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- W4297908615 abstract "We consider the one-dimensional discrete Schrodinger operator $$ bigl[H(x,omega)varphibigr](n)equiv -varphi(n-1)-varphi(n+1) + V(x + nomega)varphi(n) , $$ $n in mathbb{Z}$, $x,omega in [0, 1]$ with real-analytic potential $V(x)$. Assume $L(E,omega)>0$ for all $E$. Let $mathcal{S}_omega$ be the spectrum of $H(x,omega)$. For all $omega$ obeying the Diophantine condition $omega in mathbb{T}_{c,a}$, we show the following: if $mathcal{S}_omega cap (E',E)neq emptyset$, then $mathcal{S}_omega cap (E',E)$ is homogeneous in the sense of Carleson (see [Car83]). Furthermore, we prove, that if $G_i$, $i=1,2$ are two gaps with $1 > |G_1| ge |G_2|$, then $|G_2|lesssim expleft(-(log mathrm{dist} (G_1,G_2))^Aright)$, $Agg 1$. Moreover, the same estimates hold for the gaps in the spectrum on a finite interval, that is, for $mathcal S_{N,omega}:=cup_{xinmathbb T}mathrm{spec} H_{[-N,N]}(x,omega) $, $N ge 1 $, where $H_{[-N, N]}(x, omega)$ is the Schrodinger operator restricted to the interval $[-N,N]$ with Dirichlet boundary conditions. In particular, all these results hold for the almost Mathieu operator with $|lambda| neq 1$. For the supercritical almost Mathieu operator, we combine the methods of [GolSch08] with Jitomirskaya's approach from [Jit99] to establish most of the results from [GolSch08] with $omega$ obeying a strong Diophantine condition." @default.
- W4297908615 created "2022-10-01" @default.
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- W4297908615 date "2015-05-19" @default.
- W4297908615 modified "2023-10-06" @default.
- W4297908615 title "Homogeneity of the spectrum for quasi-periodic Schrodinger operators" @default.
- W4297908615 doi "https://doi.org/10.48550/arxiv.1505.04904" @default.
- W4297908615 hasPublicationYear "2015" @default.
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