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

Jeblick, Maximilian; Leopold, Nikolai; Pickl, Peter

doi:10.1007/s00220-019-03599-x

Cited by 27 publications

(49 citation statements)

References 77 publications

(261 reference statements)

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“…Instead of describing the condensate as the vacuum of a Fock space of fluctuations, this approach remains in the N -body setting and uses projection operators to factor out the condensate. This strategy was successfully applied to prove effective dynamics for N -boson systems in various situations, e.g., [4,8,17,30,31,34,40,41].…”

Section: A First Order Approximation To the N -Body Dynamicsmentioning

confidence: 99%

Higher Order Corrections to the Mean-Field Description of the Dynamics of Interacting Bosons

et al. 2020

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In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of N d-dimensional bosons for large N . The bosons initially form a Bose-Einstein condensate and interact with each other via a pair potential of the formWe derive a sequence of N -body functions which approximate the true many-body dynamics in L 2 (R dN )-norm to arbitrary precision in powers of N −1 . The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution.

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Section: A First Order Approximation To the N -Body Dynamicsmentioning

confidence: 99%

Higher Order Corrections to the Mean-Field Description of the Dynamics of Interacting Bosons

et al. 2020

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“…Remark VI.2. The functional β a was already used in [21,22,23,11,12,18,19,2] and others to derive the Hartree and Gross-Pitaevskii equation.…”

Section: The Strategymentioning

confidence: 99%

“…(i) β a measures if the non-relativistic particles exhibit condensation. 11 Ocassionally, we use the bra-ket notation p ϕt j = |ϕt(xj) ϕt(xj)| = |ϕt ϕt|j. 12 This is a simple consequence of W ( √ N αt) being unitary and W * ( √ N αt)a(k) = a(k)W * ( √ N αt) + √ N W * ( √ N αt)αt(k), see (130).…”

Section: The Strategymentioning

confidence: 99%

Mean-Field Limits of Particles in Interaction with Quantized Radiation Fields

Leopold¹,

Pickl²

2018

Springer Proceedings in Mathematics &Amp; Statistics

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We report on a simple strategy to treat mean-field limits of quantum mechanical systems in which a large number of particles weakly couple to a second-quantized radiation field. Extending the method of counting, introduced in [21], with ideas inspired by [16] and [6] leads to a technique that can be seen as a combination of the method of counting and the coherent state approach. It is similar to the coherent state approach but might be slightly better suited to systems in which a fixed number of particles couple to radiation. The strategy is effective and provides explicit error bounds. As an instructional example we derive the Schrödinger-Klein-Gordon system of equations from the Nelson model with ultraviolet cutoff. Furthermore, we derive explicit bounds on the rate of convergence of the one-particle reduced density matrix of the non-relativistic particles in Sobolev norm. More complicated models like the Pauli-Fierz Hamiltonian can be treated in a similar manner [14].MSC class: 35Q40, 81Q05, 82C10

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“…In [9,Appendix A.2] it is shown in detail how γ j , j ∈ {b, c, d, e, f } (see [5,Sect. 6.4] for the estimate of γ f ) can be bounded in terms of ψ N , mψ N , |E N (ψ N ) − E GP (u)| and N −η , in order to obtain (6).…”

Section: A and U T Hmentioning

confidence: 99%