Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems

Erdős, László; Schlein, Benjamin; Yau, Horng‐Tzer

doi:10.1007/s00222-006-0022-1

Cited by 290 publications

(692 citation statements)

References 18 publications

(25 reference statements)

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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%

“…To show the uniqueness of the solution of the infinite hierarchy (2.2), on the other hand, more information on the limiting densities ( ) ∞, is needed; more precisely, uniqueness was proven in [11] under the assumption that…”

Section: Resolution Of the Correlation Structure For Large Potentialmentioning

confidence: 99%

“…Uniqueness of the solution to the infinite hierarchy. In Section 9 of [11] we proved the following theorem, which states the uniqueness of solution to the infinite hierarchy (4.4) in the space of densities satisfying the a priori bound (4.3). The proof of this theorem is based on a diagrammatic expansion of the solution of (4.4).…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…□ When dealing with the limit points ( ) ∞, , for which we have stronger a priori estimates, we will make use of the following lemma, whose proof can be found in [11] (Lemma 8.2). In fact, if this is not the case, we can use the argument outlined in Section 4 (in the proof of Theorem 3.1) and based on the analysis of Section 9 (which does not use any assumption on the derivatives of ) to approximate .…”

Section: Poincaré-sobolev Type Inequalitiesmentioning

confidence: 99%

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

Erdős¹,

Schlein

Yau

2009

J. Amer. Math. Soc.

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142

246

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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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Section: Resolution Of the Correlation Structure For Large Potentialmentioning

confidence: 99%

Section: Resolution Of the Correlation Structure For Large Potentialmentioning

confidence: 99%

Section: Proof Of the Main Theoremmentioning

confidence: 99%

Section: Poincaré-sobolev Type Inequalitiesmentioning

confidence: 99%

See 2 more Smart Citations

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

Erdős¹,

Schlein

Yau

2009

J. Amer. Math. Soc.

Self Cite

142

246

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show abstract

“…which has been studied by many authors; see [48,7,20,1,16] for some pioneer works (in these works, the convergence (11) was derived using the BBGKY hierarchy, a method that is less quantitative than our approach).…”

Section: Theorem 2 (Kinetic Estimate) Letmentioning

confidence: 99%

Norm Approximation for Many-Body Quantum Dynamics and Bogoliubov Theory

Nam

Napiórkowski

2017

Advances in Quantum Mechanics

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Derivation of the Gross‐Pitaevskii hierarchy for the dynamics of Bose‐Einstein condensate

Erdős¹,

Schlein

Yau

2006

Comm Pure Appl Math

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143

336

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Consider a system of N bosons on the three-dimensional unit torus interacting via a pair potential N 2 V (N (x i − x j )) where x = (x 1 , . . . , x N ) denotes the positions of the particles. Suppose that the initial data ψ N ,0 satisfies the conditionwhere H N is the Hamiltonian of the Bose system. This condition is satisfied if ψ N ,0 = W N φ N ,0 where W N is an approximate ground state to H N and φ N ,0 is regular. Let ψ N ,t denote the solution to the Schrödinger equation with Hamiltonian H N . Gross and Pitaevskii proposed to model the dynamics of such a system by a nonlinear Schrödinger equation, the Gross-Pitaevskii (GP) equation. The GP hierarchy is an infinite BBGKY hierarchy of equations so that if u t solves the GP equation, then the family of k-particle density matrices k |u t u t | solves the GP hierarchy. We prove that as N → ∞ the limit points of the k-particle density matrices of ψ N ,t are solutions of the GP hierarchy. Our analysis requires that the N -boson dynamics be described by a modified Hamiltonian that cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical interparticle distance. Our proof can be extended to a modified Hamiltonian that only forbids at least n particles from coming close together for any fixed n.

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Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems

Cited by 290 publications

References 18 publications

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

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

Norm Approximation for Many-Body Quantum Dynamics and Bogoliubov Theory

Derivation of the Gross‐Pitaevskii hierarchy for the dynamics of Bose‐Einstein condensate

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