Bound states in straight quantum waveguides with combined boundary conditions

Quantum waveguides with Dirichlet boundary conditions were extensively studied (e.g. [1], [2], [3], [4], [5], [6] and references therein). Their spectral properties essentially depends on the geometry of the waveguide, especially the existence of bound states induced by curvature [1], [2], [3] or by coupling of straight waveguides through windows [4],[5] were shown. The waveguides with Neumann boundary condition were also investigated in several papers (e.g. [7], [8]). The possible next generalization are waveguides with combined Dirichlet and Neumann boundary conditions on different parts of the boundary. Some very simple combinations of these conditions appear due to the symmetry of special configurations in systems studied e.g. in [4], [5] and [8]. Such “combined” systems might be also of interest directly in nanoscopic physics if interphases modelled by different conditions could be realized. The presence of different boundary conditions also gives rise to nontrivial spectral properties like existence of bound states. In the present paper, we consider two simple cases of straight planar waveguide of constant width with combined boundary conditions. We show the examples with and without the presence of bound states. The systems we are going to study are sketched on Fig. 1. We consider a Schrödinger particle whose motion is confined to a planar strip of width d. For definiteness we assume that it is placed to the upper side of the x−axis. On the part of the boundary the Neumann condition is imposed (thin lines in the picture),while on the other part the Dirichlet one holds (thick lines). The length of the overlay of Neumann boundaries is 2δ and it is placed to both sides of y−axis in both cases. We shall denote this configuration space by Ω = R× (0, d) and its particular parts by ΩI = (−∞,−δ)× (0, d), ΩII = (−δ, δ)× (0, d) and ΩIII = (δ,∞) × (0, d). As we are going to prove several statements that are valid for more general combination of boundary conditions, let us define several objects. Let there is a finite number of points on the boundary ∂Ω, where boundary condition is changing, which we denote Pk = 〈xk, yk〉, k = 1, . . . ,M . We can choose the numbering so as yk = d for k = 1, . . . ,M ′ and x1 < x2 < . . . < xM ′ and yk = 0 for k = M ′ + 1, . . . ,M and