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Derivation of the Gross-Pitaevskii Equation from Quantum Dynamics of Many-Body Systems

László Erdős, Benjamin Schlein, Horng‐Tzer Yau

Published Aug 2, 2005• 1 citation
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Abstract

We prove rigorously that the one-particle density matrix of interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the Gross-Pitaevskii equation in suitable scaling limits. The result is extended to k-particle density matrices for all positive integer k.

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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 $H_N$ denote the Hamiltonian of the system and let $\psi_{N,t}$ be the solution to the Schrodinger equation. Suppose that the initial data $\psi_{N,0}$ satisfies the energy condition \[ \leq C N >. \] and that the one-particle density matrix converges to a projection as $N \to \infty$. Then, we prove that the $k$-particle density matrices of $\psi_{N,t}$ factorize in the limit $N \to \infty$. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schrodinger equation with the coupling constant proportional to the scattering length of the potential $V$. In \cite{ESY}, we proved the same statement under the condition that the interaction potential $V$ is sufficiently small; in the present work we extend this result to large potentials.

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We present a microscopic derivation of the time-dependent Gross-Pitaevskii equation starting from an interacting N-particle system of Bosons. We prove convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the respective Gross-Pitaevskii equation. Our work extends a previous result by one of us (P.P.[44]) to interaction potentials which need not to be nonnegative, but may have a sufficiently small negative part. One key estimate in our proof is an operator inequality which was first proven by Jun Yin, see [49].

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Consider a system of NNN bosons in three dimensions interacting via a repulsive short range pair potential N2V(N(xi−xj))N^2V(N(x_i-x_j))N2V(N(xi​−xj​)), where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let HNH_NHN​ denote the Hamiltonian of the system and let ψN,t\psi_{N,t}ψN,t​ be the solution to the Schrodinger equation. Suppose that the initial data ψN,0\psi_{N,0}ψN,0​ satisfies the energy condition ≤CN>. \leq C N >. ≤CN>. and that the one-particle density matrix converges to a projection as N→∞N \to \inftyN→∞. Then, we prove that the kkk-particle density matrices of ψN,t\psi_{N,t}ψN,t​ factorize in the limit N→∞N \to \inftyN→∞. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schrodinger equation with the coupling constant proportional to the scattering length of the potential VVV. In \cite{ESY}, we proved the same statement under the condition that the interaction potential VVV is sufficiently small; in the present work we extend this result to large potentials.
Consider a system of NNN bosons in three dimensions interacting via a repulsive short range pair potential N2V(N(xi−xj))N^2V(N(x_i-x_j))N2V(N(xi​−xj​)), where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let HNH_NHN​ denote the Hamiltonian of the system and let ψN,tψ_{N,t}ψN,t​ be the solution to the Schrödinger equation. Suppose that the initial data ψN,0ψ_{N,0}ψN,0​ satisfies the energy condition <ψN,0,HNkψN,0>≤CkNk < ψ_{N,0}, H_N^k ψ_{N,0} > \leq C^k N^k <ψN,0​,HNk​ψN,0​>≤CkNk for $k=1,2,... $. We also assume that the kkk-particle density matrices of the initial state are asymptotically factorized as N→∞N\to\inftyN→∞. We prove that the kkk-particle density matrices of ψN,tψ_{N,t}ψN,t​ are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant given by the scattering length of the potential VVV. We also prove the same conclusion if the energy condition holds only for k=1k=1k=1 but the factorization of ψN,0ψ_{N,0}ψN,0​ is assumed in a stronger sense.
We derive rigorously, for both R2{\Bbb R}^2R2 and [−L,L]×2[{-}L,L]^{\times 2}[−L,L]×2, the cubic nonlinear Schr\"odinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.
We derive rigorously, for both R^2 and [-L, L]^2, the cubic nonlinear Schrodinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.
We present a microscopic derivation of the time-dependent Gross-Pitaevskii equation starting from an interacting N-particle system of Bosons. We prove convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the respective Gross-Pitaevskii equation. Our work extends a previous result by one of us (P.P.[44]) to interaction potentials which need not to be nonnegative, but may have a sufficiently small negative part. One key estimate in our proof is an operator inequality which was first proven by Jun Yin, see [49].