  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.
