Waveguides with combined Dirichlet and Robin boundary conditions

Abstract

We consider the Laplacian in a curved two-dimensional strip of constant width squeezed between two curves, subject to Dirichlet boundary conditions on one of the curves and variable Robin boundary conditions on the other. We prove that, for certain types of Robin boundary conditions, the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Laplacian in a Dirichlet-Robin annulus determined by the geometry of the strip. Moreover, we show that an appropriate combination of the geometric setting and boundary conditions leads to a Hardy-type inequality in infinite strips. As an application, we derive certain stability of the spectrum for the Laplacian in Dirichlet-Neumann strips along a class of curves of sign-changing curvature, improving in this way an initial result of Dittrich and Kriz.

20 pages, LaTeX with 1 EPS figure; to appear in Mathematical Physics, Analysis and Geometry

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Cited by in corpus (7)

• Geometrical structure of Laplacian eigenfunctions
• From a Particle in a Box to the Uncertainty Relation in a Quantum Dot and to Reflecting Walls for Relativistic Fermions
• Curvature-induced bound states in Robin waveguides and their asymptotical properties
• Trapped modes in finite quantum waveguides
• Resonant alteration of propagation in guiding structures with complex Robin parameter and its magnetic-field-induced restoration
• Thin waveguides with Robin boundary conditions
• Straight Quantum Waveguide with Robin Boundary Conditions