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Bound states in curved quantum layers. (English) Zbl 0988.81034
The subject of the paper is the quantum mechanics of a particle constrained to a curved layer of constant width built over a non-compact surface embedded in $$R^3$$. The assumption of a hard-wall boundary implies that the Hamiltonian is identical to the Dirichlet Laplacian on the layer. This operator can be split into a free transverse part and a longitudinal part consisting of the Laplace-Beltrami operator of the surface and an attractive potential depending on its principal curvatures. The latter implies the existence of geometrically induced bound states, for which sufficient conditions are derived (among them the existence of geodesic polar coordinates and vanishing of the curvatures at infinity).

##### MSC:
 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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