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Asymptotic behavior of the eigenvalues of the Schrödinger operator in thin closed tubes. (English. Russian original) Zbl 1152.35452
Math. Notes 83, No. 4, 463-477 (2008); translation from Mat. Zametki 83, No. 4, 503-519 (2008).
Summary: 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.

MSC:
35P20 Asymptotic distributions of eigenvalues in context of PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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