Mean field dynamics of boson stars

Elgart, Alexander; Schlein, Benjamin

doi:10.1002/cpa.20134

Cited by 249 publications

(242 citation statements)

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“…Spohn [22] introduced a new approach to this problem using the BBGKY hierarchy. Recent progresses on mean-field limit of quantum dynamics have been based on the BBGKY hierarchy and we mention only a few: the Coulomb potential case [3,10], the pseudo-relativistic Hamiltonian with Newtonian interaction [7], and the delta function interaction in one dimension by Adami, Bardos, Golse and Teta [1] [2]. In next section, we review the BBGKY hierarchy and the two-scale nature of the eigenfunctions of interacting Bose systems.…”

Section: The Main Resultsmentioning

confidence: 99%

Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate

Erdős¹,

2010

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Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N 2 V (N (x i − x j )), where x = (x 1 , . . . , x N ) denotes the positions of the particles. Let H N denote the Hamiltonian of the system and let ψ N,t be the solution to the Schrödinger equation. Suppose that the initial data ψ N,0 satisfies the energy condition. .. We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N → ∞. We prove that the k-particle density matrices of ψ N,t are also asymptotically factorized and the one particle orbital wave function solves the GrossPitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant given by the scattering length of the potential V . We also prove the same conclusion if the energy condition holds only for k = 1 but the factorization of ψ N,0 is assumed in a stronger sense.

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Section: The Main Resultsmentioning

confidence: 99%

Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate

Erdős¹,

2010

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“…Therefore, to conclude the proof of (1.7), it suffices to show that: 1) every limit point of the family { ( ) , } =1 is a solution of the infinite hierarchy (2.2), and 2) the solution to (2.2) is unique. This strategy has already been used to derive the nonlinear Hartree equations for the effective dynamics of so-called mean-field systems (see [27,13,4,9]) to derive the cubic nonlinear Schrödinger equation with different (and simpler) scalings of the interaction potential (see [8,11]) and to derive the nonlinear Schrödinger equation in a one-dimensional setting (see [1,2]). We remark that the first derivation of the Hartree equation was obtained using a different method in [17,14].…”

Section: Resolution Of the Correlation Structure For Large Potentialmentioning

confidence: 99%

Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential

Erdős¹,

Schlein

Yau

2009

J. Amer. Math. Soc.

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Consider a system of N N bosons in three dimensions interacting via a repulsive short range pair potential N 2 V ( N ( x i − x j ) ) N^2V(N(x_i-x_j)) , where x = ( x 1 , … , x N ) \mathbf {x}=(x_1, \ldots , x_N) denotes the positions of the particles. Let H N H_N denote the Hamiltonian of the system and let ψ N , t \psi _{N,t} be the solution to the Schrödinger equation. Suppose that the initial data ψ N , 0 \psi _{N,0} satisfies the energy condition \[ ⟨ ψ N , 0 , H N ψ N , 0 ⟩ ≤ C N \langle \psi _{N,0}, H_N \psi _{N,0} \rangle \leq C N \] and that the one-particle density matrix converges to a projection as N → ∞ N \to \infty . Then, we prove that the k k -particle density matrices of ψ N , t \psi _{N,t} factorize in the limit N → ∞ N \to \infty . Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic nonlinear Schrödinger equation with the coupling constant proportional to the scattering length of the potential V V . In a recent paper, we proved the same statement under the condition that the interaction potential V V is sufficiently small. In the present work we develop a new approach that requires no restriction on the size of the potential.

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“…These bosons are also subject to a time-independent external potential V (x), see [6] for more details. In the particular case V (x) = −m, problem (1.2) was studied in [5] as an effective dynamical description for an N-body quantum system of relativistic bosons with twobody interaction given by Newtonian gravity, it leads to a Chandrasekhar type theory of boson stars. For solitary waves of problem (1.2) with V (x) = −m, a ground state is a minimizer of the energy functional…”

Section: Introductionmentioning

confidence: 99%