Global rigidity for ultra-differentiable quasiperiodic cocycles and its spectral applications

Abstract

For quasiperiodic Schrödinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schrödinger cocycle is either rotations reducible or has positive Lyapunov exponent for all irrational frequency and almost every energy [AFK11]. From spectral theory side, the "Schrödinger conjecture" [AFK11] and the "Last's intersection spectrum conjecture" have been verified [JM12a]. The proofs of above results crucially depend on the analyticity of the potentials. People are curious about if the analyticity is essential for those problems, see open problems by Fayad-Krikorian [FK09, Kri] and Jitomirskaya-Marx [JM12a, MJ17]. In this paper, we prove the above mentioned results for ultra-differentiable potentials.1 We refer to Section 2.1 for the definitions and basic results. 2 Here α is Diophantine (denote α ∈ DC(v, τ )), if there exist v > 0 and τ > 1 such thatWe also denote DC := v>0, τ >1 DC(v, τ ) the union.

Transition space for the continuity of the Lyapunov exponent of quasiperiodic Schrödinger cocycles

“…Our new Lemmas enable us to not only greatly simplify the proofs in [47], but also optimize almost all estimates in [47]. Our construction is optimal since [17] has shown that for the case of s < 2, the Lyapunov exponent is continuous.…”

“…Remark 1.1. Part (1) of Theorem 1.1 was recently proved in [17] (Theorem 6.3 of [17]), we list here for completeness. The main result of the present paper is part (2).…”

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We construct discontinuous point of the Lyapunov exponent of quasiperiodic Schrödinger cocycles in the Gevrey space G s with s > 2. In contrast, the Lyapunov exponent has been proved to be continuous in the Gevrey space G s with s < 2 [17,37]. This shows that G 2 is the transition space for the continuity of the Lyapunov exponent.

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Upper bounds on quantum dynamics in arbitrary dimension

“…Combining the approximation argument from [24, Sections 2-4] with the method of [9, Lemma 4], one obtains that if 1 ≤ s < 2 and there exists a rational p/q with |α − p/q| < 1/q 2 , then (14) holds with ̺ = (2 − s)/s for Kq s ≤ N ≤ 2Kq s (cf. [11,Proposition 5], where this fact is stated with implicit exponents). To summarise: 9,24]).…”