Olaf Post 2005 J. Phys. A: Math. Gen. 38 4917 doi:10.1088/0305-4470/38/22/015
Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case
Olaf Post
Show affiliationsWe consider a family of open sets Mε which shrinks with respect to an appropriate parameter ε to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted Dirichlet spectrum of Mε converges to the spectrum of the (differential) Laplacian on the graph with Dirichlet boundary conditions at the vertices, i.e., a graph operator without coupling between different edges. The smallness is expressed by a lower bound on the first eigenvalue of a mixed eigenvalue problem on the vertex neighbourhood. The lower bound is given by the first transversal mode of the edge neighbourhood. We also allow curved edges and show that all bounded eigenvalues converge to the spectrum of a Laplacian acting on the edge with an additional potential coming from the curvature.
- PACS
- MSC
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81Qxx General mathematical topics and methods in quantum theory
- Subjects
- Dates
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Issue 22 (3 June 2005)
Received 1 November 2004, in final form 2 February 2005
Published 18 May 2005
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Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case
Olaf Post 2005 J. Phys. A: Math. Gen. 38 4917
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