165 research outputs found
Remarks on the derivation of Gross-Pitaevskii equation with magnetic Laplacian
The effective dynamics for a Bose-Einstein condensate in the regime of high
dilution and subject to an external magnetic field is governed by a magnetic
Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the
magnetic case the proof of the derivation of the Gross-Pitaevskii equation
within the "projection counting" scheme
Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction
Using a new method it is possible to derive mean field equations from the
microscopic body Schr\"odinger evolution of interacting particles without
using BBGKY hierarchies.
In this paper we wish to analyze scalings which lead to the Gross-Pitaevskii
equation which is usually derived assuming positivity of the interaction. The
new method for dealing with mean field limits presented in [6] allows us to
relax this condition. The price we have to pay for this relaxation is however
that we have to restrict the scaling behavior to and that we have
to assume fast convergence of the reduced one particle marginal density matrix
of the initial wave function to a pure state
Effective non-linear dynamics of binary condensates and open problems
We report on a recent result concerning the effective dynamics for a mixture
of Bose-Einstein condensates, a class of systems much studied in physics and
receiving a large amount of attention in the recent literature in mathematical
physics; for such models, the effective dynamics is described by a coupled
system of non-linear Sch\"odinger equations. After reviewing and commenting our
proof in the mean field regime from a previous paper, we collect the main
details needed to obtain the rigorous derivation of the effective dynamics in
the Gross-Pitaevskii scaling limit.Comment: Corrected typos, updated reference
Mean-Field Dynamics: Singular Potentials and Rate of Convergence
We consider the time evolution of a system of identical bosons whose
interaction potential is rescaled by . We choose the initial wave
function to describe a condensate in which all particles are in the same
one-particle state. It is well known that in the mean-field limit the quantum -body dynamics is governed by the nonlinear Hartree
equation. Using a nonperturbative method, we extend previous results on the
mean-field limit in two directions. First, we allow a large class of singular
interaction potentials as well as strong, possibly time-dependent external
potentials. Second, we derive bounds on the rate of convergence of the quantum
-body dynamics to the Hartree dynamics.Comment: Typos correcte
Rate of Convergence Towards Semi-Relativistic Hartree Dynamics
We consider the semi-relativistic system of gravitating Bosons with
gravitation constant . The time evolution of the system is described by the
relativistic dispersion law, and we assume the mean-field scaling of the
interaction where and while fixed. In
the super-critical regime of large , we introduce the regularized
interaction where the cutoff vanishes as . We show that the
difference between the many-body semi-relativistic Schr\"{o}dinger dynamics and
the corresponding semi-relativistic Hartree dynamics is at most of order
for all , i.e., the result covers the sub-critical regime and
the super-critical regime. The dependence of the bound is optimal.Comment: 29 page
Everything Hits at Once: How Remote Rainfall Matters for the Prediction of the 2021 North American Heat Wave
In June 2021, Western North America experienced an intense heat wave with unprecedented temperatures and far-reaching socio-economic consequences. Anomalous rainfall in the West Pacific triggers a cascade of weather events across the Pacific, which build up a high-amplitude ridge over Canada and ultimately lead to the heat wave. We show that the response of the jet stream to diabatically enhanced ascending motion in extratropical cyclones represents a predictability barrier with regard to the heat wave magnitude. Therefore, probabilistic weather forecasts are only able to predict the extremity of the heat wave once the complex cascade of weather events is captured. Our results highlight the key role of the sequence of individual weather events in limiting the predictability of this extreme event. We therefore conclude that it is not sufficient to consider such rare events in isolation but it is essential to account for the whole cascade over different spatiotemporal scales
Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in based on spacetime norms
We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimension
, from an -body Schr\"{o}dinger equation describing a gas of
interacting bosons in the GP scaling, in the limit . The
main result of this paper is the proof of convergence of the corresponding
BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work
on the well-posedness of the Cauchy problem for GP hierarchies,
\cite{chpa2,chpa3,chpa4}, which are inspired by the solutions spaces based on
space-time norms introduced by Klainerman and Machedon in \cite{klma}. We note
that in , this has been a well-known open problem in the field. While our
results do not assume factorization of the solutions, consideration of
factorized solutions yields a new derivation of the cubic, defocusing nonlinear
Schr\"odinger equation (NLS) in .Comment: 44 pages, AMS Late
Optical lattice quantum simulator for QED in strong external fields: spontaneous pair creation and the Sauter-Schwinger effect
Spontaneous creation of electron-positron pairs out of the vacuum due to a
strong electric field is a spectacular manifestation of the relativistic
energy-momentum relation for the Dirac fermions. This fundamental prediction of
Quantum Electrodynamics (QED) has not yet been confirmed experimentally as the
generation of a sufficiently strong electric field extending over a large
enough space-time volume still presents a challenge. Surprisingly, distant
areas of physics may help us to circumvent this difficulty. In condensed matter
and solid state physics (areas commonly considered as low energy physics), one
usually deals with quasi-particles instead of real electrons and positrons.
Since their mass gap can often be freely tuned, it is much easier to create
these light quasi-particles by an analogue of the Sauter-Schwinger effect. This
motivates our proposal of a quantum simulator in which excitations of
ultra-cold atoms moving in a bichromatic optical lattice represent particles
and antiparticles (holes) satisfying a discretized version of the Dirac
equation together with fermionic anti-commutation relations. Using the language
of second quantization, we are able to construct an analogue of the spontaneous
pair creation which can be realized in an (almost) table-top experiment.Comment: 21 pages, 10 figure
Eigenfunctions at the threshold energies of magnetic Dirac operators
Discussed are modes and resonances of Dirac operators with
vector potentials . Asymptotic
limits of modes at infinity are derived when ,
, provided that has modes. In wider classes of vector
potentials, sparseness of the vector potentials which give rise to the
modes of are established. It is proved that no has
resonances if , .Comment: 25 pages, New results are adde
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