Stability of equilibria for a Hartree equation for random fields

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Abstract

We consider a Hartree equation for a random field, which describes the temporal evolution of infinitely many fermions. On the Euclidean space, this equation possesses equilibria which are not localized. We show their stability through a scattering result, with respect to localized perturbations in the not too focusing case in high dimensions d4. This provides an analogue of the results of Lewin and Sabin [22], and of Chen, Hong and Pavlović [11] for the Hartree equation on operators. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrödinger and Gross-Pitaevskii equations.

Résumé

On considère une équation de Hartree pour des champs aléatoires décrivant la dynamique d'un système infini de fermions. Sur l'espace euclidien, cette équation admet des équilibres non-localisés. On prouve la stabilité de ces derniers à travers un résultat de diffusion pour des perturbations localisées et en dimension supérieure à 4, lorsque la non-linéarité est faiblement focalisante. Cela fournit une contrepartie aléatoire aux résultats de Lewin et Sabin [22] et de Chen, Hong et Pavlović [11] qui traitent de l'équation de Hartree pour des opérateurs densités. La preuve s'appuie sur des techniques dispersives utilisées dans l'étude de la diffusion des équations de Schrödinger et de Gross-Pitaevskii.

Introduction

The present work concerns the following Hartree equation for a random field:itX=ΔX+(wE(|X|2))X. Here X:I×Rd×ΩC is a time-dependent random field over the Euclidean space Rd, and (Ω,A,dω) is the underlying probability space. The expectation is with respect to this probability space: E(|X|2)(t,x):=Ω|X|2(t,x,ω)dω. The convolution product is denoted by ⁎ and w is a real-valued pair interaction potential. By this, we mean that we only consider interactions between two particles (and no more), and that this interaction is characterized by w. Equation (1) has been introduced in [12] as an effective dynamics for a large, possibly infinite, number of fermions in a mean field regime.

Indeed, consider the evolution of a finite number of fermions interacting through the potential w. Under some mean-field hypothesis, as the number of particles tend to infinity, the system is approximated to leading order by the following system of N coupled Hartree equations on R×Rd, for an orthonormal family (uj)1jN (to be compliant with the Pauli principle):ituj=Δuj+(w(k=1N|uk|2))uj,j=1,...,N. Let us associate to this orthonormal family the operator γ=1N|ujuj|, which is the orthogonal projection onto Span{(uj)1jN}. There exists a large literature about the derivation of this system of equations and about other related approximation results. In particular, it has been shown that, if the wave function of the original fermionic system is close to a Slater determinant, then, in the mean-field limit and under sufficient conditions for w, the associated one-particle density matrix converges to the above operator γ. For the derivation of Equation (2) from many body quantum mechanics we refer to [2], [3], [4], [5], [13], [16]. Note that the so-called exchange term appearing in the Hartree-Fock equation is not present in (2), which is motivated by the fact that it is of lower order in certain regimes, see the aforementioned references.

To deal with infinitely many particles, it is customary to use the density matrices framework, which is an operator formalism. Namely, the family (uj)1jN solves Equation (2) if and only if the operator γ defined above solves the corresponding Hartree equation:itγ=[Δ+wργ,γ]. Above, [,] denotes the commutator, and ργ(x)=γ˜(x,x) is the density of particles, that is the diagonal of the integral kernel γ˜(x,y) of the operator γ. An infinite number of particles can then be modeled by a solution of (3) which is not of finite trace (the trace of the operator being, by the derivation of the model, the number of particles). Solutions of (3) with an infinite number of particles were studied previously in [7], [8], [9], [26] for example, and more recently in [10], [11], [22], [23].

In [12], the second author proposed (1) as an alternative equation to (3). It generalizes Equation (2) for a finite number of particles in the following sense. To an orthonormal family (uj)1jN, one can associate the random variable X(x,ω)=1Nuj(x)gj(ω), where (gj)1jN is any orthonormal family in Lω2. The family (uj)1jN then solves (2) if and only if the random variable X solves (1). Equation (1) is also in close correspondence with the Hartree equation for density matrices (3). Indeed, for infinite numbers of particles, taking X to be a Gaussian random field with covariance operator γ, we have that X solves (1) if and only if γ solves (3) (provided γ is taken in an appropriate metric space). We refer to [12] for how to relate the solutions of the two Equations (1) and (3) and their corresponding equilibria. One reason behind the study of (1) is that this equation shares more direct resemblances with the commonly studied nonlinear Schrödinger equation.

