Vol. 5, No. 4, 2012

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Nonconcentration in partially rectangular billiards

Luc Hillairet and Jeremy L. Marzuola

Vol. 5 (2012), No. 4, 831–854
Abstract

In specific types of partially rectangular billiards we estimate the mass of an eigenfunction of energy $E$ in the region outside the rectangular set in the high-energy limit. We use the adiabatic ansatz to compare the Dirichlet energy form with a second quadratic form for which separation of variables applies. This allows us to use sharp one-dimensional control estimates and to derive the bound assuming that $E$ is not resonating with the Dirichlet spectrum of the rectangular part.

Keywords
eigenfunctions, billiards, nonconcentration
Mathematical Subject Classification 2010
Primary: 35P20
Secondary: 35Q40, 58J51
Milestones
Received: 30 November 2010
Revised: 14 July 2011
Accepted: 24 October 2011
Published: 27 November 2012
Authors
 Luc Hillairet Laboratoire de Mathématiques Jean Leray UMR 6629 CNRS-Université de Nantes 2 rue de la Houssinière BP 92208 F-44 322 Nantes cedex 3 France Jeremy L. Marzuola Mathematics Department University of North Carolina, Chapel Hill Phillips Hall Chapel Hill, NC 27599 United States