Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrödinger operators

Abstract

This paper intends to provide new, simple, and self-contained proofs of the equivalence of various different descriptions of the uniformly hyperbolic SL$(2,\mathbb{R})$. While in the scenario of the Schrödinger cocyles, they may in turn be applied to give new and simple proofs of theorems regarding their relation with the spectral analysis of one-dimensional discrete Schrödinger operators. Concretely, this paper gives four different descriptions of uniformly hyperbolic sequences together with the detailed proofs of their equivalence. It provides concise and self-contained proof of the Johnson’s Theorem [20], both for sequence and dynamically defined potentials. In particular, we give two different proofs of the direction “uniform hyperbolicity away from the spectrum,” of which one is the standard argument in the spirit of Russell Johnson’s original proof and the other involves Combes–Thomas type of estimate [11]. Finally, the relation between the avalanche principle, discovered by Goldstein and Schlag [16], and the uniformly hyperbolic sequence is explored which yields a simple proof and a better version of the avalanche principle. Many ingredients of the proofs in this paper are new, in particular the use of asymptotic stable and unstable directions, are of independent interest and have been applied to many other related problems.