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

TL;DR

This paper extends a multiscale analysis scheme to Abelian groups and applies it to dual Hill operators with rational frequencies, showing exponential localization of eigenfunctions and spectral band-gap structure.

Contribution

It develops a generalized multiscale analysis framework for matrix functions on Abelian groups and applies it to analyze spectral properties of dual Hill operators with rational frequencies.

Findings

Eigenfunctions are exponentially localized.

Eigenvalues are described by a monotonic function E(k).

Spectral band-gap structure is characterized.

Abstract

We present an abstract multiscale analysis scheme for matrix functions $(H_{\varepsilon}(m,n))_{m,n\in \mathfrak{T}}$, 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 $H_{\tilde\omega}=-\frac{d^2}{dx^2} + \varepsilon U(\tilde\omega x)$, 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 $\varepsilon$ 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.