Curved planar quantum wires with Dirichlet and Neumann boundary conditions

Abstract

We investigate the discrete spectrum of the Hamiltonian describing a quantum particle living in the two-dimensional curved strip. We impose the Dirichlet and Neumann boundary condition on opposite sides of the strip. The existence of the discrete eigenvalue below the essential spectrum threshold depends on the sign of the total bending angle for the asymptotically straight strips.

Waveguides with Combined Dirichlet and Robin Boundary Conditions

“…Our main result is a lower bound to the spectral threshold of the Laplacian in a (bounded or unbounded) Dirichlet-Robin strip. This enables us to prove quite easily non-existence results about the discrete spectrum for certain waveguides, and generalize in this way the results of Dittrich and Kříž [9]. Moreover, we show that certain combinations of boundary conditions and geometry lead to Hardy-type inequalities for the Laplacian in unbounded strips.…”

“…We refer to [27,28] for the former and to [9,30,25] for the latter. Moreover, these types of boundary conditions are relevant to other physical systems (cf [13,8,21]).…”

“…We refer to the pioneering work [17] of Exner andŠeba and the sequence of papers [20,32,10,25,4] for the existence results under rather simple and general geometric conditions. However, it has not been noticed until the recent letter [9] of Dittrich and Kříž that the existence of eigenvalues in fact depends heavily on the geometrical setting provided the uniform Dirichlet boundary conditions are replaced by a combination of Dirichlet and Neumann ones. In particular, the discrete spectrum may be eliminated provided the Dirichlet-Neumann strip is "curved appropriately", i.e., the Neumann boundary condition is imposed on the "locally shorter" boundary curve.…”

Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions

“…This provides an insight into the mechanism which is behind the qualitative results obtained by Dittrich and Kříž in their 2002 letter [5]. Using −Δ Ωε DN as a model for the Hamiltonian of a quantum waveguide, they show that the discrete eigenvalues exist if, and only if, the reference curve γ of sign-definite κ is curved "in the right direction", namely if the Neumann boundary condition is imposed on the "locally longer" boundary (i.e.…”

Hardy inequalities in strips on ruled surfaces

“…Dittrich and Kříž [7] demonstrated that the discrete spectrum of the Laplacian in any asymptotically straight planar strip is empty provided the curvature of the boundary curves does not change sign and the Dirichlet condition on the locally shorter boundary is replaced by the Neumann one. A different proof of this result and an extension to Robin boundary conditions were performed in [14].…”