The existence of solutions of Equation (1) in Lω2Hxs is investigated in [12]. The local well-posedness is established in the case of localized initial data, as well as in the case of localized perturbations of the equilibria described below. In particular, almost everywhere in the probability space, the random variable solves the corresponding Schrödinger equation in integrated formulation. Though the results are stated and proved in the case of a Dirac potential w=δ{x=0}, their adaptation to the present case of regular interaction potentials is straightforward. Moreover, as for the defocussing nonlinear Schrödinger equation, scattering for large but localized solutions is expected for Equation (1), at least in energy critical and subcritical regimes. This has been showed in dimension 3 in the case of a Dirac potential in [12]. Nontrivial equilibria are thus non-localized, which corresponds to being not of trace-class in the framework of density matrices.

The equilibria at stake in the present paper are the following. For Equation (3), any non-negative Fourier multiplier γ:uF1(|f(ξ)|2Fu) with symbol |f|2 is a stationary solution, where F denotes the Fourier transform. Note that density operators are non-negative, we write the symbol in the form |f|2 to be able to give an analogous equilibrium in the random framework. The analogous equilibrium for Equation (1) are given by Wiener integralsYf(t,x,ω):=ξRdf(ξ)eiξ.xit(m+|ξ|2)dW(ξ), for a distribution function f:RdC (note that this equilibrium has the same law if we replace f by |f| so that we can assume f:Rd[0,+)). Above dW(ξ) denotes infinitesimal complex Gaussians characterized byE(dW(η)dW(ξ))=δηξdηdξ, and the scalar m is given by m:=Rdw(x)dxRd|f(ξ)|2dξ. We refer to [25], Chapter 1, and [1], Part I, for more information on random Gaussian fields. The function Yf is a solution of Equation (1). It is not a steady state but it is an equilibrium, for as a random field its law is invariant by time and space translations (and in particular is not localized).

In the seminal work [22], the authors show the stability of the above equilibria for the Equation (3) for density matrices in dimension 2. Important tools are dispersive estimates for orthonormal systems [14], [15]. This work has been extended to higher dimension in [11]. Note that in higher dimension, some structural hypothesis is made on the interaction potential w, to solve some technical difficulties about a singularity in low frequencies of the equation that we will identify precisely in the sequel. The stability result corresponds to a scattering property in the vicinity of these equilibria: any small and localized perturbation evolves asymptotically into a linear wave which disperses. We mention equally [10], [23] about problems of global well-posedness for the equation on density matrices.

The problem of the stability of the equilibria (4) for Equation (1) shares similarities with the stability of the trivial solution for the Gross-Pitaevskii equation itψ=Δψ+(|ψ|21)ψ. In both problems the linearized dynamics has distinct dispersive properties at low and high frequencies, making the nonlinear stability problem harder, especially in low dimensions. The proof of scattering for small data for the Gross-Pitaevskii equation was done in [19], [18], [20], [21]. We here use spaces with different regularities at low and high frequencies, inspired by [21].

The result of the present work is the stability of the equilibria (4) for Equation (1), via the proof of scattering of perturbations to linear waves in their vicinity. Our techniques however differ from those used in [11], [22], see the strategy of the proof after Theorem 1.1. We hope that the present proof provides different insights than the ones in the framework of operators, as well as relaxing some of the hypotheses on the potential w.

In what follows, ξ=(1+|ξ|2)1/2 denotes the usual Japanese bracket. Given sR, s denotes the usual ceiling function applied to s, and (s)+=max(s,0) and (s)=max(s,0) are the nonnegative and nonpositive parts of s. We write with an abuse of notation f(ξ)=f(r) with r=|ξ|, if f has spherical symmetry. The space Lω2,Hs is the set of measurable functions Z:Rd×ΩC such that Z(,ω)Hs almost surely andRd×Ωξ2s|Zˆ(ξ,ω)|2dξdω<+.

