The simplest modeling of planar quantum waveguides is the Dirichlet eigenproblem for the Laplace operator in unbounded open sets which are uniformly thin in one direction. Here we consider V-shaped guides. Their spectral properties depend essentially on a sole parameter, the opening of the V. The free energy band is a semi-infinite interval bounded from below. As soon as the V is not flat, there are bound states below the free energy band. There are a finite number of them, depending on the opening. This number tends to infinity as the opening tends to 0 (sharply bent V). In this situation, the eigenfunctions concentrate and become self-similar. In contrast, when the opening gets large (almost flat V), the eigenfunctions spread and enjoy a different self-similar structure. We explain all these facts and illustrate them by numerical simulations.
@article{EP_2012_35_a2,
author = {Monique Dauge and Yvon Lafranche and Nicolas Raymond},
title = {Quantum waveguides with corners},
journal = {ESAIM. Proceedings},
pages = {14--45},
year = {2012},
volume = {35},
doi = {10.1051/proc/201235002},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201235002/}
}
TY - JOUR
AU - Monique Dauge
AU - Yvon Lafranche
AU - Nicolas Raymond
TI - Quantum waveguides with corners
JO - ESAIM. Proceedings
PY - 2012
SP - 14
EP - 45
VL - 35
UR - http://geodesic.mathdoc.fr/articles/10.1051/proc/201235002/
DO - 10.1051/proc/201235002
LA - en
ID - EP_2012_35_a2
ER -
%0 Journal Article
%A Monique Dauge
%A Yvon Lafranche
%A Nicolas Raymond
%T Quantum waveguides with corners
%J ESAIM. Proceedings
%D 2012
%P 14-45
%V 35
%U http://geodesic.mathdoc.fr/articles/10.1051/proc/201235002/
%R 10.1051/proc/201235002
%G en
%F EP_2012_35_a2
Monique Dauge; Yvon Lafranche; Nicolas Raymond. Quantum waveguides with corners. ESAIM. Proceedings, Tome 35 (2012), pp. 14-45. doi: 10.1051/proc/201235002
