21 research outputs found
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
Design of Model For Restructure Transformation of Public Sector
Public sector such as Govt. University composed of many physical as well logical threads, which are very beneficial for public to provide services. Over times due to repeated modification of software modules, the structure of the system deteriorates and it become very complex to understand for further modification whenever requirement need to provide services to public, because it is universal truth after specific time period there is need of modification to fulfill the requirement for public. And if we repeat to modify the software module, then it is very complicated just like noodles in chowmin plate and program structure is twisted and tangled. Due to this program structure greatly decrease the scalability, reliability, efficiency, robustness and increased the complexity of software module. And it also increased the maintenance cost of software module, therefore repeated modification is not a good choice. Reengineering is good choice for this. Therefore, in this paper we will introduced a new methodology that is known as pattern based reengineer methodology, that is not only focus on only logical thread, but also focus on physical entities - reduce overall complexity. It is proved that the transformation does not alter the semantic of restructured program
(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
Relaxation of Bosons in One Dimension and the Onset of Dimensional Crossover
We study ultra-cold bosons out of equilibrium in a one-dimensional (1D)
setting and probe the breaking of integrability and the resulting relaxation at
the onset of the crossover from one to three dimensions. In a quantum Newton's
cradle type experiment, we excite the atoms to oscillate and collide in an
array of 1D tubes and observe the evolution for up to 4.8 seconds (400
oscillations) with minimal heating and loss. By investigating the dynamics of
the longitudinal momentum distribution function and the transverse excitation,
we observe and quantify a two-stage relaxation process. In the initial stage
single-body dephasing reduces the 1D densities, thus rapidly drives the 1D gas
out of the quantum degenerate regime. The momentum distribution function
asymptotically approaches the distribution of quasimomenta (rapidities), which
are conserved in an integrable system. In the subsequent long time evolution,
the 1D gas slowly relaxes towards thermal equilibrium through the collisions
with transversely excited atoms. Moreover, we tune the dynamics in the
dimensional crossover by initializing the evolution with different imprinted
longitudinal momenta (energies). The dynamical evolution towards the relaxed
state is quantitatively described by a semiclassical molecular dynamics
simulation.Comment: 32 pages, 14 figures. Minor changes are made according to the Referee
report