Theorem 1.1 Stability of equilibria for Equation (1)

Let d4. Let s=d21. Let f be a spherically symmetric function in L2(Rd)L(Rd). Assume the following hypotheses hold true:

  • Assumptions on the equilibrium:

    • (i)

      ξsfL2(Rd),

    • (ii)

      Rd|ξ|1|ff|<,

    • (iii)

      r|f|2<0 for r>0,

    • (iv)

      writing h the inverse Fourier transform of |f|2, x2αhL(Rd) for all αNd with |α|2s,

    • (v)

      (|x|1d+|x|2d)(h+h)L1(Rd).

  • Assumptions on the potential: wWs,1 is even withξwˆL2(d+2)/(d2)and(wˆ)L+wˆ(0)+C(f), where C(f)>0 is a constant depending on f.

Then there exists ϵ=ϵ(f,w) such that for any Z0Lω2Hd/21Lx2d/(d+2)Lω2 withZ0Lω2Hd/21+Z0Lx2d/(d+2)Lω2ϵ, the solution of (1) with initial datum X0=Yf(t=0)+Z0 is global. Moreover, it scatters to a linear solution in the sense that there exists Z,Z+Lω2Hd/21 such thatX(t)=Yf(t)+ei(Δm)tZ±+oLω2Hd/21(1)ast±.

Remark 1.1

Note that since w is even, its Fourier transform is real, and thus its positive and negative parts (w+ and w) are well-defined.

Relating the framework of random fields to that of density operators, from the above Theorem 1.1 one obtains a scattering result for the operator:γ=E(|XX|):=u(xE(X(x)X,uL2(Rd))), with respect to the one associated to the equilibrium Yf:γf=E(|YfYf|), which is the Fourier multiplier by |f|2(ξ). This convergence holds in Hilbert-Schmidt Sobolev spaces (where below S2 is the standard Hilbert-Schmidt norm):γHα=αγαS2.

Corollary 1.2

Under the hypotheses of Theorem 1, setting:γ±=E(|Z±Z±|)+E(|Z±Yf(t=0)|)+E(|Yf(t=0)Z±|) there holds γ±Hd/21 and the convergence:γ=γf+eiΔtγ±eiΔt+oHd21(1)ast±.

Remark 1.2

The conditions on f are satisfied by thermodynamical equilibria for bosonic or fermionic gases at a positive temperature T:|f(ξ)|2=1e|ξ|2μT1,μ<0 and |f(ξ)|2=1e|ξ|2μT+1,μR, respectively, but it is not the case of the fermionic gases at zero temperature:|f(ξ)|2=1|ξ|2μ,μ>0.

Remark 1.3

The smallness assumption on (wˆ)L, corresponds to the fact that the equation is not too focusing. The one on (wˆ(0))+ enables the equation (1) linearized around Yf to have enough dispersion. Note that these assumptions appear both in [22] and [11].

Remark 1.4

The smallness of the initial datum in Lω2,Hd/21 cannot be improved as d/21 is the critical regularity. The smallness of the initial datum in Lx2d/(d+2),Lω2 is related to the smallness of2ReE(Y¯S(t)Z0) in ΘV. Of course, if Z0 is orthogonal to the Wiener process W, this term is null. This hypothesis on the initial datum may thus be improved. The optimal space for the initial datum is unclear to us, but taking Z0 in low integrability Lebesgue spaces is related to taking the initial datum in [11], [22] in low Schatten spaces.

Remark 1.5

Let us compare briefly with the related works [11], [22]. The framework used in the present article is that of random fields, which provides a new point of view, and strengthen the understanding of equation (1). The comparison between the aforementioned results and ours is not straightforward, the connexion between the frameworks being explained in the introduction. First, the dimension treated differs, [22] providing with a complete proof in dimension 2, [11] with one for all dimensions d3, and the present paper for d4 (see the remark on lower dimensions below). All works use a linear stability result that is the invertibility of the linear response operator (see Proposition 5.7). Additional dispersive effects are used in [11], [22] by means of new Strichartz estimates in the framework of density matrices [6], [11], [14], [15]. We solely use here standard Strichartz estimates, but find that the response of the equilibrium also enjoys an improved decay at low frequencies due to randomness (Lemmas (5.3) and 5.4). This improved decay does not appear in [11], [22], and in [11] a singularity that appears at low frequencies is handled by making stronger assumptions on their interaction potential. Our analysis eventually yields a scattering result at critical regularity, as in [22], whereas [11] holds for supercritical regularity.

