6,850 research outputs found
Edge Currents for Quantum Hall Systems, I. One-Edge, Unbounded Geometries
Devices exhibiting the integer quantum Hall effect can be modeled by
one-electron Schroedinger operators describing the planar motion of an electron
in a perpendicular, constant magnetic field, and under the influence of an
electrostatic potential. The electron motion is confined to unbounded subsets
of the plane by confining potential barriers. The edges of the confining
potential barrier create edge currents. In this, the first of two papers, we
prove explicit lower bounds on the edge currents associated with one-edge,
unbounded geometries formed by various confining potentials. This work extends
some known results that we review. The edge currents are carried by states with
energy localized between any two Landau levels. These one-edge geometries
describe the electron confined to certain unbounded regions in the plane
obtained by deforming half-plane regions. We prove that the currents are stable
under various potential perturbations, provided the perturbations are suitably
small relative to the magnetic field strength, including perturbations by
random potentials. For these cases of one-edge geometries, the existence of,
and the estimates on, the edge currents imply that the corresponding
Hamiltonian has intervals of absolutely continuous spectrum. In the second
paper of this series, we consider the edge currents associated with two-edge
geometries describing bounded, cylinder-like regions, and unbounded,
strip-like, regions.Comment: 68 page
One-dimensional interpolation inequalities, Carlson--Landau inequalities and magnetic Schrodinger operators
In this paper we prove refined first-order interpolation inequalities for
periodic functions and give applications to various refinements of the
Carlson--Landau-type inequalities and to magnetic Schrodinger operators. We
also obtain Lieb-Thirring inequalities for magnetic Schrodinger operators on
multi-dimensional cylinders.Comment: 33
Celebrating Cercignani's conjecture for the Boltzmann equation
Cercignani's conjecture assumes a linear inequality between the entropy and
entropy production functionals for Boltzmann's nonlinear integral operator in
rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities
and spectral gap inequalities, this issue has been at the core of the renewal
of the mathematical theory of convergence to thermodynamical equilibrium for
rarefied gases over the past decade. In this review paper, we survey the
various positive and negative results which were obtained since the conjecture
was proposed in the 1980s.Comment: This paper is dedicated to the memory of the late Carlo Cercignani,
powerful mind and great scientist, one of the founders of the modern theory
of the Boltzmann equation. 24 pages. V2: correction of some typos and one
ref. adde
Spectral Decay of Time and Frequency Limiting Operator
For fixed the Prolate Spheroidal Wave Functions (PSWFs)
form a basis with remarkable properties for the space of band-limited functions
with bandwidth . They have been largely studied and used after the seminal
work of D. Slepian, H. Landau and H. Pollack. Many of the PSWFs applications
rely heavily of the behavior and the decay rate of the eigenvalues
of the time and frequency limiting operator, which
we denote by Hence, the issue of the accurate estimation of the
spectrum of this operator has attracted a considerable interest, both in
numerical and theoretical studies. In this work, we give an explicit integral
approximation formula for these eigenvalues. This approximation holds true
starting from the plunge region where the spectrum of starts to
have a fast decay. As a consequence of our explicit approximation formula, we
give a precise description of the super-exponential decay rate of the
Also, we mention that the described approximation scheme
provides us with fairly accurate approximations of the with low
computational load, even for very large values of the parameters and
Finally, we provide the reader with some numerical examples that illustrate the
different results of this work.Comment: arXiv admin note: substantial text overlap with arXiv:1012.388
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