65 research outputs found
Disordered Bose Einstein Condensates with Interaction in One Dimension
We study the effects of random scatterers on the ground state of the
one-dimensional Lieb-Liniger model of interacting bosons on the unit interval
in the Gross-Pitaevskii regime. We prove that Bose Einstein condensation
survives even a strong random potential with a high density of scatterers. The
character of the wave function of the condensate, however, depends in an
essential way on the interplay between randomness and the strength of the
two-body interaction. For low density of scatterers or strong interactions the
wave function extends over the whole interval. High density of scatterers and
weak interaction, on the other hand, leads to localization of the wave function
in a fragmented subset of the interval
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
Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime
While Hartree\u2013Fock theory is well established as a fundamental approximation for interacting fermions, it has been unclear how to describe corrections to it due to many-body correlations. In this paper we start from the Hartree\u2013Fock state given by plane waves and introduce collective particle\u2013hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian. We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann\u2013Brueckner\u2013type upper bound to the ground state energy. Our result justifies the random-phase approximation in the mean-field scaling regime, for repulsive, regular interaction potentials
On the Maximal Excess Charge of the Chandrasekhar-Coulomb Hamiltonian in Two Dimensions
We show that for the straightforward quantized relativistic Coulomb
Hamiltonian of a two-dimensional atom -- or the corresponding magnetic quantum
dot -- the maximal number of electrons does not exceed twice the nuclear
charge. It result is then generalized to the presence of external magnetic
fields and atomic Hamiltonians. This is based on the positivity of |\bx|
T(\bp) + T(\bp) |\bx| which -- in two dimensions -- is false for the
non-relativistic case T(\bp) = \bp^2, but is proven in this paper for T(\bp)
= |\bp|, i.e., the ultra-relativistic kinetic energy
Critical Temperature and Energy Gap for the BCS Equation
We derive upper and lower bounds on the critical temperature and the
energy gap (at zero temperature) for the BCS gap equation, describing
spin 1/2 fermions interacting via a local two-body interaction potential
. At weak coupling and under appropriate
assumptions on , our bounds show that and
for some explicit coefficients , and
depending on the interaction and the chemical potential . The ratio
turns out to be a universal constant, independent of both and
. Our analysis is valid for any ; for small , or low density,
our formulas reduce to well-known expressions involving the scattering length
of .Comment: RevTeX4, 23 pages. Revised version, to appear in Phys. Rev.
Ground state energy of the low density Hubbard model
We derive a lower bound on the ground state energy of the Hubbard model for
given value of the total spin. In combination with the upper bound derived
previously by Giuliani, our result proves that in the low density limit, the
leading order correction compared to the ground state energy of a
non-interacting lattice Fermi gas is given by , where
denotes the density of the spin-up (down) particles, and is
the scattering length of the contact interaction potential. This result extends
previous work on the corresponding continuum model to the lattice case.Comment: LaTeX2e, 18 page
A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity
We revisit and prove some convexity inequalities for trace functions
conjectured in the earlier part I. The main functional considered is
\Phi_{p,q}(A_1,A_2,...,A_m) = (trace((\sum_{j=1}^m A_j^p)^{q/p}))^{1/q} for m
positive definite operators A_j. In part I we only considered the case q=1 and
proved the concavity of \Phi_{p,1} for 0 < p \leq 1 and the convexity for p=2.
We conjectured the convexity of \Phi_{p,1} for 1< p < 2. Here we not only
settle the unresolved case of joint convexity for 1 \leq p \leq 2, we are also
able to include the parameter q\geq 1 and still retain the convexity. Among
other things this leads to a definition of an L^q(L^p) norm for operators when
1 \leq p \leq 2 and a Minkowski inequality for operators on a tensor product of
three Hilbert spaces -- which leads to another proof of strong subadditivity of
entropy. We also prove convexity/concavity properties of some other, related
functionals.Comment: Proof of a conjecture in math/0701352. Revised version replaces
earlier draft. 18 pages, late
Onsager's Inequality, the Landau-Feynman Ansatz and Superfluidity
We revisit an inequality due to Onsager, which states that the (quantum)
liquid structure factor has an upper bound of the form (const.) x |k|, for not
too large modulus of the wave vector k. This inequality implies the validity of
the Landau criterion in the theory of superfluidity with a definite, nonzero
critical velocity. We prove an auxiliary proposition for general Bose systems,
together with which we arrive at a rigorous proof of the inequality for one of
the very few soluble examples of an interacting Bose fluid, Girardeau's model.
The latter proof demonstrates the importance of the thermodynamic limit of the
structure factor, which must be taken initially at k different from 0. It also
substantiates very well the heuristic density functional arguments, which are
also shown to hold exactly in the limit of large wave-lengths. We also briefly
discuss which features of the proof may be present in higher dimensions, as
well as some open problems related to superfluidity of trapped gases.Comment: 28 pages, 2 figure, uses revtex
The BCS Functional for General Pair Interactions
The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed
attention as a description of fermionic gases interacting with local pairwise
interactions. We present here a rigorous analysis of the BCS functional for
general pair interaction potentials. For both zero and positive temperature, we
show that the existence of a non-trivial solution of the nonlinear BCS gap
equation is equivalent to the existence of a negative eigenvalue of a certain
linear operator. From this we conclude the existence of a critical temperature
below which the BCS pairing wave function does not vanish identically. For
attractive potentials, we prove that the critical temperature is non-zero and
exponentially small in the strength of the potential.Comment: Revised Version. To appear in Commun. Math. Phys
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