Fractional Hardy–Lieb–Thirring and Related Inequalities for Interacting Systems

Lundholm, Douglas; Nam, Phan Thành; Portmann, Fabian

doi:10.1007/s00205-015-0923-5

Cited by 31 publications

(59 citation statements)

References 54 publications

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“…gives a sharp condition ensuring that the mean field energy for ρ is finite, and if also ρ ∈ L 1+ s d (R d ) then the next-order term is finite too and (2.2) follows by Lemma 16 from [LNP16], which result extends the Lieb-Oxford inequality from s = 1, d = 3, to 0 < s < d.…”

Section: Preliminaries and Notationmentioning

confidence: 82%

“…5.3] (see also [Li79], [Li83], [LO81]. Translated to our notation, the bound proved in [LNP16,Lem. 16] states that for cost c(x, y) = |x − y| −s for 0 < s < d, and for any transport plan γ N ∈ P sym ((…”

Section: A12 Properties Of Wmentioning

confidence: 97%

“…Regarding Lemma A.6, in this case it can be directly implemented on manifolds, by using the decomposition in dyadic scales and the same type of positive definiteness criteria as in the appendix. Lemma A.7 is provable by a similar localization argument, reducing it to a problem on (R d , g(x 0 )) up to small error, and all the tools [HS02], [LNP16] used in the appendix can be extended to this case.…”

Section: Control Of the Error Terms In The Localization Estimatementioning

confidence: 99%

“…The present section produces a lower bound for the E xc N,c (µ) energy in the case of costs d of a special form, inspired by the above constructions. This is based on the proof of a rough fractional Lieb-Oxford inequality from [LNP16,Appendix], which itself is inspired by [LSY94,Lem. 5.3] (see also [Li79], [Li83], [LO81].…”

Section: A12 Properties Of Wmentioning

confidence: 99%

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Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs

Cotar

Petrache

2019

Advances in Mathematics

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Motivated by a problem arising from Density Functional Theory, we provide the sharp next-order asymptotics for a class of multimarginal optimal transport problems with cost given by singular, long-range pairwise interaction potentials. More precisely, we consider an N -marginal optimal transport problem with N equal marginals supported on R d and with cost of the form i =j |x i − x j | −s . In this setting we determine the second-order term in the N → ∞ asymptotic expansion of the minimum energy, for the long-range interactions corresponding to all exponents 0 < s < d. We also prove a small oscillations property for this second-order energy term. Our results can be extended to a larger class of models than power-law-type radial costs, such as non-rotationally-invariant costs. The key ingredient and main novelty in our proofs is a robust extension and simplification of the Fefferman-Gregg decomposition [F85], [Ge89], extended here to our class of kernels, and which provides a unified method valid across our full range of exponents. Our first result generalizes a recent work of Lewin, Lieb and Seiringer [LLS17], who dealt with the second-order term for the Coulomb case s = 1, d = 3 .1 and we will prove the following asymptotic expansion for F OT N,c (µ). For all 0 < s < d, if µ has density ρ ∈ L 1+s/d (R d ), then we have (see Theorem 1.1 below)Moreover we will prove that the strictly positive constant C(s, d) is independent of the choice of the marginal dµ(x) = ρ(x)dx and therefore can be interpreted as arising in an independent model problem. Furthermore, we will derive a small oscillations principle (with respect to N ) forwhich could also be interpreted as a rough third order asymptotic bound for F OT N,c (µ) with c as in (1.3). Our methods are extendable to more general costs, as non-rotationally-invariant ones. See Remark 1.3 below.For d = 3, s = 1 , the F OT N,c (µ) was introduced in the physics literature in the context of Density Functional Theory (DFT) by Seidl, Perdew, Levy, Gori-Giorgi, and Savin [Sd99, SPL99, SGGS07], without them being aware of its meaning in optimal transport. Namely, for s = 1, d = 3 , (1.1) is a natural semiclassical limit to the famous Hohenberg-Kohn (HK) functional from quantum mechanics, originally introduced by Hohenberg-Kohn in [HK64], and rigorously proved by Levy and Lieb in [Ly79], [Li83]. The connection to optimal transport was mathematically established later by [CFK11] and [BdPGG12] for N = 2 , later further extended to N = 3 in [BdP17], and recently independently proved for all N ≥ 2 by [CFK17] and [Lw17].In the process of establishing our two main results, we were required to prove, as key new tools for them, a set of additional secondary results of independent interest and of possible use to other settings, leading to generalized versions of the Fefferman-Gregg decomposition and positive definiteness criteria, as described in Section 4.1. What we use is a decomposition of positive definite kernels following the strategy established for s = 1, d = 3 by Fefferman [F85],...

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Section: Preliminaries and Notationmentioning

confidence: 82%

Section: A12 Properties Of Wmentioning

confidence: 97%

Section: Control Of the Error Terms In The Localization Estimatementioning

confidence: 99%

Section: A12 Properties Of Wmentioning

confidence: 99%

See 2 more Smart Citations

Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs

Cotar

Petrache

2019

Advances in Mathematics

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“…Up to the value of C, this is the same as the inequality for non-interacting fermions used by Lieb and Thirring [15,16] in their proof of stability of matter. (For other recent work on Lieb-Thirring inequalities for interacting particles, see [17][18][19][20]. )…”

Section: Introductionmentioning

confidence: 99%

Triviality of a model of particles with point interactions in the thermodynamic limit

Moser

Seiringer

2016

Lett Math Phys

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We consider a model of fermions interacting via point interactions, defined via a certain weighted Dirichlet form. While for two particles the interaction corresponds to infinite scattering length, the presence of further particles effectively decreases the interaction strength. We show that the model becomes trivial in the thermodynamic limit, in the sense that the free energy density at any given particle density and temperature agrees with the corresponding expression for non-interacting particles.

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