The effective Hamiltonian in curved quantum waveguides under mild regularity assuptions

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F12%3A00196871" target="_blank" >RIV/68407700:21340/12:00196871 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1142/S0129055X12500183" target="_blank" >http://dx.doi.org/10.1142/S0129055X12500183</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1142/S0129055X12500183" target="_blank" >10.1142/S0129055X12500183</a>

Result language
angličtina
Original language name
The effective Hamiltonian in curved quantum waveguides under mild regularity assuptions
Original language description
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 measuringthe 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
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Publication year
2012
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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