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Commun Math Phys
Commun Math Phys
Communications in Mathematical Physics
0010-3616
1432-0916
Springer Berlin Heidelberg Berlin/Heidelberg

4677
10.1007/s00220-023-04677-x
Article
Effective Dynamics of Extended Fermi Gases in the High-Density Regime
http://orcid.org/0000-0002-6177-7716
Fresta Luca fresta@iam.uni-bonn.de

1
Porta Marcello 2
Schlein Benjamin 3
1 grid.10388.32 0000 0001 2240 3300 Hausdorff Center for Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
2 grid.5970.b 0000 0004 1762 9868 SISSA, Via Bonomea 265, 34136 Trieste, Italy
3 grid.7400.3 0000 0004 1937 0650 Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
Communicated by H. Spohn.

20 3 2023
20 3 2023
2023
401 2 17011751
20 8 2022
15 2 2023
© The Author(s) 2023
https://creativecommons.org/licenses/by/4.0/ Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
We study the quantum evolution of many-body Fermi gases in three dimensions, in arbitrarily large domains. We consider both particles with non-relativistic and with relativistic dispersion. We focus on the high-density regime, in the semiclassical scaling, and we consider a class of initial data describing zero-temperature states. In the non-relativistic case we prove that, as the density goes to infinity, the many-body evolution of the reduced one-particle density matrix converges to the solution of the time-dependent Hartree equation, for short macroscopic times. In the case of relativistic dispersion, we show convergence of the many-body evolution to the relativistic Hartree equation for all macroscopic times. With respect to previous work, the rate of convergence does not depend on the total number of particles, but only on the density: in particular, our result allows us to study the quantum dynamics of extensive many-body Fermi gases.

http://dx.doi.org/10.13039/501100000781 European Research Council AdG CLaQS StG MaMBoQ, n.80290 Porta Marcello Schlein Benjamin http://dx.doi.org/10.13039/501100001711 Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung 200160 Fresta Luca http://dx.doi.org/10.13039/100018011 National Centres of Competence in Research SwissMAP Dynamical and energetic properties of Bose-Einstein condensates Schlein Benjamin issue-copyright-statement© Springer-Verlag GmbH Germany, part of Springer Nature 2023
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pmcIntroduction

In the last years there has been substantial progress in the derivation of effective equations for interacting fermions in the mean-field regime. In this scaling limit, we consider systems of N particles, confined in a region Λ⊂R3 with volume of order one, interacting through a weak two-body potential with range comparable to the size of Λ. Denoting by Vext the trapping potential and by V the interaction, the Hamilton operator takes the form1.1 HNmf(Vext)=∑j=1N[-ε2Δxj+Vext(xj)]+1N∑i<jNV(xi-xj)

and, in accordance with the Pauli principle, it acts on La2(R3N), the subspace of L2(R3N) consisting of functions that are antisymmetric with respect to permutations. In (1.1), we set ε=N-1/3. Together with the factor N-1 in front of the potential energy, this choice guarantees that all terms in the Hamilton operator are, typically, of order N. In fact, because of the fermionic statistics, the expectation of ∑j=1N-Δxj on states trapped in a volume of order one is at least of order N5/3; this can be verified with the Lieb–Thirring inequality, see e.g. [23, Chapter 4].

To describe low-energy properties of (1.1), we introduce the Hartree–Fock theory, defined by restricting (1.1) to Slater determinants, i.e., to wave functions of the form1.2 ψSlater(x1,⋯,xN)=1N!det(fi(xj))1≤i,j≤N

where {fj}j=1N is an orthonormal family in the one-particle space L2(R3). Slater determinants are an example of quasi-free states: they are completely characterized by their one-particle reduced density matrixωN=Ntr2,⋯,N|ψSlater⟩⟨ψSlater|=∑j=1N|fj⟩⟨fj|,

coinciding with the orthogonal projection onto the N-dimensional subspace of L2(R3), spanned by the orbitals {fj}j=1N. In particular, the energy of the Slater determinant (1.2) is given by the Hartree–Fock energy functional1.3 EHF(ωN)=⟨ψSlater,HNmf(Vext)ψSlater⟩=tr[-ε2Δ+Vext]ωN+12N∫dxdyV(x-y)[ωN(x;x)ωN(y;y)-|ωN(x;y)|2].

The interaction contributes to (1.3) through the direct term, proportional to the product of the particle densities ωN(x;x) and ωN(y;y) and through the exchange term, proportional to |ωN(x;y)|2. The Hartree–Fock energy ENHF, obtained minimizing (1.3) over all rank-N orthogonal projections ωN, provides a good approximation for the ground state energy of (1.1) as N≫1; see [2, 19] for the case of Coulomb systems. Recently, the large N asymptotics of the correlation energy, defined as the difference between the many-body ground state energy and the Hartree–Fock ground state energy, has been determined in [6–8, 14, 20].

It is natural to ask what happens when the external traps are switched off; does the Hartree–Fock theory also describe the resulting many-body evolution ψN,tmf=e-iHNmf(0)t/εψN, generated by the translation invariant Hamiltonian HNmf(0)? Here, the presence of the parameter ε=N-1/3 guarantees that ψN,tmf undergoes macroscopic changes for times t of order one. The convergence of the one-particle reduced density matrix1.4 γN,t(1)=Ntr2,⋯,N|ψN,t⟩⟨ψN,t|

associated with ψN,tmf towards the solution of the time-dependent Hartree–Fock equation1.5 iε∂tωN,t=[-ε2Δ+V∗ρt-Xt,ωN,t]

has been established in [17], for analytic potentials and for short times. Here ρt(x)=N-1ωN,t(x;x) is the density associated with ωN,t and Xt is the exchange operator, defined by its integral kernel Xt(x;y)=N-1V(x-y)ωt(x;y). More recently, in [11] (and later in [28], following a different approach), this convergence has been generalized to a much larger class of interaction potentials and to all times. This result and the techniques that were used to derive it provide the starting point for the present work. The initial data considered in [11] are assumed to satisfy suitable semiclassical estimates, which appear as a natural characterization of trapped equilibrium states, in the mean-field regime. Furthermore, in [11] it is also shown that, for bounded potentials, the exchange term in (1.5) is subleading, compared with the direct term, and that the many-body dynamics can also be approximated by the Hartree equation1.6 iε∂tωN,t=[-ε2Δ+V∗ρt,ωN,t].

Notice that the result of [11] holds in the sense of convergence of density matrices; convergence in L2-norm for homogeneous Fermi gases has been recently obtained in [9], via the rigorous bosonization techniques developed in [6–8] (in this case, ωN is translation invariant which implies, in particular, that ωN,t=ωN is stationary).

The result of [11] has been extended to fermions with relativistic dispersion (known as pseudo-relativistic fermions) in [12] and to quasi-free mixed states in [5]. See also [13] for a review. All these works consider bounded interaction potentials. As for unbounded potentials, the time-dependent Hartree–Fock equation for particles interacting through a Coulomb potential has been derived in [29], under the assumption that a suitable semiclassical structure of the initial datum propagates along the flow of the Hartree–Fock equation. Recently, the propagation of the semiclassical structure has been proven in [15], for mixed states and for a class of singular potentials that includes a suitably regularized version of the Coulomb interaction. In the absence of semiclassical scaling, that is, setting ε=1 in the previous discussion, convergence to the time-dependent Hartree–Fock equation has been shown in [4] for bounded potentials, and then extended to Coulomb potentials in [18] (see also [3]).

Notice that both the Hartree–Fock equation (1.5) and the Hartree equation (1.6) still depend on the number of particles N. In the limit N→∞, the Hartree–Fock and the Hartree dynamics are known to converge to the Vlasov equation, a classical effective evolution equation. The first proof of convergence from the quantum many-body dynamics to the Vlasov dynamics has been obtained in [27] for analytic potentials, and then extended in [32] to a much larger class of interactions. Next, convergence from the Hartree–Fock to the Vlasov equation has been proved in [24, 25]. All these results hold in a weak sense. Bounds on the rate of convergence from the Hartree–Fock equation to the Vlasov equation have been first obtained in [1], and more recently in [10] for a larger class of initial data and of interaction potentials. Unbounded interaction potentials, including the Coulomb interaction, have been considered in [21]. Finally, let us mention the result [22], where convergence from the Hartree equation to the Vlasov equation is proven for local perturbations of the equilibrium state of extended Fermi gases at fixed density, in the high-density regime. This last setting will be related to the one considered in the present work.

