262 research outputs found
Hot Topics in Cold Gases
Since the first experimental realization of Bose-Einstein condensation in
cold atomic gases in 1995 there has been a surge of activity in this field.
Ingenious experiments have allowed us to probe matter close to zero temperature
and reveal some of the fascinating effects quantum mechanics has bestowed on
nature. It is a challenge for mathematical physicists to understand these
various phenomena from first principles, that is, starting from the underlying
many-body Schr\"odinger equation. Recent progress in this direction concerns
mainly equilibrium properties of dilute, cold quantum gases. We shall explain
some of the results in this article, and describe the mathematics involved in
understanding these phenomena. Topics include the ground state energy and the
free energy at positive temperature, the effect of interparticle interaction on
the critical temperature for Bose-Einstein condensation, as well as the
occurrence of superfluidity and quantized vortices in rapidly rotating gases.Comment: Plenary lecture given at the XVI International Congress on
Mathematical Physics, Prague, August 3-8, 200
Stability of the 2+2 fermionic system with point interactions
We give a lower bound on the ground state energy of a system of two fermions
of one species interacting with two fermions of another species via point
interactions. We show that there is a critical mass ratio m_c \approx 0.58 such
that the system is stable, i.e., the energy is bounded from below, for m \in
[m_c, m_c^{-1}]. So far it was not known whether this 2+2 system exhibits a
stable region at all or whether the formation of four-body bound states causes
an unbounded spectrum for all mass ratios, similar to the Thomas effect. Our
result gives further evidence for the stability of the more general N+M system.Comment: LaTeX, 12 pages; typos corrected, references and 2 figures added; to
appear in Math. Phys. Anal. Geo
Triviality of a model of particles with point interactions in the thermodynamic limit
We consider a model of fermions interacting via point interactions, defined
via a certain weighted Dirichlet form. While for two particles the interaction
corresponds to infinite scattering length, the presence of further particles
effectively decreases the interaction strength. We show that the model becomes
trivial in the thermodynamic limit, in the sense that the free energy density
at any given particle density and temperature agrees with the corresponding
expression for non-interacting particles.Comment: LaTeX, 20 pages; final version, to appear in Lett. Math. Phy
Atoms with bosonic "electrons" in strong magnetic fields
We study the ground state properties of an atom with nuclear charge and
bosonic ``electrons'' in the presence of a homogeneous magnetic field .
We investigate the mean field limit with fixed, and identify
three different asymptotic regions, according to , , and
. In Region 1 standard Hartree theory is applicable. Region 3 is
described by a one-dimensional functional, which is identical to the so-called
Hyper-Strong functional introduced by Lieb, Solovej and Yngvason for atoms with
fermionic electrons in the region ; i.e., for very strong magnetic
fields the ground state properties of atoms are independent of statistics. For
Region 2 we introduce a general {\it magnetic Hartree functional}, which is
studied in detail. It is shown that in the special case of an atom it can be
restricted to the subspace of zero angular momentum parallel to the magnetic
field, which simplifies the theory considerably. The functional reproduces the
energy and the one-particle reduced density matrix for the full -particle
ground state to leading order in , and it implies the description of the
other regions as limiting cases.Comment: LaTeX2e, 37 page
Uniqueness and Non-Degeneracy of Minimizers of the Pekar Functional on a Ball
We consider the Pekar functional on a ball in R^3. We prove uniqueness of
minimizers, and a quadratic lower bound in terms of the distance to the
minimizer. The latter follows from non-degeneracy of the Hessian at the
minimum
Stability of a fermionic particle system with point interactions
We prove that a system of fermions interacting with an additional
particle via point interactions is stable if the ratio of the mass of the
additional particle to the one of the fermions is larger than some critical
. The value of is independent of and turns out to be less than
. This fact has important implications for the stability of the unitary
Fermi gas. We also characterize the domain of the Hamiltonian of this model,
and establish the validity of the Tan relations for all wave functions in the
domain.Comment: LaTeX, 29 pages, 2 figures; typos corrected, explanations and
references added; to appear in Commun. Math. Phy
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