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Two-microlocal
regularity of quasimodes on the torus
Fabricio Macià and Gabriel Rivière |
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Vol. 11 (2018), No. 8, 2111–2136
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DOI: 10.2140/apde.2018.11.2111
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Abstract |
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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. |
Keywords
quasimodes, Schrödinger operator, semiclassical measures,
time-dependent Schrödinger equation
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Mathematical Subject Classification 2010
Primary: 58J51, 35P20, 35Q41, 58J50
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Milestones
Received: 29 August 2017
Revised: 17 January 2018
Accepted: 10 April 2018
Published: 6 June 2018
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Authors |
| Fabricio Macià | |
| Universidad Politécnica de
Madrid ETSI Navales Madrid Spain |
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| Gabriel Rivière | |
| Laboratoire Paul Painlevé (U.M.R.
CNRS 8524) U.F.R. de Mathématiques Université Lille 1 Villeneuve d’Ascq France |
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