Reducibility for a linear wave equation with Sobolev smooth fast driven potential

TL;DR

This paper proves a reducibility result for a linear wave equation with a fast oscillating, Sobolev regular potential, extending previous work to more general assumptions and showing the equation can be simplified to a time-independent form.

Contribution

It extends reducibility results to equations with Sobolev regular potentials and non-trivial smooth potentials, broadening the applicability of previous theories.

Findings

The original wave equation can be conjugated to a time-independent, block-diagonal form.

The eigenfunctions of the Schrödinger operator exhibit localization near exponential subspaces.

The results hold for large frequency vectors from a large measure Cantor set.

Abstract

We prove a reducibility result for a linear wave equation with a time quasi-periodic driving on the one dimensional torus. The driving is assumed to be fast oscillating, but not necessarily of small size. Provided that the external frequency vector is sufficiently large and chosen from a Cantor set of large measure, the original equation is conjugated to a time-independent, block-diagonal one. With the present paper we extend the previous work \cite{FM19} to more general assumptions: we replace the analytic regularity in time with Sobolev one; the potential in the Schr\"odinger operator is a non-trivial smooth function instead of the constant one. The key tool to achieve the result is a localization property of each eigenfunction of the Schr\"odinger operator close to a subspace of exponentials, with a polynomial decay away from the latter.