Quantum strips on surfaces
arXiv:math-ph/0204049 · doi:10.1016/S0393-0440(02)00146-8
Abstract
Motivated by the theory of quantum waveguides, we investigate the spectrum of the Laplacian, subject to Dirichlet boundary conditions, in a curved strip of constant width that is defined as a tubular neighbourhood of an infinite curve in a two-dimensional Riemannian manifold. Under the assumption that the strip is asymptotically straight in a suitable sense, we localise the essential spectrum and find sufficient conditions which guarantee the existence of geometrically induced bound states. In particular, the discrete spectrum exists for non-negatively curved strips which are studied in detail. The general results are used to recover and revisit the known facts about quantum strips in the plane. As an example of non-positively curved quantum strips, we consider strips on ruled surfaces.
17 pages
References in corpus (3)
Cited by in corpus (16)
- Curvature induced toroidal bound states
- Quantum mechanics of a constrained particle and the problem of prescribed geometry-induced potential
- PT-symmetric models in curved manifolds
- Twisting versus bending in quantum waveguides
- The adiabatic limit of Schrödinger operators on fibre bundles
- Quantum strips in higher dimensions
- Ruled strips with asymptotically diverging twisting
- Effective quantum dynamics on the Möbius strip
- Existence of Bound States for Layers Built Over Hypersurfaces of Euclidean Space
- Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions
- Hardy inequalities in strips on ruled surfaces
- Submanifold Dirac Operator with Torsion
- Differential Geometry of Rotation Minimizing Frames, Spherical Curves, and Quantum Mechanics of a Constrained Particle
- On an Algebraic Essential of Submanifold Quantum Mechanics
- Location of the nodal set for thin curved tubes
- On the Discrete Spectrum of Generalized Quantum Tubes