Geodesic
Asymptotic Behavior of the Eigenvalues of the Schr\"odinger Operator in Thin Closed Tubes
Matematičeskie zametki, Tome 83 (2008) no. 4, pp. 503-519

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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, Dirichlet condition, Laplace operator.
Mots-clés : small perturbation 
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     author = {V. V. Grushin},
     title = {Asymptotic {Behavior} of the {Eigenvalues} of the {Schr\"odinger} {Operator} in {Thin} {Closed} {Tubes}},
     journal = {Matemati\v{c}eskie zametki},
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V. V. Grushin. Asymptotic Behavior of the Eigenvalues of the Schr\"odinger Operator in Thin Closed Tubes. Matematičeskie zametki, Tome 83 (2008) no. 4, pp. 503-519. http://geodesic.mathdoc.fr/item/MZM_2008_83_4_a2/