6,850 research outputs found

    Edge Currents for Quantum Hall Systems, I. One-Edge, Unbounded Geometries

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    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

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    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

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    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

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    For fixed c,c, the Prolate Spheroidal Wave Functions (PSWFs) ψn,c\psi_{n, c} form a basis with remarkable properties for the space of band-limited functions with bandwidth cc. 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 (λn(c))n≥0(\lambda_n(c))_{n\geq 0} of the time and frequency limiting operator, which we denote by Qc.\mathcal Q_c. 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 Qc\mathcal Q_c 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 λn(c).\lambda_n(c). Also, we mention that the described approximation scheme provides us with fairly accurate approximations of the λn(c)\lambda_n(c) with low computational load, even for very large values of the parameters cc and n.n. 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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