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Transactions of the American Mathematical Society

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A multi-scale analysis scheme on Abelian groups with an application to operators dual to Hill's equation


Authors: David Damanik, Michael Goldstein and Milivoje Lukic
Journal: Trans. Amer. Math. Soc. 369 (2017), 1689-1755
MSC (2010): Primary 35J10
DOI: https://doi.org/10.1090/tran/6712
Published electronically: May 6, 2016
MathSciNet review: 3581217
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Abstract: We present an abstract multi-scale 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 $ \vert\cdot \vert$. 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

$\displaystyle [H_{\tilde \omega } y](x)= -y''(x) + \varepsilon U(\tilde \omega x) y(x), \quad x \in \mathbb{R}, $

where $ U(\theta )$ is a real smooth function on the torus $ \mathbb{T}^\nu $, $ \tilde \omega = (\tilde \omega _1,\dots ,\tilde \omega _\nu )\in \mathbb{R}^\nu $ is a vector with rational components, and $ \varepsilon \in \mathbb{R}$ 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.

Although we are mostly interested in the case of analytic $ U$, we need to analyze, for technical reasons, the functions $ U$ with sub-exponentially decaying Fourier coefficients in the current work.

The main novelty for the experts in this field is that, while all known multi-scale schemes run for an irrational Diophantine frequencies vector, we can run the analysis for a rational vector. The main reason why we are able to do this is that the abstract scheme in the method by Damanik and Goldstein (2014) is flexible enough to accommodate this case. In the current work we verify that one can put the case in question in the setup of the above-mentioned work.

The verification carried out in the current work also gives a strong indication that the method of Damanik and Goldstein can be applied to limit-periodic potentials.


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Additional Information

David Damanik
Affiliation: Department of Mathematics, Rice University, 6100 South Main Street, Houston, Texas 77005-1892
Email: damanik@rice.edu

Michael Goldstein
Affiliation: Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4
Email: gold@math.toronto.edu

Milivoje Lukic
Affiliation: Department of Mathematics, Rice University, 6100 South Main Street, Houston Texas 77005-1892 – and – Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4
Email: mlukic@math.toronto.edu

DOI: https://doi.org/10.1090/tran/6712
Received by editor(s): November 12, 2014
Received by editor(s) in revised form: March 4, 2015
Published electronically: May 6, 2016
Additional Notes: The first author was partially supported by a Simons Fellowship and NSF grants DMS–0800100, DMS–1067988, and DMS–1361625
The second author was partially supported by NSERC
The third author was partially supported by NSF grant DMS–1301582
Article copyright: © Copyright 2016 by the authors

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