Hyperbolicity of the Trace Map for a Strongly Coupled Quasiperiodic Schrodinger Operator

TL;DR

This paper proves that the trace map for a strongly coupled silver ratio Schrödinger operator is hyperbolic, leading to smooth, coinciding fractal dimensions of the spectrum as the coupling varies.

Contribution

It establishes hyperbolicity of the trace map for large coupling, connecting dynamical properties to spectral dimension regularity in quasiperiodic operators.

Findings

Non-wandering set is hyperbolic at large coupling

Spectral dimensions are smooth functions of coupling

Fractal dimensions of spectrum coincide and are regular

Abstract

We consider the trace map associated with the silver ratio Schrodinger operator as a diffeomorphism on the invariant surface associated with a given coupling constant and prove that the non-wandering set of this map is hyperbolic if the coupling is sufficiently large. As a consequence, for this values of the coupling constant, the local and global Hausdorff dimension and the local and global box counting dimension of the spectrum of this operator all coincide and are smooth functions of the coupling constant.