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Sep 2012

Volume 53, Issue 9 (partial)

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Lieb-Thirring inequality for a model of particles with point interactions

Rupert L. Frank and Robert Seiringer

J. Math. Phys. 53, 095201 (2012); http://dx.doi.org/10.1063/1.3697416 (11 pages)

Online Publication Date: 14 May 2012

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We consider a model of quantum-mechanical particles interacting via point interactions of infinite scattering length. In the case of fermions we prove a Lieb-Thirring inequality for the energy, i.e., we show that the energy is bounded from below by a constant times the integral of the particle density to the power math.
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03.65.Sq Semiclassical theories and applications
02.30.Rz Integral equations
03.65.Db Functional analytical methods
03.65.Nk Scattering theory

Relativistic Scott correction in self-generated magnetic fields

László Erdős, Søren Fournais, and Jan Philip Solovej

J. Math. Phys. 53, 095202 (2012); http://dx.doi.org/10.1063/1.3697417 (26 pages)

Online Publication Date: 14 May 2012

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We consider a large neutral molecule with total nuclear charge Z in a model with self-generated classical magnetic field and where the kinetic energy of the electrons is treated relativistically. To ensure stability, we assume that Zα < 2/π, where α denotes the fine structure constant. We are interested in the ground state energy in the simultaneous limit Z → ∞, α → 0 such that κ = Zα is fixed. The leading term in the energy asymptotics is independent of κ, it is given by the Thomas-Fermi energy of order Z7/3 and it is unchanged by including the self-generated magnetic field. We prove the first correction term to this energy, the so-called Scott correction of the form S(αZ)Z2. The current paper extends the result of Solovej et al. [Commun. Pure Appl. Math. LXIII, 39–118 (2010)] on the Scott correction for relativistic molecules to include a self-generated magnetic field. Furthermore, we show that the corresponding Scott correction function S, first identified by Solovej et al. [Commun. Pure Appl. Math. LXIII, 39–118 (2010)], is unchanged by including a magnetic field. We also prove new Lieb-Thirring inequalities for the relativistic kinetic energy with magnetic fields.
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31.15.bt Statistical model calculations (including Thomas-Fermi and Thomas-Fermi-Dirac models)
33.15.Pw Fine and hyperfine structure

Critical rotational speeds for superfluids in homogeneous traps

M. Correggi, F. Pinsker, N. Rougerie, and J. Yngvason

J. Math. Phys. 53, 095203 (2012); http://dx.doi.org/10.1063/1.3697418 (45 pages)

Online Publication Date: 14 May 2012

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We present an asymptotic analysis of the effects of rapid rotation on the ground state properties of a superfluid confined in a two-dimensional trap. The trapping potential is assumed to be radial and homogeneous of degree larger than two in addition to a quadratic term. Three critical rotational velocities are identified, marking, respectively, the first appearance of vortices, the creation of a “hole” of low density within a vortex lattice, and the emergence of a giant vortex state free of vortices in the bulk. These phenomena have previously been established rigorously for a “flat” trap with fixed boundary but the “soft” traps considered in the present paper exhibit some significant differences, in particular the giant vortex regime, that necessitate a new approach. These differences concern both the shape of the bulk profile and the size of vortices relative to the width of the annulus where the bulk of the superfluid resides. Close to the giant vortex transition the profile is of Thomas-Fermi type in “flat” traps, whereas it is gaussian for soft traps, and the “last” vortices to survive in the bulk before the giant vortex transition are small relative to the width of the annulus in the former case but of comparable size in the latter.
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03.75.Kk Dynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow
05.30.Jp Boson systems

Symmetry of extremals of functional inequalities via spectral estimates for linear operators

Jean Dolbeault, Maria J. Esteban, and Michael Loss

J. Math. Phys. 53, 095204 (2012); http://dx.doi.org/10.1063/1.4704911 (18 pages)

Online Publication Date: 14 May 2012

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We prove new symmetry results for the extremals of the Caffarelli-Kohn-Nirenberg inequalities in any dimension larger or equal than 2 , in a range of parameters for which no explicit results of symmetry were previously known.
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02.30.Sa Functional analysis
02.30.Tb Operator theory
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