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arXiv:1004.2596v1[math.AP]15Apr2010 CONCENTRATION OF SYMMETRIC EIGENFUNCTIONS
 

Summary: arXiv:1004.2596v1[math.AP]15Apr2010
CONCENTRATION OF SYMMETRIC EIGENFUNCTIONS
DANIEL AZAGRA AND FABRICIO MACI`A
Abstract. In this article we examine the concentration and oscillation effects
developed by high-frequency eigenfunctions of the Laplace operator in a com-
pact Riemannian manifold. More precisely, we are interested in the structure
of the possible invariant semiclassical measures obtained as limits of Wigner
measures corresponding to eigenfunctions. These measures describe simul-
taneously the concentration and oscillation effects developed by a sequence
of eigenfunctions. We present some results showing how to obtain invariant
semiclassical measures from eigenfunctions with prescribed symmetries. As an
application of these results, we give a simple proof of the fact that in a mani-
fold of constant positive sectional curvature, every measure which is invariant
by the geodesic flow is an invariant semiclassical measure.
1. Introduction
The analysis of the concentration and oscillation properties of high-frequency
solutions to Schr¨odinger equations is a central theme in the study of the correspon-
dence principle in quantum mechanics.
Special attention has been devoted to the analysis of the high-frequency behav-
ior of eigenfunctions of the Laplace-Beltrami operator M of a smooth compact

  

Source: Azagra Rueda, Daniel - Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid

 

Collections: Mathematics