129 research outputs found
Differential Form of the Skornyakov--Ter-Martirosyan Equations
The Skornyakov--Ter-Martirosyan three-boson integral equations in momentum
space are transformed into differential equations. This allows us to take into
account quite directly the Danilov condition providing self-adjointness of the
underlying three-body Hamiltonian with zero-range pair interactions. For the
helium trimer the numerical solutions of the resulting differential equations
are compared with those of the Faddeev-type AGS equations.Comment: 4 pages, 2 figure
General relations for quantum gases in two and three dimensions. Two-component fermions
We derive exact relations for spin-1/2 fermions with zero-range or
short-range interactions, in continuous space or on a lattice, in or in
, in any external potential. Some of them generalize known relations
between energy, momentum distribution , pair distribution function
, derivative of the energy with respect to the scattering length
. Expressions are found for the second order derivative of the energy with
respect to (or to in ). Also, it is found that the leading
energy corrections due to a finite interaction range, are proportional to the
effective range in (and to in ) with exprimable
model-independent coefficients, that give access to the subleading short
distance behavior of and to the subleading tail of .
This applies to lattice models for some magic dispersion relations, an example
of which is given. Corrections to exactly solvable two-body and three-body
problems are obtained. For the trapped unitary gas, the variation of the
finite- and finite energy corrections within each energy
ladder is obtained; it gives the frequency shift and the collapse time of the
breathing mode. For the bulk unitary gas, we compare to fixed-node Monte Carlo
data, and we estimate the experimental uncertainty on the Bertsch parameter due
to a finite .Comment: Augmented version: with respect to published version, subsection V.K
added (minorization of the contact by the order parameter). arXiv admin note:
text overlap with arXiv:1001.077
Lower Spectral Branches of a Particle Coupled to a Bose Field
The structure of the lower part (i.e. -away below the two-boson
threshold) spectrum of Fr\"ohlich's polaron Hamiltonian in the weak coupling
regime is obtained in spatial dimension . It contains a single polaron
branch defined for total momentum , where is a bounded domain, and, for any , a
manifold of polaron + one-boson states with boson momentum in a bounded
domain depending on . The polaron becomes unstable and dissolves into the
one boson manifold at the boundary of . The dispersion laws and
generalized eigenfunctions are calculated
The Existence of Pair Potential Corresponding to Specified Density and Pair Correlation
Given a potential of pair interaction and a value of activity, one can
consider the Gibbs distribution in a finite domain . It is well known that for small values of activity there exist
the infinite volume () limiting Gibbs distribution
and the infinite volume correlation functions. In this paper we consider the
converse problem - we show that given and , where
is a constant and is a function on , which are
sufficiently small, there exist a pair potential and a value of activity, for
which is the density and is the pair correlation function
Geometric expansion of the log-partition function of the anisotropic Heisenberg model
We study the asymptotic expansion of the log-partition function of the
anisotropic Heisenberg model in a bounded domain as this domain is dilated to
infinity. Using the Ginibre's representation of the anisotropic Heisenberg
model as a gas of interacting trajectories of a compound Poisson process we
find all the non-decreasing terms of this expansion. They are given explicitly
in terms of functional integrals. As the main technical tool we use the cluster
expansion method.Comment: 38 page
On - Component Models on Cayley Tree: The General Case
In the paper we generalize results of paper [12] for a - component models
on a Cayley tree of order . We generalize them in two directions: (1)
from to any (2) from concrete examples (Potts and SOS models)
of component models to any - component models (with nearest neighbor
interactions). We give a set of periodic ground states for the model. Using the
contour argument which was developed in [12] we show existence of different
Gibbs measures for -component models on Cayley tree of order .Comment: 8 page
Spin dependent point potentials in one and three dimensions
We consider a system realized with one spinless quantum particle and an array
of spins 1/2 in dimension one and three. We characterize all the
Hamiltonians obtained as point perturbations of an assigned free dynamics in
terms of some ``generalized boundary conditions''. For every boundary condition
we give the explicit formula for the resolvent of the corresponding
Hamiltonian. We discuss the problem of locality and give two examples of spin
dependent point potentials that could be of interest as multi-component
solvable models.Comment: 15 pages, some misprints corrected, one example added, some
references modified or adde
Bound states in a quasi-two-dimensional Fermi gas
We consider the problem of N identical fermions of mass M and one
distinguishable particle of mass m interacting via short-range interactions in
a confined quasi-two-dimensional (quasi-2D) geometry. For N=2 and mass ratios
M/m<13.6, we find non-Efimov trimers that smoothly evolve from 2D to 3D. In the
limit of strong 2D confinement, we show that the energy of the N+1 system can
be approximated by an effective two-channel model. We use this approximation to
solve the 3+1 problem and we find that a bound tetramer can exist for mass
ratios M/m as low as 5 for strong confinement, thus providing the first example
of a universal, non-Efimov tetramer involving three identical fermions.Comment: 5 pages, 4 figure
Three-body problem for ultracold atoms in quasi-one-dimensional traps
We study the three-body problem for both fermionic and bosonic cold atom
gases in a parabolic transverse trap of lengthscale . For this
quasi-one-dimensional (1D) problem, there is a two-body bound state (dimer) for
any sign of the 3D scattering length , and a confinement-induced scattering
resonance. The fermionic three-body problem is universal and characterized by
two atom-dimer scattering lengths, and . In the tightly bound
`dimer limit', , we find , and is linked
to the 3D atom-dimer scattering length. In the weakly bound `BCS limit',
, a connection to the Bethe Ansatz is established, which
allows for exact results. The full crossover is obtained numerically. The
bosonic three-body problem, however, is non-universal: and
depend both on and on a parameter related to the sharpness of
the resonance. Scattering solutions are qualitatively similar to fermionic
ones. We predict the existence of a single confinement-induced three-body bound
state (trimer) for bosons.Comment: 20 pages, 6 figures, accepted for publication in PRA, appendix on the
derivation of an integral formula for the Hurvitz zeta functio
Relaxation times for Hamiltonian systems
Usually, the relaxation times of a gas are estimated in the frame of the
Boltzmann equation. In this paper, instead, we deal with the relaxation problem
in the frame of the dynamical theory of Hamiltonian systems, in which the
definition itself of a relaxation time is an open question. We introduce a
lower bound for the relaxation time, and give a general theorem for estimating
it. Then we give an application to a concrete model of an interacting gas, in
which the lower bound turns out to be of the order of magnitude of the
relaxation times observed in dilute gases.Comment: 26 page
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