Remark 1.6

The stability is not a consequence of a strong convergence to 0 of the perturbation in Hd/21 almost everywhere in probability, but of the dispersion for the linear dynamics implying in particular local convergence. The solution converges back to the same equilibrium Yf. In particular, it does not trigger modulational instabilities.

Remark 1.7

In [21], which solves scattering for the Gross-Pitaevskii equation in dimension greater than 4, a normal form is needed to close the argument and in particular to deal with the low-frequency singularity. We do not require the use of a normal form transformation because of random cancellations. First, exact computations of the linearized equation around the equilibrium enable us to lose only 1/2 derivatives on V:=E(|X|2)E(|Yf|2) in the low frequencies, while in [21] it is required to lose 1 full derivative on the nonlinearity of the Gross-Pitaevskii equation, which contains quadratic and cubic terms. The second issue is that V, even if it contains a linear and a quadratic term in the perturbation XYf, behaves in terms of Lebesgue integration as |XYf|2, which makes the analysis easier. Note that the potential of interaction plays a role to gain derivatives in high frequencies, we do not use it to gain derivatives in low frequencies.

Remark 1.8

The strategy of the present paper does not directly apply to dimensions 2 and 3, as dispersive effects are weaker. The proof fails at Proposition 6.1. In dimension 2, the first two Picard iterations display a singularity at low frequency, and would have to be treated separately. This has been successfully done in [22]. In dimension 3, we believe that only the first iteration would have such singularity. In [11], a related singularity at low frequency is compensated by hypotheses on the interaction potential. We believe that our strategy could be modified along the lines of [22] for d=3.

Remark 1.9

The function f may be referred as momentum distribution function for the equilibrium. In the case of|f|2(ξ)=1/(e(|ξ|2μ)T+1) where T is the temperature and μ the chemical potential, it may be referred as the Fermi-Dirac distribution. Even in the defocussing case, some other equilibria than the ones described by a momentum distribution function can have instabilities. Two plane waves, which are orthogonal in probability, propagating in opposite directions, are linearly unstable, which is showed In Section 9. We do not claim nonlinear instability as we did not prove that a non trivial codimensional stability cannot arise. However, we believe that it would follow from the analysis of the linearized around the equilibrium equation since high and low frequencies interact through the cubic term of the nonlinearity. Note that the equilibria of Theorem 1.1 can be seen as a superposition of infinitely many plane waves propagating in different directions, hence this shows the importance of regularity of the underlying function f.

The strategy of the proof is the following. First note that the dynamics of Equation (1) near the above equilibria is somewhat similar to that of the Gross-Pitaevskii equation. We use a more direct fixed point argument which does not involve iterations of the wave operator as in [11], [22], and dispersion properties at low and high frequencies which are inspired from [21].

We start by reducing the proof to finding a correct functional framework for our contraction argument. Namely, instead of solving an equation for the perturbation Z=XYf, we solve a fixed point for the perturbation and the induced potential Z,V where V=E(|Z|2)+2ReE(Y¯fZ)), at the same time. The idea behind this is that even if V contains a linear term in Z, it behaves more like a quadratic term in Z (in the sense of the Lebesgue spaces to which they belong) and thus, we can put it in better spaces regarding dispersion.

The fixed point is solved in a classical way by finding the right Banach space Θ for Z,V and proving suitable estimates on the linear and nonlinear terms. The Lipschitz-continuity of the quadratic part is treated in a classical fashion, in the sense that it requires that Θ is included in classical Lebesgue, Besov, or Sobolev spaces.

The difficulty comes from the linear term. It can be writtenL=(0L20L1), see (12), where L1 corresponds to the analogue of the linear term in [11], [22]. The invertibility and continuity of 1L1 had been dealt with in both papers and their treatment is more or less sufficient for our argument. Note that this term is the linear response of the equilibrium, related to the so-called Lindhard function [17], [24].

But L2 is where singularities in low frequencies occurs. To get the continuity of L2, V needs to be in a space that compensates this singularity, namely inhomogeneous Besov spaces, with two levels of regularity, one for the low frequencies and one for the high frequencies. But V contains E(|Z|2) and we cannot close the fixed point argument for Z in a space that compensates singularities in low frequencies. This is where we use bilinear estimates on inhomogeneous Besov spaces coming from the scattering for Gross-Pitaevskii literature, [21].

