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arXiv:1608.05326 · Communications in Mathematical Physics

Derivation of the Time Dependent Gross–Pitaevskii Equation in Two Dimensions

Maximilian Jeblick, Nikolai Leopold, Peter Pickl

Published Nov 1, 2019• 29 citations
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Abstract

Abstract We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting from an interacting N-particle system of bosons. We consider the interaction potential to be given either by $$W_\beta (x)=N^{-1+2 \beta }W(N^\beta x)$$ Wβ(x)=N-1+2βW(Nβx) , for any $$\beta >0$$ β>0 , or to be given by $$V_N(x)=e^{2N} V(e^N x)$$ VN(x)=e2NV(eNx) , for some spherical symmetric, nonnegative and compactly supported $$W,V \in L^\infty ({\mathbb {R}}^2,{\mathbb {R}})$$ W,V∈L∞(R2,R) . In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential $$V_N$$ VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.

DOI
10.1007/s00220-019-03599-x

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Abstract We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting from an interacting N-particle system of bosons. We consider the interaction potential to be given either by Wβ(x)=N−1+2βW(Nβx)W_\beta (x)=N^{-1+2 \beta }W(N^\beta x)Wβ​(x)=N−1+2βW(Nβx) Wβ(x)=N-1+2βW(Nβx) , for any β>0\beta >0β>0 β>0 , or to be given by VN(x)=e2NV(eNx)V_N(x)=e^{2N} V(e^N x)VN​(x)=e2NV(eNx) VN(x)=e2NV(eNx) , for some spherical symmetric, nonnegative and compactly supported W,V∈L∞(R2,R)W,V \in L^\infty ({\mathbb {R}}^2,{\mathbb {R}})W,V∈L∞(R2,R) W,V∈L∞(R2,R) . In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential VNV_NVN​ VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.
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].
Abstract We study the dynamics of a system of N interacting bosons in a disc-shaped trap, which is realised by an external potential that confines the bosons in one spatial dimension to an interval of length of order ε\varepsilon ε ε . The interaction is non-negative and scaled in such a way that its scattering length is of order ε/N\varepsilon /Nε/N ε / N , while its range is proportional to (ε/N)β(\varepsilon /N)^{\beta }(ε/N)β ( ε / N ) β with scaling parameter β∈(0,1]\beta \in (0,1]β∈(0,1] β ∈ ( 0 , 1 ] . We consider the simultaneous limit (N,ε)→(∞,0)(N,\varepsilon )\rightarrow (\infty ,0)(N,ε)→(∞,0) ( N , ε ) → ( ∞ , 0 ) and assume that the system initially exhibits Bose–Einstein condensation. We prove that condensation is preserved by the N-body dynamics, where the time-evolved condensate wave function is the solution of a two-dimensional non-linear equation. The strength of the non-linearity depends on the scaling parameter β\beta β β . For β∈(0,1)\beta \in (0,1)β∈(0,1) β ∈ ( 0 , 1 ) , we obtain a cubic defocusing non-linear Schrödinger equation, while the choice β=1\beta =1β=1 β = 1 yields a Gross–Pitaevskii equation featuring the scattering length of the interaction. In both cases, the coupling parameter depends on the confining potential.
We consider the dynamics of N interacting bosons initially exhibiting Bose–Einstein condensation. Due to an external trapping potential, the bosons are strongly confined in two spatial directions, with the transverse extension of the trap being of order ε. The non-negative interaction potential is scaled such that its scattering length is positive and of order (N/ε2)−1 and the range of the interaction scales as (N/ε2)−β for β ∈ (0, 1). We prove that in the simultaneous limit N → ∞ and ε → 0, the condensation is preserved by the dynamics and the time evolution is asymptotically described by a cubic defocusing nonlinear Schrödinger equation in one dimension, where the strength of the nonlinearity depends on the interaction and on the confining potential. This is the first derivation of a lower-dimensional effective evolution equation for singular potentials scaling with β≥12 and lays the foundations for the derivation of the physically relevant one-dimensional Gross–Pitaevskii equation (β = 1). For our analysis, we adapt an approach by Pickl to the problem with strong confinement.
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.
We derive the time-dependent dipolar Gross-Pitaevskii (GP) equation from the N-body Schr{ö}dinger equation. More precisely we show a norm approximation for the solution of the many body equation as well as the convergence of its one-body reduced density matrix towards the orthogonal projector onto the solution of the dipolar GP equation. We consider the interpolation regime where interaction potential is scaled like N3β−−1w(Nβ(x−−y))N^{3β--1} w(N^β(x -- y))N3β−−1w(Nβ(x−−y)), the range of validity of βββ depends on the stability of the ground state problem. In particular we can prove the convergence on the one-body density matrix assuming $\widehat{w} ≥\ge≥ 0$ and β<3/8β< 3/8β<3/8.
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.