The results described above (with the exception of [22]) apply to the mean-field limit, where particles are initially trapped in a volume of order one. To describe the physically important case of extended gases, let us now consider N fermions moving in a large region Λ⊂R3, at high density ϱ=N/|Λ|≫1. If the potential has range of order one, each particle interacts, at any given time, with order ϱ other particles. Furthermore, the kinetic energy of the N particles is now of the order ϱ2/3N. Therefore, to describe an extended Fermi gas at high density, we consider the Hamilton operator1.7 HN=∑j=1N-ε2Δj+ε3∑i<jNV(xi-xj)onLa2(R3N)

where we set ε=ϱ-1/3 to make sure that both terms are of order N. In contrast with the mean-field regime (where one has ε=N-1/3), ε is now small but independent of N. We will be interested in the many-body evolution governed by the Schrödinger equation1.8 iε∂tψN,t=HNψN,t

for initial data ψN,0=ψN that are close (in an appropriate sense) to a Slater determinant with reduced density matrix ωN, describing a quasi-free state of N fermions in a large domain Λ⊂R3, with density of particles of order ε-3. To be more precise, we will assume that the density, averaged against a function decaying on length scales of order one, is such that1.9 supz∈R3∫dyωN(y;y)1+|y-z|4≤Cε-3.

Additionally, we will assume the initial data to exhibit a local semiclassical structure, captured by localized commutator bounds for ωN with the position operator and with the momentum operator, that are expected to hold true for equilibrium states of confined systems at (or close to) zero temperature and at density of order ε-3.

The present setting can also be viewed as a Kac regime, see [30] for a review. The Kac scaling interpolates between short-range interactions and mean-field potentials: in the Kac regime, one considers a system of particles with density of order one, and one rescales the interaction potential by a suitable parameter γ, keeping the L1-norm of the potential fixed. The parameter γ is large with respect to the typical interparticle spacing, which is of order one, but it is independent of the size of the system. For γ large enough, it is reasonable to expect the predictions of the mean-field approximation to hold true at a local scale, thanks to the fact that each particle interacts with O(γ3) other particles, with strength γ-3. Still, the fact that the thermodynamic limit is taken at fixed potential range allows one to correct unphysical features of the mean-field approximation, notably the non-convexity of the thermodynamic potentials [30]. The many-body Hamiltonian in the Kac scaling is:HNKac=∑j=1N-Δj+γ-3∑i<jNV((xi-xj)/γ).

We look at the evolution of the system for times τ=O(γ): being the Fermi velocity of order one, on this time scale every particle covers a distance comparable with the range of the interaction potential. Thus, writing τ=γt, the time evolution of the system is described by the Schrödinger equation:1.10 iγ-1∂tψN,t=HNKacψN,t.

Let us denote by Uγ the unitary operator on L2(R3N) implementing the space rescaling: UγψN(x1,…,xN)=γ3N/2ψN(γx1,…,γxN). Applying the transformation to both sides of (1.10), we have:1.11 iγ-1∂tUγψN,t=UγHNKacUγ∗UγψN,t=(∑j=1N-γ-2Δj+γ-3∑i<jNV(xi-xj))UγψN,t,

where now UγψN,t is a wave function describing a quantum system with density O(γ3). Thus, the dynamics (1.11) is equivalent to the one generated by (1.7), after setting ε=γ-1.

Although (1.7) does not describe a mean-field regime (particles typically interact with ε-3 other particles and the size of the potential is of the order ε3, with ε now independent of N), for small ε>0 we can still expect a local averaging mechanism to take place and thus that the many-body dynamics (1.8) can be approximated by the time-dependent Hartree equation1.12 iε∂tωN,t=[-ε2Δ+V∗ρt,ωN,t],

with ωN,0=ωN and where nowρt(x):=ε3ωN,t(x;x).

In our main theorem we compare the one-particle reduced density matrix γN,t(1) associated with the solution of (1.8) with the solution of (1.12), and we show that1.13 ‖γN,t(1)-ωN,t‖HSN1/2≤Cε1/2,

for short macroscopic times t of order one in ε, and for a constant C independent of ε and of N,|Λ|. This result should be compared with the trivial estimates1.14 ‖γN,t(1)‖HS≤N1/2,‖ωN,t‖HS=N1/2.

It is important to notice that the rate of convergence, on the r.h.s. of (1.13), only depends on the parameter ε, and not on the volume of the system: in particular, our theorem applies to the setting in which the limit |Λ|→∞ is taken before the limit ε→0. To the best of our knowledge, this is the first derivation of the time-dependent Hartree equation for an interacting, extended Fermi gas (notice, however, that the dynamics of tracer particles or impurities moving through an extended ideal gas has been considered in [16, 26]). Compared with (1.5), in (1.12) we are neglecting the exchange term: as in the mean-field setting [11], for the class of bounded potentials considered in the present paper the exchange term turns out to be of smaller order, and cannot be resolved with our current estimates.

Furthermore, we also consider the case of massive pseudo-relativistic fermions, evolving with Hamiltonian:1.15 HNrel=∑j=1N1-ε2Δj+ε3∑i<jNV(xi-xj).

In this case, the relevant effective evolution equation is the pseudo-relativistic Hartree equation,1.16 iε∂tωN,t=[1-ε2Δ+V∗ρt,ωN,t].

For mean-field fermions, the validity of the pseudo-relativistic Hartree equation has been proved in [12]. Here, we show the validity of the bound (1.13) for extended pseudo-relativistic fermions, for all times. It is worth pointing out that the estimate (1.13) is useful to study the average of extensive operators O, for which ‖O‖HS∼|Λ|12. In fact, the difference of the traces per unit volume of O evolved with the many-body and with the Hartree dynamics is bounded as:1.17 |1|Λ|trO(γN,t(1)-ωN,t)|≤‖O‖HS|Λ|12‖γN,t(1)-ωN,t‖HS|Λ|12≤C(Nε)12|Λ|12≤Cε-1

since N/|Λ|∼ε-3. This bound should be compared with the trivial estimate, of order ε-3/2, implied by the normalizations (1.14).

Technically, the main challenge we have to face consists in showing that the Hartree equation, both in the non-relativistic and in the relativistic case, preserves the local semiclassical structure assumed on the initial data ωN. The fact that the commutator bounds for ωN,t are localized in space makes the proof of the propagation of the semiclassical structure much more involved than in the mean-field regime. In particular, to achieve this we need to propagate the bound (1.9) on the density of particles along the solution of the Hartree equation. For non-relativistic fermions we are able to propagate this estimate for small times of order 1 in ε. For the pseudo-relativistic case, on the other hand, we can take advantage of the boundedness of the group velocity of the particles to show that (1.9) remains true for all times (of order 1 in ε).

The article is organized as follows. In Sect. 2 we state our main results, Theorem 2.3 (non-relativistic case) and Theorem 2.5 (relativistic case). Both theorems are stated for initial data satisfying the estimates of Assumption 2.1. These assumptions are verified for the free Fermi gas and for coherent states in Appendix A. In Sect. 3 we introduce the basic tools of our analysis, namely the fermionic Fock space and Bogoliubov transformations. In Sect. 4 we prove our main result; the proof is based on the adaptation of the method of [11] to extended systems, which crucially relies on the propagation of the local semiclassical structure along the flow of the Hartree equation, as stated in Theorem 4.1 for the non-relativistic case. The proof of Theorem 4.1 is given in Sect. 5. In Sect. 6 we extend the propagation of the local semiclassical structure to the pseudo-relativistic case. Finally, in Appendix A we discuss examples of fermionic states satisfying the local semiclassical structure, while in Appendix B we prove the closeness of the Hartree and the Hartree–Fock dynamics.