The paper is organized as follows. In Section 2, we state a few definitions and known results that we use in the rest of the paper. In Section 3, we set up the fixed point argument. In Section 4, estimates related to some direct embeddings are given. The linear terms are studied in Section 5 where explicit formulas, continuity estimates and invertibility conditions are obtained. Estimates for the quadratic terms are proven in Section 6, and estimates for the source terms are showed in Section 7. Theorem 1.1 is then proved in Section 8. The last Section 9 is devoted to an instability result in a defocussing case when the momentum distribution function is not smooth. In the appendix A we prove Corollary 1.2.

Notation 1.3 Fourier transform

We define the Fourier transform with the following constants: for gS,gˆ(ξ)=F(g)(ξ)=Rdg(x)eixξdx, and the inverse Fourier transform byF1(g)(x)=(2π)dRdg(ξ)eixξdξ.

Notation 1.4 Time-space norms

For p,q[1,], we denote by Ltp,Lxq=Lp,Lq the spaceLp(R,Lq(Rd)).

For p,q[1,],sR, we denote by Ltp,Wxs,q=Lp,Ws,q the spaceLp(R,Ws,q(Rd)). In the case q=2 we also write it Lp,Hs or Ltp,Hxs.

When p=q, we may write Lt,xp for Ltp,Lxp.

For p,q[1,],σ,σ˜R, we denote by Ltp,(Bqσ,σ˜)x=Lp,Bqσ,σ˜ the spaceLp(R,Bqσ,σ˜(Rd)). The proper definition of inhomogeneous Besov spaces is given in Section 2, Definition 2.3.

Notation 1.5 Probability-time-space norms

For p,q[1,], we denote by Lω2,Ltp,Lxq=Lω2,Lp,Lq the spaceL2(Ω,Lp(R,Lq(Rd))).

For p,q[1,],sR, we denote by Lω2,Ltp,Wxs,q=Lω2,Lp,Ws,q the spaceL2(Ω,Lp(R,Ws,q(Rd))). In the case q=2 we also write it Lω2,Lp,Hs or Lω2,Ltp,Hxs.

For p,q[1,],σ,σ˜R, we denote by Lω2,Ltp,(Bqσ,σ˜)x=Lω2,Lp,Bqσ,σ˜ the spaceL2(Ω,Lp(R,Bqσ,σ˜(Rd))).

Notation 1.6 Time-space-probability norms

For p,q[1,],sR, we denote by Ltp,Wxs,q,Lω2=Lp,Ws,q,Lω2 the space(1x)s/2Lp(R,Lq(Rd,L2(Ω))). In the case q=2 we also write it Lp,Hs,Lω2, and note that Lp,Hs,Lω2=LpLω2Hs. In the case s=0, we also write it Lp,Lq,Lω2.

For p,q[1,],σ,σ˜R, we denote by Ltp,(Bqσ,σ˜)x,Lω2=Lp,Bqσ,σ˜,Lω2 the space induced by the normgLp,Bqσ,σ˜,Lω2=(j<022jσgjLxq,Lω22+j022jσ˜gjLxq,Lω22)1/2Lp(R).

Section snippets

Toolbox

In this section, we present existing results in the literature, either classical or more recent ones such as dispersive estimates and matters related to Littlewood-Paley decomposition.

Take η a smooth function with support included in the annulus {ξRd||ξ|(1/2,2)}, and define for jZ, ηj(ξ)=η(2jξ). We assume that on Rd{0}, jηj=1. For any tempered distribution fS(R), we write fj=Δjf where Δj is the Fourier multiplier by ηj i.e. fˆj=ηjfˆ.

Notation 2.1

We have f=jZfj and we call this the Littlewood-Paley

Set-up

In this section, we reduce the problem to finding a correct functional setting for our fixed point problem.

To lighten the notation, and f being fixed, we sometimes write Y instead of Yf.

Writing X=Y+Z, the perturbation Z satisfiesitZ=(m)Z+w(2ReE(YZ)+E(|Z|2))(Y+Z). Let an initial perturbation Z0Lω2,Hs, we have that Z solves the Cauchy problem{itZ=(m)Z+w(2ReE(YZ)+E(|Z|2))(Y+Z)Z|t=0=Z0 if and only if the couple perturbation/induced potential (Z,V) solves the Cauchy problem{itZ=(m)Z+wV

Embeddings and Strichartz estimates

In this section, we check assumption 1 in Proposition 3.1, and dispersive estimates for the linear flow which induce assumption 4. In the whole section, d3.

