Exponential decay of Laplacian eigenfunctions

D. S. Grebenkov
Laboratory of Condensed Matter Physics, 
CNRS - Ecole Polytechnique 91128 Palaiseau France
Email: denis.grebenkov@polytechnique.edu

This talk is dedicated to low-frequency eigenfunctions of the 
Laplace operator in bounded Euclidean domains. In spite of a 
common picture of eigenfunctions as oscillating waves, the 
geometrical structure of Laplacian eigenfunctions is much 
richer and more complicated [1]. We consider a bounded domain 
of arbitrary shape with elongated "branches" of variable 
cross-sectional profiles. When an eigenvalue is smaller than 
a prescribed threshold (which is determined by the shape of 
the branch), the corresponding eigenfunction is proved to 
have an upper bound decaying exponentially fast along each 
branch [2,3]. This behavior is demonstrated for Dirichlet 
and Robin boundary conditions on the branch boundary. We 
also discuss how the exponential decay leads to localization 
or trapping of eigenmodes in finite quantum waveguides. In 
particular, we obtain a sufficient condition which determines 
the minimal length of branches for getting a trapped eigenmode. 
Varying the branch lengths may switch certain eigenmodes from 
non-trapped to trapped or, equivalently, the waveguide state 
from conducting to insulating [4].

References:

[1] D. S. Grebenkov and B.-T. Nguyen, Geometrical structure of 
Laplacian eigenfunctions, SIAM Reviews 55, 601-667 (2013).

[2] B.-T. Nguyen, D. S. Grebenkov, and A. L. Delitsyn, On the 
Exponential Decay of Laplacian Eigenfunctions in Planar Domains 
with Branches, Contemp. Math. 630, 337-348 (2014). 

[3] A. L. Delitsyn, B.-T. Nguyen, and D. S. Grebenkov, Exponential 
decay of Laplacian eigenfunctions in domains with branches of 
variable cross-sectional profiles, Eur. Phys. J. B 85, 371 (2012). 

[4] A. L. Delitsyn, B.-T. Nguyen, and D. S. Grebenkov, Trapped 
modes in finite quantum waveguides, Eur. Phys. J. B. 85, 176 (2012).