Main Result

Let Λ⊂R3 denote a Lebesgue measurable domain of volume |Λ|, such that diam(Λ)≤C|Λ|1/3. We consider an initial wave function ψN∈La2(R3N), ‖ψN‖2=1, describing N particles localized in Λ (in a sense that will be made precise below). We set ε=ϱ-1/3, with the average density ϱ=N/|Λ|. We are interested in the high-density regime, where ϱ≫1 or, equivalently, ε≪1 (independently of N,|Λ|).

Let us denote by γN(1) the reduced one-particle density matrix of ψN, with the integral kernel2.1 γN(1)(x;y)=N∫dx2…dxNψN(x,x2,⋯,xN)ψN(y,x2,⋯,xN)¯.

In the high-density regime, the ground state of the system is expected to be well approximated by a Slater determinant2.2 ψSlater=f1∧⋯∧fN,⟨fi,fj⟩=δij,

for a suitable choice of orthonormal functions fi. This is the content of the Hartree–Fock approximation; in the case of the extended, homogeneous Fermi gas interacting via the Coulomb potential, the validity of the Hartree–Fock approximation has been proved in [19]. The closeness of ψN to a Slater determinant is expressed in terms of closeness of the reduced one-particle density matrix. The reduced one-particle density matrix of the Slater determinant (2.2) is given by a rank-N orthogonal projectorωN=∑i=1N|fi⟩⟨fi|,

and we shall suppose that1N‖γN(1)-ωN‖HS2≪1.

We shall now introduce some important assumptions on ωN, which are expected to hold for a wide class of confined systems at equilibrium. We shall check them in Appendix A, for the free Fermi gas and for coherent states. Strictly speaking, coherent states do not really fit our setting, since they are not projections. However, in Appendix A we will consider a family of coherent states that can be viewed as approximate projections.

For any z∈R3, any t∈R and any n∈N, let us define the localization operator as:2.3 Wz(n)(t):=11+|x^(t)-z|4n,Wz(n):=Wz(n)(0),

where x^(t) denotes the free evolution of the position operator2.4 x^(t)=e-iεtΔx^e+iεtΔ=x^-2iεt∇.

Let us also introduce the weight function:2.5 XΛ(z):=1+dist(z,Λ)4,

with dist(x,Λ)=infy∈Λ|x-y|. Next, we collect the assumptions we shall make on the reference Slater determinant.

Assumption 2.1

(Assumptions on the initial datum). There exist n∈N and T1≥0 such that the following holds true:2.6 supt∈[0,T1]supz∈R3supp∈R3|p|≤ε-1XΛ(z)1+|p|‖Wz(n)(t)[eip·x^,ωN]‖tr≤Cε-2,

and2.7 supt∈[0,T1]supz∈R3XΛ(z)‖Wz(n)(t)[ε∇,ωN]‖tr≤Cε-2.

Furthermore,2.8 supt∈[0,T1]supz∈R3XΛ(z)‖Wz(n)(t)ωN‖tr≤Cε-3,

and2.9 supt∈[0,T1]supz∈R3trWz(1)(t)ωN≤Cε-3.

Remark 2.2

(i) The presence of the weight function XΛ in Assumption 2.1 introduces a weak form of localization of ωN in Λ.

(ii) The class of initial data we are considering has particle density bounded by ε-3. This is quantified, in a weak sense, by the estimate (2.9). In particular, this estimate, together with its propagation by the Hartree dynamics, will be useful to estimate the size of the Hartree potential, that involves the convolution of the interaction potential with the particle density.

(iii) In the assumptions (2.6)–(2.9), we could move the t-dependence from the localization operator to the density matrix, replacing Wz(n)(t) by Wz(n) and ωN by its free evolution ωN,tfree=e-iεtΔωNeiεtΔ. Stating assumptions on the free-evolved ωN,tfree rather than only on the initial density matrix ωN allows us to implicitly control also the momentum distribution associated with the initial data ωN. This will not be needed for pseudo-relativistic particles, because of the boundedness of their group velocity.

(iv) We shall refer to the first two estimates in Assumption 2.1 as the local semiclassical structure of the initial datum. The semiclassical commutator structure of Eq. (2.6) is relevant for momenta below the typical Fermi velocity which is of order ε-1. This is why we restrict the supremum to the set |p|≲ε-1. For higher momenta, we do not expect any gain from the commutator and we will just rely on (2.8).

We are now ready to state our main result. We shall separate the cases of non-relativistic and pseudo-relativistic fermions.

Theorem 2.3

(Main result: non-relativistic case). Let ωN be a rank-N orthogonal projector on L2(R3), satisfying Assumption 2.1 for some n∈N and T1>0. Suppose that V∈L1(R3) is such that2.10 supα:|α|≤8n∫R3dp(1+|p|max(4n,7))|∂pαV^(p)|<∞.

Let ψN∈La2(R3N), such that:2.11 ‖γN(1)-ωN‖tr≤CεδN,for someδ>0.

Let ψN,t=e-iHNt/εψN, with HN given by (1.7), and let γN,t(1) be the reduced one-particle density matrix of ψN,t. Let ωN,t be the solution of the time-dependent Hartree equation:2.12 iε∂tωN,t=[-ε2Δ+ρt∗V,ωN,t],ωN,0=ωN.

Then, there exist 0<T<T1 and C>0, independent of ε and N such that, for all t∈[0,T]:2.13 ‖γN,t(1)-ωN,t‖HS≤Cmax{ε12,εδ2}N12,‖γN,t(1)-ωN,t‖tr≤Cmax{ε12,εδ2}N.

Remark 2.4

(i) The N-dependence of the estimates (2.11), (2.13) is the natural one. In particular, these estimates allow us to quantify the closeness of the expectation values of extensive operators.

(ii) As discussed in the introduction, the result should be compared with the trivial estimates ‖γN,t(1)‖HS≤N1/2, ‖ωN,t‖HS=N1/2, ‖γN,t(1)‖tr=N, ‖ωN,t‖tr=N. Normalizing the Hilbert–Schmidt norm and the trace norm by |Λ|1/2 and |Λ| respectively, and recalling that N=ϱ|Λ| with ϱ=O(ε-3), our main theorem proves convergence of the many-body evolution towards the nonlinear Hartree equation as the density goes to infinity, uniformly in the system size |Λ|.

(iii) We expect the Hartree–Fock equation to give a better approximation of the many-body quantum dynamics. However, similarly to the mean-field setting [11], the difference between the Hartree and the Hartree–Fock dynamics is smaller than the error on the r.h.s. of (2.13) and thus it cannot be resolved with our present techniques. See Appendix B, Proposition B.1, for a proof of the closeness of the Hartree and the Hartree–Fock dynamics.

The time T>0 appearing in Theorem 2.3 is related to the validity of a suitable non-concentration estimate for the solution of the time-dependent Hartree equation, which quantifies the number of particles in bounded regions of space. We prove this bound in Proposition 5.2 for short macroscopic times of order 1 in ε.

Next, let us consider pseudo-relativistic fermions. In this case, we only need assumptions on norms of the initial projector ωN and of its commutators, multiplied with the multiplication operator Wz(n) (in the non-relativistic case, the assumptions involved instead the free evolution Wz(n)(t) of Wz(n)). For this reason, in the next theorem we will only require Assumption 2.1 to hold with T1=0 (which implies t=0 in (2.6)–(2.9)). The other important difference, compared with the non-relativistic case, is that, thanks to the boundedness of the group velocity of the particles, we can establish convergence towards Hartree dynamics for all fixed times t∈R (rather than only for short times).

Theorem 2.5

(Main result: pseudo-relativistic case). Let ωN be a rank-N orthogonal projector on L2(R3), satisfying Assumption 2.1 for some n∈N and for T1=0. Assume that V∈L1(R3) is such that2.14 supα:|α|≤8n∫R3dp(1+|p|7)|∂pαV^(p)|<∞.

Let ψN be as in Theorem 2.3, let ψN,t=e-iHNrelt/εψN, with HNrel given by Eq. (1.15), and let γN,t(1) be the reduced one-particle density matrix of ψN,t. Let ωN,t be the solution of the time-dependent pseudo-relativistic Hartree equation:2.15 iε∂tωN,t=[1-ε2Δ+ρt∗V,ωN,t],ωN,0=ωN.

