Beschreibung:
<jats:title>Abstract</jats:title>
<jats:p>We establish a quantitative version of strong almost reducibility result for <jats:inline-formula>
<jats:tex-math><?CDATA $\mathrm{S}\mathrm{L}(2,\mathbb{R})$?></jats:tex-math>
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</jats:inline-formula> quasi-periodic cocycle close to a constant in the Gevrey class. We prove that, if the frequency is Diophantine, the long range operator has pure point spectrum with sub-exponentially decaying eigenfunctions for almost all phases; for the one dimensional quasi-periodic Schrödinger operators with small Gevrey potentials, the length of spectral gaps decays sub-exponentially with respect to its labelling; and the spectrum is an interval for discrete Schrödinger operators acting on <jats:inline-formula>
<jats:tex-math><?CDATA ${\mathbb{Z}}^{d}$?></jats:tex-math>
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<mml:msup>
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<mml:mi>d</mml:mi>
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<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonac98edieqn2.gif" xlink:type="simple" />
</jats:inline-formula> with small separable potentials.</jats:p>