Proposition 4.1

The space ΘZ×ΘZ is embedded in ΘV as in for all u,vΘZ,E(uv)ΘVuΘZvΘZ.

Proof

We have that Ltd+2,Lxd+2×Ltd+2,Lxd+2 is embedded in Lt(d+2)/2,Lx(d+2)/2.

It remains to prove that L4,Bq0,14×L4,Bq0,14 is embedded in L2,B21/2,0. The temporal part of the norm works by Hölder inequality. We are left with proving the embedding Bq0,14×Bq0,14 in B21/2,0

Linear term

We study in this section the linearized operator L defined in (12). We prove that 1L is continuous, invertible with continuous inverse on ΘZ×ΘV. We recall that1L=(1L201L1), hence it is sufficient to prove that 1L1 is continuous, invertible with continuous inverse from ΘV to ΘV and that L2 is continuous from ΘV to ΘZ. We start with the continuity of L2.

Proposition 5.1

The operatorL2:VWV(Y)=i0tS(ts)[(wV(s))Y(s)]ds, is continuous from ΘV to ΘZ.

Proposition 5.1 is a corollary of the following estimates.

Proposition 5.2

Let

The remaining quadratic term

We have already dealt with the quadratic terms E(|Z|2) and WV(Z) in Proposition 4.1, Proposition 4.3 respectively, it remains to prove assumption 5 in Proposition 3.1. In what follows, we assume d4, it is the only part of the paper that does not work in dimension 3.

Proposition 6.1

There exists C>0 such that for all (Z,V)Θ,2ReE(Y¯WV(Z))ΘVCVΘVZΘZ.

Proof

The proof relies on a duality argument. We first prove the L2B21/2,0 estimate. Let UL2,B21/2,0. We write U=U1+U2 withU1=j<0UjandU2=j0Uj where we use the

A space for the initial datum

One has the following compatibility result between the space for the perturbation at initial time, and the leading order term for the solution and the potential as given in (11). Here, we prove assumption 2 in Proposition 3.1 for dimension higher than 4 but the proof can be adapted to dimension 3.

Lemma 7.1

There exists a universal constant C>0 such that for all Z0Lω2,Hd/21Lx2d/(d+2),Lω2, one has CZ0ΘZ×ΘV withCZ0ΘZ×ΘVC(Z0Lω2,Hd21+Z0Lx2dd+2,Lω2).

Proof

We have that S(t)Z0 belongs to ΘZ because of

Proof of Theorem 1.1

We recall that the proof of Theorem 1.1 relies on finding a solution to the fixed point equation (9) for the perturbation Z and the induced potential V. According to (10), the fixed point equation can be written in the form:(ZV)=(IdL)1(CZ0+Q(ZV)). This is now a standard routine to solve the above equation thanks to the various estimates derived previously. To solve the above equation, one defines Φ[Z0](Z,V) as the mappingΦ[Z0]:ΘZ×ΘVΘZ×ΘV(ZV)(IdL)1(CZ0+Q(ZV)). Let us denote by Θ0=Lω2Hd/21

Example of instability for a rough momentum distribution function

We study here the linearization of the dynamics near the superposition of two waves which are orthogonal in probability and propagate in opposite directions. This corresponds to an equilibrium of the form (4) with a rough momentum distribution function f. Even in the defocussing case wˆ0, the form of f is involved to ensure linear stability. Indeed, we will prove linear instability for the present example.

Consider a potential w satisfying, without loss of generality if the equation is

Acknowledgements

C. Collot is supported by the ERC-2014-CoG 646650 SingWave. A-S de Suzzoni is supported by ESSED ANR-18-CE40-0028. Part of this work was done when C. Collot was visiting IHÉS and he thanks the institute.

References (26)

  • J.M. Chadam

    The time-dependent Hartree-Fock equations with Coulomb two-body interaction

    Commun. Math. Phys.

    (1976)
  • T. Chen et al.

    Global well-posedness of the nls system for infinitely many fermions

    Arch. Ration. Mech. Anal.

    (2017)
  • A.S. de Suzzoni

    An equation on random variables and systems of fermions

  • Cited by (9)

    View all citing articles on Scopus
    View full text