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The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions
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SYSNO ASEP 0384331 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions Author(s) Krejčiřík, David (UJF-V) RID
Šediváková, Helena (UJF-V)Number of authors 2 Source Title Reviews in Mathematical Physics - ISSN 0129-055X
Roč. 24, č. 7 (2012), 1250018/1-1250018/39Number of pages 39 s. Language eng - English Country SG - Singapore Keywords quantum waveguides ; thin-width limit ; effective Hamiltonian ; twisting versus bending ; norm-resolvent convergence ; Dirichlet Laplacian ; curved tubes ; relatively parallel frame ; Steklov approximation Subject RIV BE - Theoretical Physics R&D Projects LC06002 GA MŠMT - Ministry of Education, Youth and Sports (MEYS) GAP203/11/0701 GA ČR - Czech Science Foundation (CSF) Institutional support UJF-V - RVO:61389005 UT WOS 000306590500005 DOI https://doi.org/10.1142/S0129055X12500183 Annotation The Dirichlet Laplacian in a curved three-dimensional tube built along a spatial (bounded or unbounded) curve is investigated in the limit when the uniform cross-section of the tube diminishes. Both deformations due to bending and twisting of the tube are considered. We show that the Laplacian converges in a norm-resolvent sense to the well-known one-dimensional Schrödinger operator whose potential is expressed in terms of the curvature of the reference curve, the twisting angle and a constant measuring the asymmetry of the cross-section. Contrary to previous results, we allow the reference curves to have non-continuous and possibly vanishing curvature. For such curves, the distinguished Frenet frame standardly used to define the tube need not exist and, moreover, the known approaches to prove the result for unbounded tubes do not work. To establish the norm-resolvent convergence under the minimal regularity assumptions, we use an alternative frame defined by a parallel transport along the curve and a refined smoothing of the curvature via the Steklov approximation. Workplace Nuclear Physics Institute Contact Markéta Sommerová, sommerova@ujf.cas.cz, Tel.: 266 173 228 Year of Publishing 2013
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