Then, for all t∈R:2.16 ‖γN,t(1)-ωN,t‖HS≤Cexp(expCt)max{ε12,εδ2}N12,‖γN,t(1)-ωN,t‖tr≤Cexp(expCt)max{ε12,εδ2}N.

The rest of the paper is devoted to the proof of Theorems 2.3, 2.5. The proofs of the two theorems are very similar, except for the propagation of the local semiclassical structure. We will discuss in detail the proof for the non-relativistic case, and only in Sect. 6 we will come back to the pseudo-relativistic case, to adapt the propagation of the local semiclassical structure (the rest of the proof applies unchanged). In Sect. 3 we introduce the Fock space formalism, which will allow us to efficiently describe the fluctuation of the many-body evolution around the nonlinear effective dynamics as the creation and annihilation of particles around a suitable time-dependent state. The proof of Theorem 2.3 is given in Sect. 4. It relies on a bound for the growth of the number of fluctuations, proven in Proposition 4.4. In turn, this result crucially relies on the propagation of the local semiclassical structure along the flow of the Hartree equation. This is established in Sect. 5 for non-relativistic fermions and in Sect. 6 for pseudo-relativistic fermions.

Fock Space Representation

To prove Theorem 2.3 we switch to a Fock space formulation of the problem.

Second quantization

We define the fermionic Fock space F over L2(R3) as:F=C⊕⨁n≥1F(n),F(n):=La2(R3n).

Vectors in the Fock space correspond to infinite sequences of functions (ψ(n))n with ψ(n)∈La2(R3n). A simple example is the vacuum state, Ω=(1,0,…,0,…).

Given ψ∈F, ψ=(ψ(0),ψ(1),…,ψ(n),…), we define the scalar product:⟨ψ1,ψ2⟩=∑n≥0⟨ψ1(n),ψ2(n)⟩L2(R3n).

Equipped with this natural inner product, the Fock space F is a Hilbert space. We henceforth denote by ‖·‖ the norm induced by this inner product, and with a slight abuse of notation we shall use the same notation to denote the operator norm of linear operators acting on F.

It is convenient to introduce creation and annihilation operators, acting on F. Let f∈L2(R3). We define the creation operator a∗(f) and the annihilation operator a(f) asa∗(f)ψ(n)(x1,…,xn):=1n∑j=1n(-1)j-1f(xj)ψ(n-1)(x1,…,xj-1,xj+1,…,xn)a(f)ψ(n)(x1,…,xn):=n+1∫R3dxf(x)¯ψ(n+1)(x,x1,…,xn),

for any ψ∈F. From a physics viewpoint, the creation operator creates a particle with wave function f, while the annihilation operator annihilates a particle with wave function f. These definitions are supplemented by the requirement a(f)Ω=0.

It is not difficult to see that a∗(f)=a(f)∗. Also, the creation and annihilation operators satisfy the canonical anticommutation relations:{a(f),a(g)}={a∗(f),a∗(g)}=0,{a(f),a∗(g)}=⟨f,g⟩L2(R3).

These relations imply that ‖a(f)‖≤‖f‖2, ‖a∗(f)‖≤‖f‖2 (it is not difficult to see that these bounds are sharp, that is the norms of the operators are actually equal to ‖f‖2). In the following sections, we will also make use of operator-valued distributions, e.g., ax∗ and ax for x∈R3, such thata∗(f)=∫R3dxf(x)ax∗,a(f)=∫R3dxf(x)¯ax.

The creation and annihilation operators can be used to define the second quantization of observables. For instance, consider the number operator N, acting on a given Fock space vector as (Nψ)(n)=nψ(n). In terms of the operator-distributions, it can be written as:N=∫R3dxax∗ax.

More generally, for a given one-particle operator J on L2(R3) we define its second quantization dΓ(J) as the operator on the Fock space acting as follows:dΓ(J)↾F(n)=∑j=1nJ(j)

where J(j)=1⊗(n-j)⊗J⊗1⊗(j-1). If J has the integral kernel J(x; y), we can write dΓ(J) as:dΓ(J)=∫(R3)2dxdyJ(x;y)ax∗ay.

In the next lemma we collect some bounds for the second quantization of one-particle operators, that will play an important role in the proof of our main result. Their proofs can be found in [11, Lemma 3.1].

Lemma 3.1

Let J be a bounded operator on L2(R3). We have, for any ψ∈F:|⟨ψ,dΓ(J)ψ⟩|≤‖J‖op⟨ψ,Nψ⟩,dΓ(J)ψ≤JopNψ.

Let J be a Hilbert–Schmidt operator. We then have, for any ψ∈F:dΓ(J)ψ≤JHSN1/2ψ∫(R3)2dxdx′J(x;x′)axax′ψ≤‖J‖HSN1/2ψ∫(R3)2dxdx′J(x;x′)ax∗ax′∗ψ≤2‖J‖HS(N+1)1/2ψ.

Let J be a trace class operator. We then have, for any ψ∈F:dΓ(J)ψ≤2Jtr‖ψ‖∫(R3)2dxdyJ(x;x′)axax′ψ≤2Jtr‖ψ‖∫(R3)2dxdyJ(x;x′)ax∗ax′∗ψ≤2Jtr‖ψ‖.

Given a Fock space vector ψ∈F, we define its reduced one-particle density matrix as the non-negative trace class operator γψ(1) on L2(R3) with integral kernel:3.1 γψ(1)(x;y)=⟨ψ,ay∗axψ⟩.

If ψ is an N-particle state, it is not difficult to check that this definition agrees with (2.1). Furthermore, given a one-particle observable J, we have:3.2 ⟨ψ,dΓ(J)ψ⟩=∫(R3)2dxdyJ(x;y)⟨ψ,ax∗ayψ⟩=trJγψ(1),

an identity which motivates the definition (3.1). In particular, trγψ(1)=⟨ψ,Nψ⟩ is the expected number of particles in ψ.

Next, we lift the many-body Hamiltonian to the Fock space, as follows. We define the second quantization of HN as HN=0⊕⨁n≥1HN(n), whereHN(n)=∑i=1n-ε2Δi+ε3∑i<jnV(xi-xj).

In terms of the operator-valued distributions, we can write:3.3 HN=ε2∫R3dx∇ax∗∇ax+ε32∫(R3)2dxdyV(x-y)ax∗ay∗ayax.

The time evolution of a state in the Fock space is defined as ψt=e-iHNt/εψ. On N-particle states, this coincides with the solution of the Schrödinger equation (1.8).

Bogoliubov transformations

The Fock-space representation of the problem is particularly convenient also in view of the representation of Slater determinants via Bogoliubov transformations, defined here.

Given an orthonormal family (fi)i=1N, the corresponding Slater determinant can be represented in Fock space as follows:a∗(f1)⋯a∗(fN)Ω=(0,0,…,0,f1∧⋯∧fN,0,…)

where the only nontrivial entry is the N-th. A crucial fact for our analysis, as was the case in previous work starting from [11], is the existence of a unitary operator RωN:F→F with the following two properties:RωNΩ=a∗(f1)⋯a∗(fN)Ω

and, for any g∈L2(R3),3.4 RωN∗a(g)RωN=a(uNg)+a∗(v¯Ng¯)

where uN=1-ωN and vN=∑j=1N|fj¯⟩⟨fj|. Equivalently, consider an orthonormal basis (fi)i≥1 of L2(R3) obtained completing the orthonormal family f1,…,fN in an arbitrary way. We have:3.5 RωN∗a(fj)RωN=a(fj)ifj>Na∗(fj)ifj≤N.

It follows that RωN∗=RωN=RωN-1. The map RωN is known as Bogoliubov transformation, and it acts as a particle-hole transformation. It allows us to switch to a new representation of the system, where the new vacuum is the Slater determinant associated to the reduced density ωN. The new creation operators, given by (3.5), create excitations around the Slater determinant, which are either particles outside the determinant or holes in it. The proof of the existence of the unitary operator RωN with the properties listed above can be found, for example, in [31].

