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Asymptotic behavior of eigenvalues of the Laplace operator in thin infinite tubes. (English. Russian original) Zbl 1181.35150
Math. Notes 85, No. 5, 661-673 (2009); translation from Mat. Zametki 85, No. 5, 687-701 (2009).
Summary: 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.

##### MSC:
 35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs 35J10 Schrödinger operator, Schrödinger equation 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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