Curved planar quantum wires with Dirichlet and Neumann boundary conditions

The spectral properties of curved quantum wires with Dirichlet boundary condition were widely investigated (e.g. [1], [2], [3]). It was shown that any small curvature of the tube in dimensions 2 and 3 produces at least one positive eigenvalue below the essential spectrum threshold. The problem of the existence of such eigenvalues in the straight quantum waveguides with a combination of Dirichlet and Neumann boundary conditions were also studied (e.g. [4], [5]). In the present paper we consider curved planar quantum wires, where Dirichlet boundary condition is imposed on one side of the wire, while the Neumann boundary condition is imposed on the opposite side. Both boundary conditions represent an impenetrable wall in the sense that there is no current through the boundary. They can model two types of interphases in a solid, e.g. in a superconductor, in principle. It is worth to know whether the presence of two types of interphases leads to new nanoscopic phenomena. We prove that the existence of the discrete eigenvalue essentially depends on the direction of the total bending of the strip. Roughly speaking at least one bound state always exists if the Neumann boundary condition is imposed on the ”outer side” of the boundary, i.e. the one which is locally longer. On the other hand we show that there is no eigenvalue below the essential spectrum threshold provided that the curvature does not change its direction and the Neumann boundary condition is imposed on the inner part of the boundary. We consider a Schrödinger particle whose motion is confined to a curved planar strip of the width d. For definiteness, let a curve Γ : R → R be a C-diffeomorfism of the real axis onto Γ(R). Without loss of generality we can assume Γ̇1(s) 2 + Γ̇2(s) 2 = 1, so s is the arc length of the curve.