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

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

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