The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions

0384331 - ÚJF 2013 RIV SG eng J - Článek v odborném periodiku
Krejčiřík, David - Šediváková, Helena
The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions.
Reviews in Mathematical Physics. Roč. 24, č. 7 (2012), 1250018/1-1250018/39. ISSN 0129-055X. E-ISSN 1793-6659
Grant CEP: GA MŠMT LC06002; GA ČR GAP203/11/0701
Institucionální podpora: RVO:61389005
Klíčová slova: quantum waveguides * thin-width limit * effective Hamiltonian * twisting versus bending * norm-resolvent convergence * Dirichlet Laplacian * curved tubes * relatively parallel frame * Steklov approximation
Kód oboru RIV: BE - Teoretická fyzika
Impakt faktor: 1.092, rok: 2012

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.
Trvalý link: http://hdl.handle.net/11104/0217004