Sharp $ \frac12 $-Hölder continuity of the Lyapunov exponent atthe bottom of the spectrum for a class of Schrödinger cocycles

Jordi-Lluís Figueras; Thomas Ohlson Timoudas

doi:10.3934/dcds.2020189

Article Contents

2020, Volume 40, Issue 7: 4519-4531. Doi: 10.3934/dcds.2020189

This issue Previous Article Persistence of invariant tori for almost periodically forced reversible systems Next Article Large spectral gap induced by small delay and its application to reduction

Sharp $ \frac12 $-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles

Jordi-Lluís Figueras^1, , and
Thomas Ohlson Timoudas^2,

1.
Department of Mathematics, Uppsala University, Box 480, 75106 Uppsala, Sweden
2.
Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden

^* Corresponding author: Jordi-Lluís Figueras
^* Corresponding author: Jordi-Lluís Figueras

Received: October 2019

Published: April 2020

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Abstract

We consider the setting for the disappearance of uniform hyperbolicity as in Bjerklöv and Saprykina (2008 Nonlinearity 21), where it was proved that the minimum distance between invariant stable and unstable bundles has a linear power law dependence on parameters. In this scenario we prove that the Lyapunov exponent is sharp $ \frac12 $-Hölder continuous.

In particular, we show that the Lyapunov exponent of Schrödinger cocycles with a potential having a unique non-degenerate minimum is sharp $ \frac12 $-Hölder continuous below the lowest energy of the spectrum, in the large coupling regime.

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Mathematics Subject Classification: Primary: 37C55, 37C60, 37D20; Secondary: 37D25, 37D45.

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