Geodesic
Quantum waveguides with corners
Monique Dauge1 ; Yvon Lafranche1 ; Nicolas Raymond1
1 IRMAR (UMR CNRS 6625), Université de Rennes 1, France
ESAIM. Proceedings, Tome 35 (2012), pp. 14-45.

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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.
DOI : 10.1051/proc/201235002

Monique Dauge 1 ; Yvon Lafranche 1 ; Nicolas Raymond 1

1 IRMAR (UMR CNRS 6625), Université de Rennes 1, France 
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Monique Dauge; Yvon Lafranche; Nicolas Raymond. Quantum waveguides with corners. ESAIM. Proceedings, Tome 35 (2012), pp. 14-45. doi : 10.1051/proc/201235002. http://geodesic.mathdoc.fr/articles/10.1051/proc/201235002/

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