More generally, for every t∈R, we can associate a Bogoliubov transformation RωN,t to the solution of the time-dependent Hartree equation (2.12). Then RωN,tΩ is the Slater determinant with reduced one-particle density matrix ωN,t andRωN,t∗a(g)RωN,t=a(uN,tg)+a∗(v¯N,tg¯)

with uN,t,vN,t defined similarly as uN,vN after (3.4).

Proof of the Main Result

Here we shall prove our main result, Theorem 2.3. It will be a corollary of an estimate for the growth of the number operator evolved with a suitable fluctuation dynamics, Proposition 4.1, proven in the next section.

Bound on the growth of fluctuations

Let ωN,t be the solution of the time-dependent Hartree equation. We introduce the fluctuation dynamics4.1 UN(t;s):=RωN,t∗e-iHN(t-s)/εRωN,s.

Given an N-particle state ψ, we define the corresponding fluctuation vector ξ=RωN∗ψ. Then, we rewrite the many-body evolution of ψ asψt=e-iHNt/εRωNξ=RωN,tUN(t;0)ξ.

To show our main theorem we need to prove that, for N-particle initial data ψ close to the Slater determinant with reduced one-particle density matrix ωN, the evolution ψt remains close to the Slater determinant with reduced one-particle density matrix ωN,t. This will follow, if we can control the growth of the expectation of the number of particles4.2 ⟨UN(t;0)ξ,NUN(t;0)ξ⟩.

To reach this goal, a key ingredient is the propagation of the semiclassical structure, introduced in Assumption 2.1, along the flow of the Hartree equation. This is the content of the next theorem.

Theorem 4.1

(Propagation of the local semiclassical structure). Under the same assumptions of Theorem 2.3, the following is true. There exist C>0 and T>0 such that:4.3 supt∈[0,T]supz∈R3XΛ(z)‖Wz(n)ωN,t‖tr≤Cε-3,

and4.4 supt∈[0,T]supp:|p|≤ε-1supz∈R3XΛ(z)1+|p|‖Wz(n)[eip·x^,ωN,t]‖tr≤Cε-2,

Remark 4.2

The requirement that ωN is a rank-N projection is not needed in this theorem. It could be replaced by 0≤ωN≤1, trωN=N.

The proof of Theorem 4.1 is postponed to Sect. 5. From Theorem 4.1, we obtain the following corollary, which will be used to control the growth of the expectation (4.2) and thus to prove our main result, Theorem 2.3.

Corollary 4.3

(Bounds for commutators with regular functions). Under the same assumptions of Theorem 4.1, the following is true. Let:4.5 F(x)=∫R3dpeip·xF^(p),∫dp(1+|p|)|∂pkF^(p)|≤Cfor allk≤8n.

Let Fz(x)=F(x-z). Then, the following bound holds true:4.6 supt∈[0,T]supz∈R3XΛ(z)‖[ωN,t,Fz(x^)]‖tr≤Cε-2.

Proof

Let χ(p) be a smooth, non-increasing, compactly supported function, equal to 1 for |p|≤ε-1-1 and equal to 0 for |p|>ε-1. We write:4.7 F(x)=F(≤)(x)+F(>)(x),F(≤)(x)=∫R3dpeip·xχ(p)F^(p),F(>)(x)=∫R3dpeip·x(1-χ(p))F^(p),

so that4.8 ‖[ωN,t,Fz(x^)]‖tr≤‖[ωN,t,Fz(≤)(x^)]‖tr+‖[ωN,t,Fz(>)(x^)]‖tr.

Let us define, for ♯=≤,>:4.9 f(♯)(x):=(1+|x|4n)F(♯)(x),g(♯)(x):=(1+|x|4n)2F(♯)(x),

together with fz♯(x):=f♯(x-z) and gz♯(x):=g♯(x-z). Notice that both f^(≤) and g^(≤) are supported on {p∈R3||p|≤ε-1}. By the assumptions (4.5) and the smoothness of χ,4.10 ‖(1+|·|)f^(♯)‖1≤C,‖g^(♯)‖1≤C.

Consider the first term on the r.h.s. of (4.8). We have:4.11 ‖[ωN,t,Fz(≤)(x^)]‖tr=‖[ωN,t,Wz(n)fz(≤)(x^)]‖tr≤‖[ωN,t,Wz(n)]fz(≤)(x^)‖tr+‖Wz(n)[ωN,t,fz(≤)(x^)]‖tr.

Consider the second term on the r.h.s. of (4.11). We estimate it as:4.12 ‖Wz(n)[ωN,t,fz(≤)(x^)]‖tr≤∫dp|f^(≤)(p)|‖Wz(n)[ωN,t,eip·x^]‖tr.

Therefore, using (4.4), the contribution of this term to the final bound (4.6) is:4.13 supt∈[0,T]supz∈R3XΛ(z)‖Wz(n)[ωN,t,fz(≤)(x^)]‖tr≤supt∈[0,T]supz∈R3∫|p|≤ε-1dp|f^(≤)(p)|(1+|p|)XΛ(z)1+|p|‖Wz(n)[ωN,t,eip·x^]‖tr≤Csupt∈[0,T]supz∈R3supp:|p|≤|ε|-1XΛ(z)1+|p|‖Wz(n)[ωN,t,eip·x^]‖tr≤Cε-2,

where we used (4.10) and the fact that f^(≤) is supported on {p∈R3||p|≤ε-1}. Consider now the first term on the r.h.s. of (4.11). We estimate it as:‖[ωN,t,Wz(n)]fz(≤)(x^)‖tr=‖[ωN,t,Wz(n)]Wz(n)gz(≤)(x^)‖tr≤C‖[ωN,t,Wz(n)]Wz(n)‖tr,

where we used that g(♯) is bounded, which follows from (4.10). We then have:4.14 ‖[ωN,t,Wz(n)]Wz(n)‖tr≤∫dp|W(n)^(p)|‖[ωN,t,eip·x^]Wz(n)‖tr≤∫|p|≤ε-1dp|W(n)^(p)|‖[ωN,t,eip·x^]Wz(n)‖tr+∫|p|>ε-1dp|W(n)^(p)|‖[ωN,t,eip·x^]Wz(n)‖tr,

where we denoted the Fourier transform of (1+|·|4n)-1 by W(n)^. The contribution of the first term to the final bound (4.6) is estimated exactly as before. We get:4.15 supt∈[0,T]supz∈R3XΛ(z)∫|p|≤ε-1dp|W(n)^(p)|‖[ωN,t,eipx^]Wz(n)‖tr≤Cε-2.

Consider now the second term in (4.14). We estimate it as:4.16 ∫|p|>ε-1dp|W(n)^(p)|‖[ωN,t,eip·x^]Wz(n)‖tr≤2∫|p|>ε-1dp|W(n)^(p)|‖ωN,tWz(n)‖tr.

Using that∫|p|>ε-1dp|W(n)^(p)|≤ε∫dp|p||W(n)^(p)|≤Cε,

and using (4.3) to estimate the last trace norm in (4.16), we get:4.17 supt∈[0,T]supz∈R3XΛ(z)∫|p|>ε-1dp|W(n)^(p)|‖[ωN,t,eip·x^]Wz(n)‖tr≤Cε-2.

Putting together (4.13), (4.15), (4.17), we find:4.18 supt∈[0,T]supz∈R3XΛ(z)‖[ωN,t,Wz(n)]fz(≤)(x^)‖tr≤Cε-2.

Together with (4.11), the bounds (4.13), (4.18) imply:4.19 supt∈[0,T]supz∈R3XΛ(z)‖[ωN,t,Fz(≤)(x^)]‖tr≤Cε-2.

This proves the desired estimate for the first term on the r.h.s. of (4.8). Consider now the second term on the r.h.s. of (4.8). By opening the commutator and by using the invariance of the norm under hermitian conjugation, we get:4.20 ‖[ωN,t,Fz(>)(x^)]‖tr≤2‖ωN,tFz(>)(x^)‖tr≤2‖ωN,tWz(n)fz(>)(x^)‖tr.

