This article is cited in 10 papers

Asymptotic Behavior of Eigenvalues of the Laplace Operator in Thin Infinite Tubes

V. V. Grushin

Moscow State Institute of Electronics and Mathematics

Abstract: 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, Frenet equations, Schrödinger operator, Maslov canonical operator, quantum waveguide.

UDC: 517.958

Received: 05.10.2006

DOI: 10.4213/mzm3886

 English version: Mathematical Notes, 2009, 85:5, 661–673

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