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The quintic NLS as the mean field limit of a Boson gas with three-body interactions
Thomas Chen,Nataša Pavlović +1 more
TLDR
In this paper, it was shown that in the limit of infinite particle number, the BBGKY hierarchy of $k$-particle marginals converges to a limiting (Gross-Pitaevskii) hierarchy for which they proved existence and uniqueness of solutions.Abstract:
We investigate the dynamics of a boson gas with three-body interactions in dimensions $d=1,2$. We prove that in the limit of infinite particle number, the BBGKY hierarchy of $k$-particle marginals converges to a limiting (Gross-Pitaevskii (GP)) hierarchy for which we prove existence and uniqueness of solutions. Factorized solutions of the GP hierarchy are shown to be determined by solutions of a quintic nonlinear Schrodinger equation. Our proof is based on, and extends, methods of Erdos-Schlein-Yau, Klainerman-Machedon, and Kirkpatrick-Schlein-Staffilani.read more
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Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation
Xuwen Chen,Justin Holmer +1 more
TL;DR: In this article, the dynamics of N bosons in 1D were considered and the authors derived rigorously the 1D focusing cubic NLS with a quadratic trap as the \(N \rightarrow \infty}\) limit of the N-body dynamic and hence justify the mean field limit and prove the propagation of chaos for the focusing quantum many-body system.
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Derivation of the Cubic NLS and Gross–Pitaevskii Hierarchy from Manybody Dynamics in d = 3 Based on Spacetime Norms
Thomas Chen,Nataša Pavlović +1 more
TL;DR: Chen and Pavlovic as discussed by the authors derived the defocusing cubic Gross-Pitaevskii hierarchy in dimension d = 3, from an N-body Schrodinger equation describing a gas of interacting bosons in the GP scaling, in the limit N → ∞.
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Negative Energy Ground States for the L2-Critical NLSE on Metric Graphs
TL;DR: In this article, the existence of ground states with prescribed mass for the focusing nonlinear Schrodinger equation with L 2-critical power nonlinearity on noncompact quantum graphs was investigated.
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Second order corrections to mean field evolution for weakly interacting Bosons in the case of 3-body interactions
TL;DR: In this article, a second order correction to the mean field approximation using a kernel k(t,x,y) and derived an evolution equation for k was presented, and the global existence for the resulting evolution equation was established.
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Unconditional Uniqueness for the Cubic Gross-Pitaevskii Hierarchy via Quantum de Finetti
TL;DR: In this paper, the authors present a new, simpler proof of the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy in the quantum de Finetti theorem.
References
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Semilinear Schrodinger Equations
TL;DR: In this article, the linear Schrodinger equation and local Cauchy problem are studied in the repulsive case and the attractive case, respectively, and the smoothing effect is considered.
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Kinetic equations from Hamiltonian dynamics: Markovian limits
TL;DR: In this paper, a variety of classical as well as quantum-mechanical models for which kinetic equations can be derived rigorously are discussed and the probabilistic nature of the problem is emphasized: the approximation of the microscopic dynamics by either a kinetic or a hydrodynamic equation can be understood as the approximate approximation of a non-Markovian stochastic process by a Markovian process.
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The classical limit for quantum mechanical correlation functions
TL;DR: For quantum systems of finitely many particles as well as for boson quantum field theories, the classical limit of the expectation values of products of Weyl operators, translated in time by the quantum mechanical Hamiltonian and taken in coherent states centered inx-andp-space around −1/2 (coordinates of a point in classical phase space) are shown to become the exponentials of coordinate functions of the classical orbit in phase space.
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Proof of Bose-Einstein condensation for dilute trapped gases.
Elliott H. Lieb,Robert Seiringer +1 more
TL;DR: This work proves theoretically for bosons with two-body repulsive interaction potentials in the dilute limit that the condensation is 100% into the state that minimizes the Gross-Pitaevskii energy functional.