1,139 research outputs found
Quantum Approximation II. Sobolev Embeddings
A basic problem of approximation theory, the approximation of functions from
the Sobolev space W_p^r([0,1]^d) in the norm of L_q([0,1]^d), is considered
from the point of view of quantum computation. We determine the quantum query
complexity of this problem (up to logarithmic factors). It turns out that in
certain regions of the domain of parameters p,q,r,d quantum computation can
reach a speedup of roughly squaring the rate of convergence of classical
deterministic or randomized approximation methods. There are other regions were
the best possible rates coincide for all three settings.Comment: 23 pages, paper submitted to the Journal of Complexit
An effective mass theorem for the bidimensional electron gas in a strong magnetic field
We study the limiting behavior of a singularly perturbed
Schr\"odinger-Poisson system describing a 3-dimensional electron gas strongly
confined in the vicinity of a plane and subject to a strong uniform
magnetic field in the plane of the gas. The coupled effects of the confinement
and of the magnetic field induce fast oscillations in time that need to be
averaged out. We obtain at the limit a system of 2-dimensional Schr\"odinger
equations in the plane , coupled through an effective selfconsistent
electrical potential. In the direction perpendicular to the magnetic field, the
electron mass is modified by the field, as the result of an averaging of the
cyclotron motion. The main tools of the analysis are the adaptation of the
second order long-time averaging theory of ODEs to our PDEs context, and the
use of a Sobolev scale adapted to the confinement operator
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