Quantum waveguides with corners

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. INTRODUCTION A quantum waveguide refer to nanoscale electronic device with a wire or thin surface shape. In the first case, one speaks of a quantum wire. The electronic density is low enough to allow a modeling of the system by a simple one-body Schrödinger operator with potential ψ 7−→ −∆ψ + V ψ in R. The structure of the device causes the potential to be very large outside and very small inside the device. As a relevant approximation, we can consider that the potential is zero in the device and infinite outside ; this can be described by a Dirichlet operator ψ 7−→ −∆ψ in Ω and ψ = 0 on ∂Ω where Ω is the open set filled by the device. We refer to [3] where we can see, at least on the numerical simulations, the analogy between a problem with a confining potential and a Dirichlet condition. These kinds of device are intended to drive electronic fluxes. But their shape may capture some bound states, i.e. eigenpairs of the Dirichlet problem: −∆ψ = λψ in Ω and ψ = 0 on ∂Ω. The topic of this paper is two-dimensional wire shaped structures, i.e. structures which coincide with strips of the form R+×(0, α) outside a ball of center 0 and radius R large enough. These structures can be called planar waveguides. More specifically, there are the bent waveguides and the broken waveguides: Bent waveguides have a constant width around some central smooth curve, see Figure 1, and the central curve of a broken waveguide is a broken line, see Figure 2. Due to the semi-infinite strips contained in such waveguide, the spectrum of the Laplacian −∆ with Dirichlet conditions is not discrete: It contains a semi-infinite interval of the form [μ,+∞) which is the energy band where electronic transport can occur. The presence of discrete spectrum at lower energy levels is not obvious, but nevertheless, frequent. A remarkable result by Duclos and Exner [10] (and generalized in [6]) tells us that if the mid-line of a planar waveguide is smooth and straight outside a compact set, then there exists bound states as soon as the line is not straight everywhere. For broken guides, a similar result holds, [11, 2]: There exist bound states as soon as the 1 2 MONIQUE DAUGE, YVON LAFRANCHE, NICOLAS RAYMOND