Vol. 11, No. 8, 2018

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Two-microlocal regularity of quasimodes on the torus

Fabricio Macià and Gabriel Rivière

Vol. 11 (2018), No. 8, 2111–2136
DOI: 10.2140/apde.2018.11.2111
Abstract

We study the regularity of stationary and time-dependent solutions to strong perturbations of the free Schrödinger equation on two-dimensional flat tori. This is achieved by performing a second microlocalization related to the size of the perturbation and by analyzing concentration and nonconcentration properties at this new scale. In particular, we show that sufficiently accurate quasimodes can only concentrate on the set of critical points of the average of the potential along closed geodesics.

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Keywords
quasimodes, Schrödinger operator, semiclassical measures, time-dependent Schrödinger equation
Mathematical Subject Classification 2010
Primary: 58J51, 35P20, 35Q41, 58J50
Milestones
Received: 29 August 2017
Revised: 17 January 2018
Accepted: 10 April 2018
Published: 6 June 2018
Authors
Fabricio Macià
Universidad Politécnica de Madrid
ETSI Navales
Madrid
Spain
Gabriel Rivière
Laboratoire Paul Painlevé (U.M.R. CNRS 8524)
U.F.R. de Mathématiques
Université Lille 1
Villeneuve d’Ascq
France