The effective Hamiltonian in curved quantum waveguides and when it does not work
Abstract:
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 diminishes.
Both deformations due to bending and twisting are considered.
We show that the Laplacian converges in a norm resolvent sense
to the well known one-dimensional Schroedinger 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 reference curves
to have non-continuous and possibly vanishing curvature.
For such curves, the distinguished Frenet frame need not exist
and, moreover, the known approaches to establish the result
do not work. We ask the question under which minimal
regularity assumptions the effective one-dimensional
approximation holds.
Our main ideas how to establish the norm-resolvent convergence
under the minimal regularity assumptions are to use an alternative
frame defined by a parallel transport along the curve and a refined
smoothing of the curvature via the Steklov approximation.
On the negative side, we construct an explicit waveguide
for which the usefulness of the spectral information provided
by the effective Hamiltonian is rather doubtful.
05.11.13
09:00
Donatella Donatelli
( University of L'Aquila )
Zero electron mass limit of a hydrodynamic model for charge-carrier transport
Abstract:
We are concerned with the rigorous analysis of the zero electron mass limit of hydrodynamic model for charge-carrier transport. The model is given by the full Navier-Stokes-Poisson system. This system has been introduced in the literature by Anile and Pennisi (see [1]) in order to describe a hydrodynamic model for charge-carrier transport in semiconductor devices. The purpose of our work is to prove rigorously zero electron mass limit in the framework of general ill prepared initial data. In this situation the velocity field and the electronic fields develop fast oscillations in time. The main idea we are using is a combination of formal asymptotic expansion and rigorous uniform estimates on the error terms. Finally we prove the strong convergence of the full Navier Stokes Poisson system towards the strong solutions of the incompressible Navier Stokes equations plus a term of fast singular oscillating gradient vector fields (see [2]).
[1] A. Anile, S. Pennisi, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors, Physical Review B, 46, no. 20 (1992), 13186--13193.
[2] L. Chen, D. Donatelli, P. Marcati, Incompressible type limit analysis of a hydrodynamic model for charge-carrier transport, SIAM J. Math. Analysis, 45, no.3 (2013), 915-933
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