By (4.10), together with the fact that f^(>)(p)=0 for |p|<ε-1-1:‖f(>)(x^)‖op≤∫dp|f^(>)(p)|≤Cε∫dp|f^(>)(p)||p|≤Cε;

using (4.3), we easily get:4.21 supt∈[0,T]supz∈R3XΛ(z)‖ωN,tWz(n)fz(>)(x^)‖tr≤Cε-2.

Plugging this estimate in (4.20), we get:supt∈[0,T]supz∈R3XΛ(z)‖[ωN,t,Fz(>)(x^)]‖tr≤Cε-2.

Combined with (4.19) and with (4.8), this concludes the proof of Corollary 4.3. □

The next result is the key to control the distance between the many-body and the effective dynamics. It relies on the propagation of the local semiclassical structure, Theorem 4.1, and on its Corollary 4.3.

Proposition 4.4

(Bound on the growth of fluctuations). Under the same assumptions of Theorem 2.3, the following is true. Let ψ be the Fock space vector associated with ψN, and let ξ=R∗ψ. Then, there exists C>0 such that:4.22 supt∈[0,T]⟨ξ,UN(t;0)∗NUN(t;0)ξ⟩≤C(Nε+⟨ξ,Nξ⟩).

The proof of Theorem 2.3 is a corollary of Proposition 4.4, and will be given in Sect. 4.2. Let us now prove Proposition 4.4

Proof of Proposition 4.4

The proof is based on a Gronwall-type argument, as in [11, 12]. For convenience, we shall use the following notationsut;x(·):=uN,t(·;x),vt;x(·):=vN,t(·;x),v¯t;x(·):=vN,t(·;x)¯.

The starting point is the following identity, for any ξ∈F:4.23 iε∂t〈ξ,UN(t;0)∗NUN(t;0)ξ〉=-4iε3Im∫dxdyV(x-y)〈ξ,UN(t;0)∗(a(v¯t;x)a(v¯t;y)a(ut;y)a(ut;x)+a∗(ut;x)a(v¯t;y)a(ut;y)a(ut;x)+a∗(ut;y)a∗(v¯t;y)a∗(v¯t;x)a(v¯t;x))UN(t;0)ξ〉+4iε3Im∫dxdyV(x-y)〈ξ,UN(t;0)∗(ωN,t(y;x)a∗(ut,y)a∗(v¯t,x))UN(t;0)ξ〉=I+II+III+IV,

with a natural identification of each term. The proof of this identity follows [11, Proof of Proposition 3.3], and it is insensitive to the form of the dispersion relation. Thus, it applies unchanged to the pseudo-relativistic case [12]. The difference with respect to [11] is that now the fluctuation dynamics is defined starting from the Hartree equation rather than the Hartree–Fock equation. This is the reason for the extra quadratic term in the right-hand side of (4.23), which in [11] is cancelled by the presence of the exchange term in the time-dependent Hartree–Fock equation. Let us briefly sketch the arguments and refer to [11, Proof of Proposition 3.3] for further details. By using the definition of the Bogoliubov transformation, one can see that4.24 iε∂tUN(t;0)∗NUN(t;0)=-2UN(t;0)∗RωN,t∗(dΓ(iε∂tωN,t)-[HN,dΓ(ωN,t)])RωN,tUN(t;0).

Plugging the Hartree equation, and using thatdΓ([-ε2Δ,ωN,t])=[dΓ(-ε2Δ),dΓ(ωN,t)],

we easily get:iε∂tUN(t;0)∗NUN(t;0)=-2UN(t;0)∗RωN,t∗(dΓ([V∗ρΛ,t,ωN,t])-[V,dΓ(ωN,t)])RωN,tUN(t;0),

where V is the second quantization of the many-body interaction. After conjugation with the Bogoliubov transformation, see (3.4), and normal ordering, we get:RωN,t∗dΓ([V∗ρΛ,t,ωN,t])RωN,t=ε3∫dxdyV(x-y)ωN,t(x,x)a∗(ut,y)a∗(u¯t,y)-h.c.,

andRωN,t∗[V,dΓ(ωN,t)]RωN,t=ε3∫dxdyV(x-y)(a(v¯t;x)a(v¯t;y)a(ut;y)a(ut;x)+a∗(ut;x)a(v¯t;y)a(ut;y)a(ut;x)+a∗(ut;y)a∗(v¯t;y)a∗(v¯t;x)a(v¯t;x))+ε3∫dxdyV(x-y)(ωN,t(x;x)a∗(ut,y)a∗(u¯t,y)-ωN,t(y;x)a∗(ut,y)a∗(v¯t,x))-h.c.

which gives the claim.

Let us now bound the terms appearing on the r.h.s. of (4.23). For brevity, we set ξt:=UN(t;0)ξ.

Bound for the term I. We rewrite the interaction potential as:4.25 V(x-y)=∫R3dzV(1)(x-z)V(2)(z-y),

where:4.26 V(1)(x):=∫R3dpeip·x1+|p|6,V(2)(x):=∫R3dpeip·x(1+|p|6)V^(p).

The function V(2) is bounded, and its regularity can be inferred from the assumption (2.10) on the potential. Accordingly, the term I is re-written asI=ε3∫(R3)2dxdyV(x-y)ξt,a(v¯t;x)a(v¯t;y)a(ut;y)a(ut;x)ξt=ε3∫R3dz∫R3dxVz(1)(x)a∗(ut;x)a∗(v¯t;x)ξt,∫R3dyVz(2)(y)a(v¯t;y)a(ut;y)ξt,

where we used the notation fz(x)=f(x-z). Next, we notice that4.27 ∫R3dxVz(1)(x)a∗(v¯t;x)a∗(ut;x)=∫(R3)2drds(∫R3dxuN,t(s;x)Vz(1)(x)v¯N,t(r;x))ar∗as∗=∫(R3)2drds(uN,tVz(1)(x^)v¯N,t)(r;s)ar∗as∗

and that4.28 ∫R3dyVz(2)(y)a(v¯t;y)a(ut;y)=∫(R3)2drds(∫R3dyuN,t(s;y)¯Vz(2)(y)vN,t(r;y))aras=∫(R3)2drds(vN,tVz(2)(x^)uN,t)(r;s)aras.

By the Cauchy–Schwarz inequality and by Lemma 3.1 we can bound the term I as:|I|≤ε3(∫R3dz‖∫R3dxVz(1)(x)a∗(v¯t;x)a∗(ut;x)ξt‖2)1/2·(∫R3dz‖∫R3dxVz(2)(x)a(v¯t;x)a(ut;x)ξt‖2)1/2≤ε3(∫R3dz‖uN,tVz(1)(x^)v¯N,t‖tr2)1/2(∫R3dz‖vN,tVz(2)(x^)uN,t‖tr2)1/2≤ε3(∫R3dz‖[ωN,t,Vz(1)(x^)]‖tr2)1/2(∫R3dz‖[ωN,t,Vz(2)(x^)]‖tr2)1/2≤ε3(∫R3dzXΛ(z)-2)∏j=1,2supz∈R3XΛ(z)‖[ωN,t,Vz(j)(x^)]‖tr,

where in the third step we used that uN,t=1-ωN,t together with the orthogonality condition uN,tv¯N,t=vN,tuN,t=0, and that ‖v¯N,t‖op=1. The trace norm of the commutator can be estimated using Corollary 4.3. In fact, the function V(1) satisfies the assumptions of Corollary 4.3, and the same is true for the function V(2), thanks to the assumptions on the potential V, Eq. (2.10). Therefore, we get:supt∈[0,T]∏j=1,2supz∈R3XΛ(z)‖[ωN,t,Vz(j)(x^)]‖tr≤Cε-4.

Using that ∫R3dzXΛ(z)-2≤C|Λ|, we find:4.29 |I|≤C|Λ|ε-1≤Cε2N.

Bound for the term II. We write:II=ε3∫R3dx〈a(ut;x)ξt,(∫R3dyVx(y)a(v¯t;y)a(ut;y))a(ut;x)ξt〉=ε3∫R3dx〈a(ut;x)ξt,(∫(R3)2drds(vN,tVx(x^)uN,t)(r,s)aras)a(ut;x)ξt〉,

recall (4.28). By the Cauchy–Schwarz inequality and by Lemma 3.1 we4.30 |II|≤ε3∫R3dx‖a(ut;x)ξt‖‖vN,tVx(x^)uN,t‖tr‖a(ut;x)ξt‖≤ε3supz∈R3‖[Vz(x^),ωN,t]‖tr∫R3dx‖a(ut;x)ξt‖2.

