Derivation of the Hartree equation for compound Bose gases in the mean field limit

Anapolitanos, Ioannis; Hott, Michael; Hundertmark, Dirk

doi:10.1142/s0129055x17500222

Cited by 40 publications

(93 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem 2.1 (as well as its mean-field counterpart in Theorem 4.1) justifies crucial information of the initial states assumed in various recent works on the dynamical problem for mixtures of condensates [34,41,2]. There, one proves that the mixture preserves its double-component condensation in the course of time evolution, if it is prepared at time t = 0 in a state of condensation and, in the GP regime, provided that the energy per particle of the initial state is given by the GP energy functional.…”

Section: Resultsmentioning

confidence: 52%

“…mn00 . Moreover, as a straightforward consequence of Assumption (A MF 2 ), each such operator is Hilbert-Schmidt: indeed, (4.26) and the same holds for K (2) and K (12) . In terms of the K's, and of h (1) and h (2) defined in (2.16), the Hessian of the Hartree functional reads…”

Section: Bogoliubov Hamiltonian the Aim Of This Section Is To Show Tmentioning

confidence: 90%

“…(ii) The ground state energy of H MF N satisfies (1) , V (2) and V (12) , we obtain the following analogue…”

Section: Energy Lower Boundmentioning

confidence: 93%

“…Ψ N ] are bounded uniformly in N . Since T (1) and T (2) have compact resolvents, up to a subsequence as N → ∞, γ Next, thanks to the operator inequality (4.5),…”

Section: Consequently Tr[t (1) γmentioning

confidence: 99%

“…Under our assumptions, H N will be bounded from below on the core domain of smooth, compactly supported functions and it is unambiguously realised as a self-adjoint operator by Friedrichs's extension, still denoted by H N . Correspondingly, we shall specialise (1.3) in two forms, the mean field Hamiltonian H MF N and the Gross-Pitaevskii Hamiltonian H GP N , in which the interaction potentials V for suitable N -independent potentials V (1) , V (2) , and V (12) . We are interested in the large-N behavior of the ground state energies 5) and the corresponding ground states.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Ground state energy of mixture of Bose gases

2019

View full text Add to dashboard Cite

We consider the asymptotic behavior of a system of multi-component trapped bosons, when the total particle number N becomes large. In the dilute regime, when the interaction potentials have the length scale of order O(N −1 ), we show that the leading order of the ground state energy is captured correctly by the Gross-Pitaevskii energy functional and that the many-body ground state fully condensates on the Gross-Pitaevskii minimizers. In the mean-field regime, when the interaction length scale is O(1), we are able to verify Bogoliubov's approximation and obtain the second order expansion of the ground state energy. While such asymptotic results have several precursors in the literature on one-component condensates, the adaption to the multi-component setting is non-trivial in various respects and the analysis will be presented in details. Contents42 4.5. Validity of Bogoliubov correction 44 Appendix A. Quantum de Finetti Theorem 3.4 47 References 50 1 In the physical literature [34, 41, 33] a different convention is preferred for α ∈ {1, 2}, namely V (α) = c −1 α V (α) for mean-field regime, and V (α) = c 2 α V (α) (cα·) for Gross-Pitaevskii regime.

show abstract

Section: Resultsmentioning

confidence: 52%

Section: Bogoliubov Hamiltonian the Aim Of This Section Is To Show Tmentioning

confidence: 90%

“…(ii) The ground state energy of H MF N satisfies (1) , V (2) and V (12) , we obtain the following analogue…”

Section: Energy Lower Boundmentioning

confidence: 93%

“…Ψ N ] are bounded uniformly in N . Since T (1) and T (2) have compact resolvents, up to a subsequence as N → ∞, γ Next, thanks to the operator inequality (4.5),…”

Section: Consequently Tr[t (1) γmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Ground state energy of mixture of Bose gases

2019

View full text Add to dashboard Cite

show abstract

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

Leopold¹,

Pickl²

2018

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

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

show abstract

Effective Non-linear Dynamics of Binary Condensates and Open Problems

Olgiati

2017

Advances in Quantum Mechanics

View full text Add to dashboard Cite

We report on a recent result concerning the effective dynamics for a mixture of Bose-Einstein condensates, a class of systems much studied in physics and receiving a large amount of attention in the recent literature in mathematical physics; for such models, the effective dynamics is described by a coupled system of non-linear Schödinger equations. After reviewing and commenting our proof in the mean-field regime from a previous paper, we collect the main details needed to obtain the rigorous derivation of the effective dynamics in the Gross-Pitaevskii scaling limit.

show abstract

Derivation of the Hartree equation for compound Bose gases in the mean field limit

Cited by 40 publications

References 17 publications

Ground state energy of mixture of Bose gases

Ground state energy of mixture of Bose gases

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

Effective Non-linear Dynamics of Binary Condensates and Open Problems

Contact Info

Product

Resources

About