Geodesic
Asymptotic Behavior of Eigenvalues of the Laplace Operator in Thin Infinite Tubes
Matematičeskie zametki, Tome 85 (2009) no. 5, pp. 687-701

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In this paper, we obtain an asymptotic expansion for the eigenvalues of the Laplace operator with zero Dirichlet conditions in tubes, i.e., in infinite bent cylinders with internal torsion under uniform contraction of their cross-sections, with respect to a small parameter characterizing the transverse dimensions of the tube. A method of reducing the problem of determining the eigenvalues to the solution of an implicit equation is proposed.
Keywords: eigenvalues of the Laplace operator, Dirichlet condition, thin infinite tube, Schrödinger operator, Maslov canonical operator, quantum waveguide.
Mots-clés : Frenet equations 
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     author = {V. V. Grushin},
     title = {Asymptotic {Behavior} of {Eigenvalues} of the {Laplace} {Operator} in {Thin} {Infinite} {Tubes}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {687--701},
     publisher = {mathdoc},
     volume = {85},
     number = {5},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_85_5_a4/}
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V. V. Grushin. Asymptotic Behavior of Eigenvalues of the Laplace Operator in Thin Infinite Tubes. Matematičeskie zametki, Tome 85 (2009) no. 5, pp. 687-701. http://geodesic.mathdoc.fr/item/MZM_2009_85_5_a4/