Geometrical Structure of Laplacian Eigenfunctions

by D. S. Grebenkov

Laboratoire de Physique de la Matiere Condensee,
CNRS - Ecole Polytechnique, Palaiseau, France

Abstract

In this talk, we present a brief overview on the 
geometrical structure of Laplacian eigenfunctions.
In particular, we discuss the problem of localization
when an eigenfunction essentially "lives" on a small 
subset of the domain and takes small values on the 
remaining part. In the high-frequency limit, this 
is a classical problem in the theory of quantum 
billiards. However, less attention has been paid to 
localization of low-frequency eigenfunctions. We
present several recent results on the exponential
decay of low-frequency eigenfunctions.

References:

1) Delitsyn, Nguyen, Grebenkov, Exponential decay 
of Laplacian eigenfunctions in domains with branches 
of variable cross-sectional profiles, Eur. Phys. J. B 
85, 371 (2012).

2) Delitsyn, Nguyen, Grebenkov, Trapped modes in finite 
quantum waveguides, Eur. Phys. J. B 85, 176 (2012).

3) Grebenkov, Nguyen, Geometrical Structure of 
Laplacian Eigenfunctions (SIAM Reviews), online
http://arxiv.org/abs/1206.1278