The trace norm of the commutator can be estimated using Corollary 4.3. The last term is estimated in terms of the number operator:∫R3dx‖a(ut;x)ξt‖2=⟨ξt,dΓ(uN,t)ξt⟩≤⟨ξt,Nξt⟩,

where we used that uN,t2=uN,t. Thus, we get, for t∈[0,T]:4.31 |II|≤Cε⟨ξt,Nξt⟩.

Bound for the term III. We writeIII=ε3∫R3dx〈a(v¯t;x)ξt,(∫(R3)2drds(uN,tVx(x^)v¯N,t)(r;s)aras)a(vt;x)ξt〉.

Proceeding as we did for the term II, we find, for t∈[0;T],4.32 |III|≤ε3supz∈R3‖[Vz(x^),ωN,t]‖tr∫R3dx‖a(vt;x)ξt‖2≤Cε⟨ξt,Nξt⟩,

where we used that v¯N,tvN,t≤1.

Bound for the term IV. Finally, we consider the term IV, containing the quadratic contributions. We rewrite the potential as in (4.25). We then get:IV=ε3∫R3dz〈ξt,∫(R3)2dxdyVz(1)(x)Vz(2)(y)ωN,t(y,x)a∗(ut,y)a∗(v¯t,x)ξt〉=ε3∫R3dz〈ξt,∫(R3)2drds(uN,tVz(2)ωN,tVz(1)v¯N,t)(r;s)ar∗as∗ξt〉

where we used that vN,t(s,x)=vN,t(x,s). By the Cauchy–Schwarz inequality and by Lemma 3.1 we obtain4.33 |IV|≤ε3∫R3dz‖∫(R3)2drds(uN,tVz(2)ωN,tVz(1)v¯N,t)(r,s)ar∗as∗ξt‖≤ε3∫R3dz‖uN,tVz(2)(x^)ωN,tVz(1)(x^)v¯N,t‖tr≤ε3(∫R3dzXΛ(z)-1)supz∈R3XΛ(z)‖Vz(2)(x^)ωN,tVz(1)(x^)‖tr

where we used that ‖uN,t‖op=‖v¯N,t‖op=1. Next, we estimate:‖Vz(2)(x^)ωN,tVz(1)(x^)‖tr≤C‖ωN,tVz(1)‖tr≤C‖ωN,tWz(n)‖tr

where we used that ‖Vz(2)‖∞≤C and that ‖(Wz(n))-1Vz(1)‖∞≤C. Hence, using the bound (2.8), we get:4.34 |IV|≤C|Λ|≤Cε3N.

Notice that, by using the orthogonality between uN,t and ωN,t, we could have written the bound in (4.33) with ‖[Vz(2)(x^),ωN,t]Vz(1)(x^)‖tr instead and eventually improved the estimate (4.34) by ε.

Conclusion. Putting together (4.23), (4.29), (4.31), (4.32), (4.34), we have:∂t⟨ξt,Nξt⟩≤CNε+C⟨ξt,Nξt⟩.

By the Gronwall lemma, for all t∈[0,T], for a constant K depending on T:⟨ξt,Nξt⟩≤K(Nε+⟨ξ,Nξ⟩)

which concludes the proof. □

Proof of Theorem 2.3

Proof of Theorem 2.3

We now prove our main result. It turns out that the distance between the many-body evolution and the Hartree equation can be quantified by the average number of particles in the fluctuation vector ξt=UN(t;0)ξ=RωN,t∗ψN,t, associated with the solution ψN,t=e-iHNt/εψN of the Schrödinger equation (1.8), with initial data ψN=RωNξ. In fact4.35 ⟨ξt,Nξt⟩=⟨ψN,t,RωN,tNRωN,t∗ψN,t⟩=⟨ψN,t,(N-2dΓ(ωN,t)+N)ψN,t⟩=2trγN,t(1)(1-ωN,t),

where we used that trγN,t(1)=N. Thus, we find4.36 ‖γN,t(1)-ωN,t‖HS2=tr|γN,t(1)-ωN,t|2=tr(γN,t(1)2+ωN,t2-2γN,t(1)ωN,t)≤2trγN,t(1)(1-ωN,t)≡⟨ξt,Nξt⟩.

In the second step we used the cyclicity of the trace, while in the last step we used that γN,t(1)≤1, ωN,t≤1 and that trγN,t(1)=trωN,t=N. On the other hand, the quantity ⟨ξ,Nξ⟩ is controlled by the distance between γN(1) and ωN, in the trace-norm topology. In fact:4.37 trγN(1)(1-ωN)=tr(γN(1)-ωN)(1-ωN)≤‖γN(1)-ωN‖tr.

Thus, (4.35) at t=0, (4.37) and the assumption (2.11) imply that4.38 ⟨ξ,Nξ⟩≤CεδN.

Hence, thanks to Proposition 4.4, we have:⟨ξt,Nξt⟩≤Cmax{εδ,ε}N;

plugging this bound in (4.36), the final claim (2.13) follows. The bound in the trace-norm topology follows by using the inequality‖γN,t(1)-ωN,t‖tr≤C(⟨ξt,(N+1)ξt⟩+‖vN,t‖HS⟨ξt,(N+1)ξt⟩1/2),

which is proven by duality in [11, Proof of Theorem 2.1], and by recalling that ‖vN,t‖HS=N1/2. This concludes the proof of Theorem 2.3. □

Propagation of the Semiclassical Structure: Nonrelativistic Case

Here we shall prove Theorem 4.1, which is the key technical ingredient to control the growth of the number of fluctuations around the Hartree dynamics, Proposition 4.4. In what follows, we will denote multi-indices in N3 by Greek letters, and we shall denote the length of a multi-index by |α|:=∑jαj. Moreover, unless otherwise specified, we will denote generic constants, possibly depending on n and T, by C and K, with the understanding that these constants can be different on different lines.

Throughout the section, we will make use of the following elementary lemma.

Lemma 5.1

(Monotonicity properties of the trace norm). Let A, B, C be bounded operators on L2(R3) such that |A|2≤|B|2. Suppose that AC and BC are trace class. Then:‖AC‖tr≤‖BC‖tr.

Proof

We write:‖AC‖tr=tr(AC)∗AC=trC∗|A|2C≤trC∗|B|2C=‖BC‖tr,

where the inequality follows from the operator monotonicity of the square root. This concludes the proof of the lemma. □

Evolution of the Localization Operators

With respect to previous works, [5, 11, 12, 29], one important difference is the presence of the localization operators Wz(n) in the semiclassical structure defined in Assumption 2.1. To propagate these bounds, we need to control the behavior of the localization operators under the Hartree dynamics U(t; s), defined byiε∂tU(t;s)=(-ε2Δ+V∗ρt)U(t;s),U(s;s)=1,

where ρt(x)=ε3ωN,t(x;x), ωN,t is the solution of the Hartree equation with initial datum ωN. To achieve this, a key role will be played by the following proposition.

Proposition 5.2

(No-concentration bound for short times). Under the assumption on the potential V of Theorem 2.3, and under the assumption of Eq. (2.9) on the initial datum ωN, the following is true. There exists T∗>0 independent of ε such that:5.1 supt∈[0,T∗]supz∈R3trWz(1)ωN,t≤ε-3C.

We will postpone the proof of Proposition 5.2 after the proof of the next proposition, where we control the propagation of the localization operators.

Proposition 5.3

(Bounds on the evolution of the localization operator). Under the same assumptions of Theorem 2.3, consider the modified Hartree generator Up(t;s), defined byiε∂tUp(t;s)=(-ε2Δ+V∗ρt+iε2p·∇)Up(t;s),Up(s;s)=1.

Then, for all z∈R3, t0∈R, for any 0≤s,t≤T and any 1≤k≤2n the following is true:5.2 Up(t;s)∗Wz(k)(t0)Up(t;s)≤CWz+εp(t-s)(k)(t0+t-s).

