Exponential decay of the size of spectral gaps for quasiperiodic Schrödinger operators
arXiv:1607.03422
Abstract
In the following we are interested in the spectral gaps of discrete quasiperiodic Schrödinger operators when the frequency is Diophantine, the potential is analytic, and in the subcritical regime. The gap-labelling theorem asserts in this context that each gap has constant rotation number, labeled by some integer. We prove that the size of these gaps decays exponentially fast with respect to their label. This refines a subexponential bound obtained previously by Sana Ben Hadj Amor. Contrary to her approach, which is based on KAM methods, the arguments in the present paper are non-perturbative, and use quantitative reducibility estimates obtained by Artur Avila and Svetlana Jitomirskaya. As a corollary of our result, we show that under the previous assumptions, the spectrum is 1/2-homogeneous.
The results of this paper will be included in a joint work with J. You, Z. Zhao and Q. Zhou in which we obtain improved estimates on the size of spectral gaps of quasiperiodic Schrödinger operators as well as results on the homogeneity of the spectrum of such operators, and explain how our results can be applied to the Toda lattice