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

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 $\mathrm{\mid }\cdot \mathrm{\mid }$. 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$U$isarealsmoothfunctiononthetorus$$\mathbb{T}$^\nu$,$\tilde\omega\in $\mathbb{R}$^\nu$ is a vector with rational components, and $\epsilon$ is a small parameter. The group in this particular case is the quotient $\mathfrak{T} = $\mathbb{Z}$^\nu/{m\in$\mathbb{Z}$^\nu:m\tilde\omega=0}$. We show that the general theory indeed applies to this special case, provided that the rational frequency vector $\tilde{\omega }$ obeys a suitable Diophantine condition in a large box of modes. Despite the fact that in this setting the orbits $k + m\omega$,$k \in $\mathbb{R}$$,$m\in$\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.