5,746 research outputs found
(Semi)classical limit of the Hartree equation with harmonic potential
Nonlinear Schrodinger Equations (NLS) of the Hartree type occur in the
modeling of quantum semiconductor devices. Their "semiclassical" limit of
vanishing (scaled) Planck constant is both a mathematical challenge and
practically relevant when coupling quantum models to classical models.
With the aim of describing the semi-classical limit of the 3D
Schrodinger--Poisson system with an additional harmonic potential, we study
some semi-classical limits of the Hartree equation with harmonic potential in
space dimension n>1. The harmonic potential is confining, and causes focusing
periodically in time. We prove asymptotics in several cases, showing different
possible nonlinear phenomena according to the interplay of the size of the
initial data and the power of the Hartree potential. In the case of the 3D
Schrodinger-Poisson system with harmonic potential, we can only give a formal
computation since the need of modified scattering operators for this long range
scattering case goes beyond current theory. We also deal with the case of an
additional "local" nonlinearity given by a power of the local density - a model
that is relevant when incorporating the Pauli principle in the simplest model
given by the "Schrodinger-Poisson-X equation". Further we discuss the
connection of our WKB based analysis to the Wigner function approach to
semiclassical limits.Comment: 26 page
Higher order Schrodinger and Hartree-Fock equations
The domain of validity of the higher-order Schrodinger equations is analyzed
for harmonic-oscillator and Coulomb potentials as typical examples. Then the
Cauchy theory for higher-order Hartree-Fock equations with bounded and Coulomb
potentials is developed. Finally, the existence of associated ground states for
the odd-order equations is proved. This renders these quantum equations
relevant for physics.Comment: 19 pages, to appear in J. Math. Phy
Coupled Hartree-Fock-Bogoliubov kinetic equations for a trapped Bose gas
Using the Kadanoff-Baym non-equilibrium Green's function formalism, we derive
the self-consistent Hartree-Fock-Bogoliubov (HFB) collisionless kinetic
equations and the associated equation of motion for the condensate wavefunction
for a trapped Bose-condensed gas. Our work generalizes earlier work by Kane and
Kadanoff (KK) for a uniform Bose gas. We include the off-diagonal (anomalous)
pair correlations, and thus we have to introduce an off-diagonal distribution
function in addition to the normal (diagonal) distribution function. This
results in two coupled kinetic equations. If the off-diagonal distribution
function can be neglected as a higher-order contribution, we obtain the
semi-classical kinetic equation recently used by Zaremba, Griffin and Nikuni
(based on the simpler Popov approximation). We discuss the static local
equilibrium solution of our coupled HFB kinetic equations within the
semi-classical approximation. We also verify that a solution is the rigid
in-phase oscillation of the equilibrium condensate and non-condensate density
profiles, oscillating with the trap frequency.Comment: 25 page
Entanglement and Second Quantization in the Framework of the Fermionic Projector
A method is developed for realizing entanglement and general second quantized
fermionic and bosonic fields in the framework of the fermionic projector.Comment: 41 pages, LaTeX, 2 figures, shortened (published version,
supplemented by appendix
On the time evolution of Wigner measures for Schrodinger equations
In this survey, our aim is to emphasize the main known limitations to the use
of Wigner measures for Schrodinger equations. After a short review of
successful applications of Wigner measures to study the semi-classical limit of
solutions to Schrodinger equations, we list some examples where Wigner measures
cannot be a good tool to describe high frequency limits. Typically, the Wigner
measures may not capture effects which are not negligible at the pointwise
level, or the propagation of Wigner measures may be an ill-posed problem. In
the latter situation, two families of functions may have the same Wigner
measures at some initial time, but different Wigner measures for a larger time.
In the case of systems, this difficulty can partially be avoided by considering
more refined Wigner measures such as two-scale Wigner measures; however, we
give examples of situations where this quadratic approach fails.Comment: Survey, 26 page
Finite Temperature Nuclear Response in Extended Random-Phase Approximation
The nuclear collective response at finite temperature is investigated for the
first time in the quantum framework of the small amplitude limit of the
extended TDHF approach, including a non-Markovian collision term. It is shown
that the collision width satisfies a secular equation. By employing a Skyrme
force, the isoscalar monopole, isovector dipole and isoscalar quadrupole
excitations in are calculated and important quantum features are
pointed out. The collisional damping due to decay into incoherent 2p-2h states
is small at low temperatures but increases rapidly at higher temperatures.Comment: 22 Latex pages including 9 figures. Phys. Rev. C (in press
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