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arXiv:1001.4894

Derivation of the Time Dependent Gross Pitaevskii Equation with External Fields

Peter Pickl

Published Jan 27, 2010• 24 citations
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Abstract

Using a new method [9] it is possible to derive mean field equations from the microscopic N-body Schrodinger evolution of interacting particles without using BBGKY hierarchies. Recently this method was used to derive the Hartree equation for singular interactions [4] and the Gross Pitaevskii equation without positivity condition on the interaction [10] where one had to restrict the scaling behaviour of the interaction. In this paper more general scalings shall be considered assuming positivity of the interaction.

DOI
10.48550/arxiv.1001.4894

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Using a new method [9] it is possible to derive mean field equations from the microscopic N-body Schrodinger evolution of interacting particles without using BBGKY hierarchies. Recently this method was used to derive the Hartree equation for singular interactions [4] and the Gross Pitaevskii equation without positivity condition on the interaction [10] where one had to restrict the scaling behaviour of the interaction. In this paper more general scalings shall be considered assuming positivity of the interaction.
We are interested in solutions ΨtΨ_tΨt​ of the Schrödinger equation of NNN interacting bosons under the influence of a time dependent external field, where the range and the coupling constant of the interaction scale with NNN in such a way, that the interaction energy per particle stays more or less constant. Let Nϕ0\mathcal{N}^{ϕ_0}Nϕ0​ be the particle number operator with respect to some ϕ0∈L2(R3→C)ϕ_0\in L^2(\mathbb{R}^3\to\mathbb{C})ϕ0​∈L2(R3→C). Assume that the relative particle number of the initial wave function N−1<Ψ0,Nϕ0Ψ0>N^{-1}< Ψ_0,\mathcal{N}^{ϕ_0}Ψ_0>N−1<Ψ0​,Nϕ0​Ψ0​> converges to one as N→∞N\to\inftyN→∞. We shall show that we can find a ϕt∈L2(R3→C)ϕ_t\in L^2(\mathbb{R}^3\to\mathbb{C})ϕt​∈L2(R3→C) such that lim⁡N→∞N−1<Ψt,NϕtΨt>=1\lim_{N\to\infty}N^{-1}< Ψ_t,\mathcal{N}^{ϕ_t}Ψ_t>=1limN→∞​N−1<Ψt​,Nϕt​Ψt​>=1 and that ϕtϕ_tϕt​ is -- dependent of the scaling of the range of the interaction -- solution of the Gross-Pitaevskii or Hartree equation. We shall also show that under additional decay conditions of ϕtϕ_tϕt​ the limit can be taken uniform in $t
In these notes, we review some recent mathematical results concerning the derivation of effective evolution equations from many body quantum mechanics. In particular, we discuss the emergence of the Hartree equation in the so-called mean field regime (for example, for systems of gravitating bosons), and we show that the Gross-Pitaevskii equation approximates the dynamics of initially trapped Bose-Einstein condensates. We explain how effective evolution equations can be derived, on the one hand, by analyzing the so called BBGKY hierarchy, describing the time-evolution of reduced density matrices, and, on the other hand, by studying the dynamics of coherent initial states in a Fock-space representation of the many body system.
We shall present a new strategy for handling mean field limits of quantum mechanical systems. The new method is simple and effective. It is simple, because it translates the idea behind the mean field description of a many particle quantum system directly into a mathematical algorithm. It is effective because the strategy yields with lesser effort better results than previously achieved. As an instructional example we treat a simple model for the time dependent Hartree equation which we derive under more general conditions than what has been considered so far. Other mean field scalings leading e.g. to the Gross-Pitaevskii equation can also be treated.
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].
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.
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.