The Hartree equation for infinite quantum systems

Sabin, Julien

doi:10.5802/jedp.111

Cited by 8 publications

(14 citation statements)

References 25 publications

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“…It is thus interesting to control such quantities for large times, but also for large M . We refer to [8,7,10] for applications of such inequalities in the study of nonlinear PDEs modelling the evolution of infinite quantum systems. The inequality (4) can also be stated in a more concise way using the language of one-body density matrices [5].…”

Section: Introductionmentioning

confidence: 99%

The Stein-Tomas inequality in trace ideals

Frank¹,

Sabin²

2016

Séminaire Laurent Schwartz — EDP Et Applications

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Section: Introductionmentioning

confidence: 99%

The Stein-Tomas inequality in trace ideals

Frank¹,

Sabin²

2016

Séminaire Laurent Schwartz — EDP Et Applications

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“…A more general statement for the Schrödinger equation can be found in [5,Proposition 5.1]. Our proof of Proposition 12 follows the same lines of argument, and thus we simply provide a sketch; for further details, we refer the reader to [69] and [5].…”

Section: (R×r )mentioning

confidence: 92%

“…In the case of the Schrödinger equation ( ) = | | 2 when ( , ) is sharp 2 -admissible, Proposition 12 can be found in [69,Lemma 9]. A more general statement for the Schrödinger equation can be found in [5,Proposition 5.1].…”

Section: (R×r )mentioning

confidence: 99%

“…Recent developments on the rigorous mathematical theory of the above system in the case = 2 have made significant use of estimate (7.1) (or variants that incorporate additional smoothing effects). We refer the reader to [19], [20], [30], [54], [55] and [69] for further background and details on these applications and to other recent developments in the theory of such systems in [41] (on finite-time blowup criteria) and [34] (on ground states). We also remark that in the case of a single particle, the system (7.4) reduces to = (−Δ) /2 + ( * | | 2 ) , and this has been the subject of numerous papers; see, for example, [22], [23], [35] and the references therein.…”

Section: Local Well-posedness For Hartree-type Infinite Systemsmentioning

confidence: 99%

“…Motivation to study estimates of the form (1.1) is plentiful. Applications to the Hartree equation modelling infinitely many fermions in a quantum system can be found in work of Chen-Hong-Pavlovic [19,20], Frank-Sabin [30], Lewin-Sabin [54,55] and Sabin [69]. Physically, the quantity =1 | Δ | 2 gives a representation of the density of a quantum system of fermions (at time ), where represents the th fermion at the initial time, and thus it is desirable to have optimal control over such a quantity in terms of the number of particles ; this corresponds to obtaining bounds of the form (1.1) with as large as possible.…”

Section: Introductionmentioning

confidence: 99%

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Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates

Bez

Lee

Nakamura

2021

Forum of Mathematics, Sigma

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We establish new Strichartz estimates for orthonormal families of initial data in the case of the wave, Klein–Gordon and fractional Schrödinger equations. Our estimates extend those of Frank–Sabin in the case of the wave and Klein–Gordon equations, and generalize work of Frank et al. and Frank–Sabin for the Schrödinger equation. Due to a certain technical barrier, except for the classical Schrödinger equation, the Strichartz estimates for orthonormal families of initial data have not previously been established up to the sharp summability exponents in the full range of admissible pairs. We obtain the optimal estimates in various notable cases and improve the previous results. The main novelty of this paper is our derivation and use of estimates for weighted oscillatory integrals, which we combine with an approach due to Frank and Sabin. Our weighted oscillatory integral estimates are, in a certain sense, rather delicate endpoint versions of known dispersive estimates with power-type weights of the form $|\xi |^{-\lambda }$ or $(1 + |\xi |^2)^{-\lambda /2}$ , where $\lambda \in \mathbb {R}$ . We achieve optimal decay rates by considering such weights with appropriate $\lambda \in \mathbb {C}$ . For the wave and Klein–Gordon equations, our weighted oscillatory integral estimates are new. For the fractional Schrödinger equation, our results overlap with prior work of Kenig–Ponce–Vega in a certain regime. Our contribution to the theory of weighted oscillatory integrals has also been influenced by earlier work of Carbery–Ziesler, Cowling et al., and Sogge–Stein. Finally, we provide some applications of our new Strichartz estimates for orthonormal families of data to the theory of infinite systems of Hartree type, weighted velocity averaging lemmas for kinetic transport equations, and refined Strichartz estimates for data in Besov spaces.

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