This article is cited in 9 papers

Asymptotic Behavior of the Eigenvalues of the Schrödinger Operator in Thin Closed Tubes

V. V. Grushin

Moscow State Institute of Electronics and Mathematics

Abstract: In the present paper, we obtain an asymptotic expansion of the eigenvalues of the Schrödinger operator with the magnetic field taken into account and with zero Dirichlet conditions in closed tubes, i.e., in closed curved cylinders with intrinsic torsion under uniform compression of the transverse cross-sections, with respect to a small parameter characterizing the tube's transverse dimensions. We propose a method for reducing the eigenvalue problem to the problem of solving an implicit equation.

Keywords: Schrödinger operator, eigenvalue problem, asymptotics, thin closed tube, small perturbation, Dirichlet condition, Laplace operator.

UDC: 517.958

Received: 17.09.2007

DOI: 10.4213/mzm4573

 English version: Mathematical Notes, 2008, 83:4, 463–477

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