A Multi-Scale Analysis Scheme on Abelian Groups with an Application to Operators Dual to Hill's Equation

Abstract

We present an abstract multiscale analysis scheme for matrix functions $(H_{\epsilon }(m,n){)}_{m,n\in \mathfrak{T}}$, where $\mathfrak{T}$ is an Abelian group equipped with a distance $|\cdot |$. This is an extension of the scheme developed by Damanik and Goldstein for the special case $\mathfrak{T}={\mathbb{Z}}^{\nu }$. Our main motivation for working out this extension comes from an application to matrix functions which are dual to certain Hill operators. These operators take the form $H_{\tilde{\omega }}=-\frac{d^2}{d{x}^2}+\epsilon U(\tilde{\omega }x)$, where $U$ is a real smooth function on the torus ${\mathbb{T}}^{\nu }$, $\tilde{\omega }\in {\mathbb{R}}^{\nu }$ is a vector with rational components, and $\epsilon$ is a small parameter. The group in this particular case is the quotient $\mathfrak{T}={\mathbb{Z}}^{\nu }/\{m\in {\mathbb{Z}}^{\nu }:m\tilde{\omega }=0\}$. We show that the general theory indeed applies to this special case, provided that the rational frequency vector $\tilde{\omega }$ obeys a suitable Diophantine condition in a large box of modes. Despite the fact that in this setting the orbits $k+m\omega$, $k\in \mathbb{R}$, $m\in {\mathbb{Z}}^{\nu }$ are not dense, the dual eigenfunctions are exponentially localized and the eigenvalues of the operators can be described as $E(k+m\omega )$ with $E(k)$ being a "nice" monotonic function of the impulse $k\ge 0$. This enables us to derive a description of the Floquet solutions and the band-gap structure of the spectrum, which we will use in a companion paper to develop a complete inverse spectral theory for the Sturm-Liouville equation with small quasi-periodic potential via periodic approximation of the frequency. The analysis of the gaps in the range of the function $E(k)$ plays a crucial role in this approach.

arXiv comment53 pages. This paper generalizes the results from arXiv:1209.4331 to general Abelian groups