2}+\epsilon V$ in ${\mathbb R}$ with almost-periodic potential $V$. We study the behaviour of the integrated density of states (IDS) $N(H^{(\epsilon)};\lambda)$ when $\epsilon\to 0$ and $\lambda$ is a fixed energy. When $V$ is quasi-periodic (i.e. is a finite sum of complex exponentials), we prove that for each $\lambda$ the IDS has a complete asymptotic expansion in powers of $\epsilon$; these powers are either integer, or in some special cases half-integer. These results are new even for periodic $V$. We also prove that when the potential is neither periodic nor quasi-periodic, there is an exceptional set $\mathcal S$ of energies (which we call $\hbox{the super-resonance set}$) such that for any $\sqrt\lambda\not\in\mathcal S$ there is a complete power asymptotic expansion of IDS, and when $\sqrt\lambda\in\mathcal S$, then even two-terms power asymptotic expansion does not exist. We also show that the super-resonant set $\mathcal S$ is uncountable, but has measure zero. Finally, we prove that the length of any spectral gap of $H^{(\epsilon)}$ has a complete asymptotic expansion in natural powers of $\epsilon$ when $\epsilon\to 0$.