Homogeneity of the spectrum for quasi-periodic Schrödinger operators

Michael Goldstein; David Damanik; Wilhelm Schlag; Mircea Voda

paper

Homogeneity of the spectrum for quasi-periodic Schrödinger operators

arXiv:1505.04904

Abstract

We consider the one-dimensional discrete Schrödinger operator $[H (x, ω) φ] (n) \equiv - φ (n - 1) - φ (n + 1) + V (x + nω) φ (n),$ $n \in Z$ , $x, ω \in [0, 1]$ with real-analytic potential $V (x)$ . Assume $L (E, ω) > 0$ for all $E$ . Let $S_{ω}$ be the spectrum of $H (x, ω)$ . For all $ω$ obeying the Diophantine condition $ω \in T_{c, a}$ , we show the following: if $S_{ω} \cap (E^{'}, E ") \neq = \emptyset$ , then $S_{ω} \cap (E^{'}, E ")$ is homogeneous in the sense of Carleson (see [Car83]). Furthermore, we prove, that if $G_{i}$ , $i = 1, 2$ are two gaps with $1 > ∣ G_{1} ∣ \geq ∣ G_{2} ∣$ , then $∣ G_{2} ∣ ≲ exp (- (lo g dist (G_{1}, G_{2}))^{A})$ , $A ≫ 1$ . Moreover, the same estimates hold for the gaps in the spectrum on a finite interval, that is, for $S_{N, ω} := \cup_{x \in T} spec H_{[- N, N]} (x, ω)$ , $N \geq 1$ , where $H_{[- N, N]} (x, ω)$ is the Schrödinger operator restricted to the interval $[- N, N]$ with Dirichlet boundary conditions. In particular, all these results hold for the almost Mathieu operator with $∣ λ ∣ \neq = 1$ . 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 $ω$ obeying a strong Diophantine condition.

Homogeneity of the spectrum for quasi-periodic Schrödinger operators

Abstract

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