Norm approximation for many-body quantum dynamics: Focusing case in low dimensions
@article{Nam2017NormAF, title={Norm approximation for many-body quantum dynamics: Focusing case in low dimensions}, author={Phan Th{\`a}nh Nam and Marcin Napi{\'o}rkowski}, journal={Advances in Mathematics}, year={2017} }
29 Citations
Higher Order Corrections to the Mean-Field Description of the Dynamics of Interacting Bosons
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In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of N…
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Dynamical Hartree–Fock–Bogoliubov Approximation of Interacting Bosons
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We consider a many-body Bosonic system with pairwise particle interaction given by $$N^{3\beta -1}v(N^\beta x)$$ N 3 β - 1 v ( N β x ) where $$0<\beta <1$$ 0 < β < 1 and v a non-negative spherically…
Norm approximation for the Fröhlich dynamics in the mean-field regime
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Blow-up profile of rotating 2D focusing Bose gases
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We consider the Gross-Pitaevskii equation describing an attractive Bose gas trapped to a quasi 2D layer by means of a purely harmonic potential, and which rotates at a fixed speed of rotation…
Derivation of the Bogoliubov Time Evolution for a Large Volume Mean-Field Limit
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The derivation of mean-field limits for quantum systems at zero temperature has attracted many researchers in the last decades. Recent developments are the consideration of pair correlations in the…
The mean-field limit of the Lieb-Liniger model
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It is shown that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence.
Beyond Bogoliubov dynamics
- Mathematics, PhysicsPure and Applied Analysis
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We consider a quantum system of N interacting bosons in the mean field scaling regime. We construct corrections to the Bogoliubov dynamics that approximate the true N-body dynamics in norm to…
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We consider the dynamics of N bosons in 1D. We assume that the pair interaction is attractive and given by $${N^{\beta-1}V(N^{\beta}.) where }$$Nβ-1V(Nβ). where $${\int V \leqslant 0}$$∫V⩽0. We…
Rigorous Derivation of the Gross-Pitaevskii Equation with a Large Interaction Potential
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Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let…
A note on 2D focusing many-boson systems
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We consider a 2D quantum system of $N$ bosons in a trapping potential $|x|^s$, interacting via a pair potential of the form $N^{2\beta-1} w(N^\beta x)$. We show that for all $0 \textless{} \beta…
Derivation of the Time Dependent Two Dimensional Focusing NLS Equation
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- 2018
We present a microscopic derivation of the two-dimensional focusing cubic nonlinear Schrödinger equation starting from an interacting N-particle system of Bosons. The interaction potential we…
Dynamics of Large Boson Systems with Attractive Interaction and a Derivation of the Cubic Focusing NLS in $\mathbb{R}^3$
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Let us consider a many-body Boson system where the particles experience a short-range two-body interparticle interaction given by $v_N(x) = N^{3\beta}v(N^\beta x)$ with $v \in C^\infty_0$, without…
Quantum Many-Body Fluctuations Around Nonlinear Schrödinger Dynamics
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We consider the many-body quantum dynamics of systems of bosons interacting through a two-body potential $${N^{3\beta-1} V (N^\beta x)}$$N3β-1V(Nβx), scaling with the number of particles N. For $${0…
Fluctuations of N-particle quantum dynamics around the nonlinear Schrödinger equation
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The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution
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We consider a 2D time-dependent quantum system of $N$-bosons with harmonic external confining and \emph{attractive} interparticle interaction in the Gross-Pitaevskii scaling. We derive stability of…