Quantum waveguides with corners

Monique Dauge, Yvon Lafranche, Nicolas Raymond

The simplest modeling of planar quantum waveguides is the Dirichlet eigenproblem for the Laplace operator $\Delta$ 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 $\theta$ 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 $\theta$ 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.

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22 dec. 2011

ESAIM: Proceedings 35 (2012) 14-45. DOI: 10.1051/proc/201235002

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