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

Luca Franzoi

doi:10.3934/dcds.2023047

Article Contents

2023, Volume 43, Issue 9: 3251-3285. Doi: 10.3934/dcds.2023047

This issue Previous Article Nonexpansive directions in the Jeandel-Rao Wang shift Next Article Non-injectivity of infinite interval exchange transformations and generalized Thue-Morse sequences

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

Luca Franzoi

NYUAD Research Institute, New York University Abu Dhabi, Abu Dhabi, UAE

Received: February 2023

Revised: April 2023

Early access: May 11, 2023

Published online: May 11, 2023

The work of the author is supported by Tamkeen under the NYU Abu Dhabi Research Institute grant CG002.

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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 [26] to more general assumptions: we replace the analytic regularity in time with Sobolev one; the potential in the Schrödinger 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 Schrödinger operator close to a subspace of exponentials, with a polynomial decay away from the latter.

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Mathematics Subject Classification: Primary: 35L10, 37K55.

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