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- W1479693586 abstract "We present an abstract multiscale analysis scheme for matrix functions $(H_{varepsilon}(m,n))_{m,nin mathfrak{T}}$, 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 $H_{tildeomega}=-frac{d^2}{dx^2} + varepsilon U(tildeomega x)$, where $U$ is a real smooth function on the torus $mathbb{T}^nu$, $tildeomegain mathbb{R}^nu$ is a vector with rational components, and $varepsilon$ is a small parameter. The group in this particular case is the quotient $mathfrak{T} = mathbb{Z}^nu/{minmathbb{Z}^nu:mtildeomega=0}$. We show that the general theory indeed applies to this special case, provided that the rational frequency vector $tildeomega$ obeys a suitable Diophantine condition in a large box of modes. Despite the fact that in this setting the orbits $k + momega$, $k in mathbb{R}$, $minmathbb{Z}^nu$ are not dense, the dual eigenfunctions are exponentially localized and the eigenvalues of the operators can be described as $E(k+momega)$ 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." @default.
- W1479693586 created "2016-06-24" @default.
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- W1479693586 date "2014-09-07" @default.
- W1479693586 modified "2023-09-27" @default.
- W1479693586 title "A Multi-Scale Analysis Scheme on Abelian Groups with an Application to Operators Dual to Hill's Equation" @default.
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- W1479693586 hasPublicationYear "2014" @default.
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