Strong scarring of logarithmic quasimodes

Eswarathasan, Suresh and Nonnenmacher, Stephane 2017. Strong scarring of logarithmic quasimodes. Annales de l'Institut Fourier 67 (6) , pp. 2307-2347. 10.5802/aif.3137

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Abstract

We consider a semiclassical (pseudo)differential operator on a compact surface (M,g), such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit γ at some energy E0. For any ϵ>0, we then explicitly construct families of quasimodes of this operator, satisfying an energy width of order ϵh|logh| in the semiclassical limit, but which still exhibit a 'strong scar' on the orbit γ, i.e. that these states have a positive weight in any microlocal neighbourhood of γ. We pay attention to optimizing the constants involved in the estimates. This result generalizes a recent result of Brooks \cite{Br13} in the case of hyperbolic surfaces. Our construction, inspired by the works of Vergini et al. in the physics literature, relies on controlling the propagation of Gaussian wavepackets up to the Ehrenfest time.

Item Type:	Article
Date Type:	Published Online
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	Association des Annales de l'Institut Fourier
ISSN:	0373-0956
Date of First Compliant Deposit:	30 March 2016
Date of Acceptance:	6 February 2017
Last Modified:	05 May 2023 23:35
URI:	https://orca.cardiff.ac.uk/id/eprint/86438

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