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Existence of bound states for layers built over hypersurfaces in \(\mathbb R^{n+1}\). (English) Zbl 1111.81059

Summary: The existence of discrete spectrum below the essential spectrum is deduced for the Dirichlet Laplacian on tubular neighborhoods (or layers) about hypersurfaces in \(\mathbb R^{n+1}\), with various geometric conditions imposed. This is a generalization of the results of Duclos, Exner, and Krejčiřík (2001) in the case of a surface in \(\mathbb R^{3}\). The key to the generalization is the notion of parabolic manifolds. An interesting case in \(\mathbb R^{3}\) — that of the layer over a convex surface — is also investigated.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
58J05 Elliptic equations on manifolds, general theory
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References:

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