82 research outputs found
Free Energy of a Dilute Bose Gas: Lower Bound
A lower bound is derived on the free energy (per unit volume) of a
homogeneous Bose gas at density and temperature . In the dilute
regime, i.e., when , where denotes the scattering length of
the pair-interaction potential, our bound differs to leading order from the
expression for non-interacting particles by the term . Here, denotes the critical density for
Bose-Einstein condensation (for the non-interacting gas), and denotes
the positive part. Our bound is uniform in the temperature up to temperatures
of the order of the critical temperature, i.e., or smaller.
One of the key ingredients in the proof is the use of coherent states to extend
the method introduced in [arXiv:math-ph/0601051] for estimating correlations to
temperatures below the critical one.Comment: LaTeX2e, 53 page
The scattering length at positive temperature
A positive temperature analogue of the scattering length of a potential
can be defined via integrating the difference of the heat kernels of
and , with the Laplacian. An upper bound on this
quantity is a crucial input in the derivation of a bound on the critical
temperature of a dilute Bose gas \cite{SU}. In \cite{SU} a bound was given in
the case of finite range potentials and sufficiently low temperature. In this
paper, we improve the bound and extend it to potentials of infinite range.Comment: LaTeX, 6 page
The excitation spectrum for weakly interacting bosons in a trap
We investigate the low-energy excitation spectrum of a Bose gas confined in a
trap, with weak long-range repulsive interactions. In particular, we prove that
the spectrum can be described in terms of the eigenvalues of an effective
one-particle operator, as predicted by the Bogoliubov approximation.Comment: LaTeX, 32 page
The Dynamics of the One-Dimensional Delta-Function Bose Gas
We give a method to solve the time-dependent Schroedinger equation for a
system of one-dimensional bosons interacting via a repulsive delta function
potential. The method uses the ideas of Bethe Ansatz but does not use the
spectral theory of the associated Hamiltonian
On the maximal ionization of atoms in strong magnetic fields
We give upper bounds for the number of spin 1/2 particles that can be bound
to a nucleus of charge Z in the presence of a magnetic field B, including the
spin-field coupling. We use Lieb's strategy, which is known to yield N_c<2Z+1
for magnetic fields that go to zero at infinity, ignoring the spin-field
interaction. For particles with fermionic statistics in a homogeneous magnetic
field our upper bound has an additional term of order
.Comment: LaTeX2e, 8 page
Microscopic Derivation of the Ginzburg-Landau Model
We present a summary of our recent rigorous derivation of the celebrated Ginzburg–Landau (GL) theory, starting from the microscopic Bardeen–Cooper–Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof
Energy Cost to Make a Hole in the Fermi Sea
The change in energy of an ideal Fermi gas when a local one-body potential is
inserted into the system, or when the density is changed locally, are important
quantities in condensed matter physics. We show that they can be rigorously
bounded from below by a universal constant times the value given by the
semiclassical approximation.Comment: 4 pages, final version published in Phys. Rev. Let
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