Remark 5.4

Using the inequalities:5.3 cWz(k/2)2(t0)≤Wz(k)(t0)≤CWz(k/2)2(t0)

Equation (5.2) also implies:Up(t;s)∗Wz(k/2)2(t0)Up(t;s)≤CWz+εp(t-s)(k/2)2(t0+t-s).

These inequalities are of the form |A|2≤|B|2, and will be extensively used in combination with Lemma 5.1.

The proof of Proposition 5.3 relies on the following technical lemma.

Lemma 5.5

Let k∈N and let F:R3→C be such that:‖DjF‖∞:=maxα:|α|=j‖∂αF‖∞<∞for allj≤2k.

Then, the following is true:5.4 ‖11+|x^(t)|2kF(x^)(1+|x^(t)|2k)-F(x^)‖op≤Ck(max0≤j≤2k‖DjF‖∞)|t|ε(1+t2kε2k),

for some constants Ck.

Proof of Lemma 5.5

We have:5.5 11+|x^(t)|2kF(x^)(1+|x^(t)|2k)-F(x^)=11+|x^(t)|2k[F(x^),|x^(t)|2k].

Next, we write:5.6 [F(x^),|x^(t)|2k]=|x^(t)|2[F(x^),|x^(t)|2(k-1)]+[F(x^),|x^(t)|2]|x^(t)|2(k-1).

Consider the last commutator. We have:[F(x^),|x^(t)|2]=∑i=13(x^i(t)[F(x^),x^i(t)]+[F(x^),x^i(t)]x^i(t))=∑i=13(2x^i(t)[F(x^),x^i(t)]+[[F(x^),x^i(t)],x^i(t)]).

Recalling that x^i(t)=x^i-i2tε∂i, we get:[F(x^),x^i(t)]=i2tε∂iF(x^),[[F(x^),x^i(t)],x^i(t)]=-4t2ε2∂i2F(x^).

Similarly, we can rewrite [F(x^),|x^(t)|2n] as a sum of terms, involving 0≤j≤2k-1 operators x^i(t) on the left times a partial derivative of F(x^) of order 2k-j, multiplied by a factor (2t)2k-jε2k-j. It is not difficult to see that:5.7 ‖11+|x^(t)|2k[F(x^),|x^(t)|2k]‖op≤∑j=02k-1Cj|t|2k-jε2k-jsupα:|α|=2k-j‖11+|x^(t)|2k|x^(t)|j∂αF(x^)‖op≤Ck(max0≤j≤2k‖DjF‖∞)|t|ε(1+t2kε2k).

This concludes the proof of (5.4). □

Remark 5.6

As a consequence of the assumption in Eq. (2.9), the function V∗ρt satisfies the hypotheses of Lemma 5.5 for 0≤t≤T, 1≤k≤max(2n,3). In fact, for j≤max(4n,7) we have:5.8 ‖DjV∗ρt‖∞≤‖DjV(1+|·|4)‖∞‖W(1)∗ρt‖∞≤Cj‖W(1)∗ρt‖∞,

where we used the non-negativity of the density ρt and the assumption (2.10) on the potential V. Next, by Eq. (5.1):5.9 ‖W(1)∗ρt‖∞=supz∈R3ε3∫dy11+|z-y|4ωN,t(y,y)≡supz∈R3ε3trWz(1)ωN,t≤CT,

which proves the boundedness of ‖DjV∗ρt‖∞ for any j≤max(4n,7).

We are now ready to prove Proposition 5.3.

Proof of Proposition 5.3

Let Up0(t;s)=ei(ε2Δ-iε2p·∇)(t-s)/ε be the modified free dynamics and notice thatUp0(t;s)∗x^Up0(t;s)=x^(t-s)+εp(t-s).

We introduce the Hartree dynamics in the interaction picture as:5.10 UpI(t;s):=Up0(t;0)∗Up(t;s)Up0(s;0),

satisfying the following evolution equation:5.11 iε∂tUpI(t;s)=Up0(t;0)∗(V∗ρt)Up0(t;0)UpI(t;s),UpI(s;s)=1.

Let us start by proving the bound (5.2). Recalling (5.3), we write:5.12 Up(t;s)∗Wz(k/2)(t0)2Up(t;s)=Up(t;s)∗Up0(t;0)Up0(t;0)∗Wz(k/2)(t0)2Up0(t;0)Up0(t;0)∗Up(t;s)≡Up0(s;0)UpI(t;s)∗Wz+εpt(k/2)(t0+t)2UpI(t;s)Up0(s;0)∗.

By the arbitrariness of z and t0, we will focus on the following operator:UpI(t;s)∗Wz(k/2)(t0)2UpI(t;s).

Let ϕ∈L2(R3). We write:5.13 ⟨ϕ,UpI(t;s)∗Wz(k/2)(t0)2UpI(t;s)ϕ⟩=⟨ϕ,Wz(k/2)(t0)2ϕ⟩-iε∫stdτiε∂τ⟨ϕ,UpI(τ;s)∗Wz(k/2)(t0)2UpI(τ;s)ϕ⟩.

To compute the derivative, recall (5.11). We get:5.14 iε∂τ⟨ϕ,UpI(τ;s)∗Wz(k/2)(t0)2UpI(τ;s)ϕ⟩=⟨UpI(τ;s)ϕ,[Wz(k/2)(t0)2,Up0(τ;0)∗(V∗ρτ)Up0(τ;0)]UpI(τ;s)ϕ⟩=⟨Up0(τ;0)UpI(τ;s)ϕ,[Wz-εpτ(k/2)(t0-τ)2,V∗ρτ]Up0(τ;0)UpI(τ;s)ϕ⟩.

Let us now focus on the commutator appearing in the scalar product. We introduce the short-hand notations:5.15 x~:=x^(t0-τ)-z+εpτ,W(k/2):=Wz-εpτ(k/2)(t0-τ).

We write:5.16 [(W(k/2))2,V∗ρτ]=W(k/2)[W(k/2),V∗ρτ]+[W(k/2),V∗ρτ]W(k/2).

Consider the first term. We rewrite:|⟨ϕ,W(k/2)[W(k/2),V∗ρτ]ϕ⟩|≤‖W(k/2)ϕ‖‖[W(k/2),V∗ρτ]ϕ‖=‖W(k/2)ϕ‖‖(W(k/2)(V∗ρτ)(W(k/2))-1-V∗ρτ)W(k/2)ϕ‖≤‖(W(k/2)(V∗ρτ)(W(k/2))-1-V∗ρτ)‖op‖W(k/2)ϕ‖2.

The function V∗ρτ satisfies the assumptions of Lemma 5.5, recall Remark 5.6. Hence, recalling also (5.3):5.17 |⟨ϕ,W(k/2)[W(k/2),V∗ρτ]ϕ⟩|≤Cε‖W(1)∗ρτ‖∞|t0-τ|⟨ϕ,W(k)ϕ⟩≤Kε|t0-τ|⟨ϕ,W(k)ϕ⟩.

The same bound holds for the second term in (5.16). Therefore:5.18 |⟨ϕ,[(W(k/2))2,V∗ρτ]ϕ⟩|≤Cε|t0-τ|⟨ϕ,W(k)ϕ⟩.

Let us now come back to (5.13). The identity (5.14) implies:⟨ϕ,UpI(t;s)∗Wz(k/2)(t0)2UpI(t;s)ϕ⟩≤⟨ϕ,Wz(k/2)(t0)2ϕ⟩+Cε∫stdτ|⟨Up0(τ;0)UpI(τ;s)ϕ,[Wz-εpτ(k/2)(t0-τ)2,V∗ρτ]Up0(τ;0)UpI(τ;s)ϕ⟩|,

which we estimate as, using the bound (5.18) with ϕ replaced by Up0(τ;0)UpI(τ;s)ϕ:⟨ϕ,UpI(t;s)∗Wz(k/2)(t0)2UpI(t;s)ϕ⟩≤⟨ϕ,Wz(k/2)(t0)2ϕ⟩+C∫stdτ|t0-τ|⟨Up0(τ;0)UpI(τ;s)ϕ,Wz-εpτ(k)(t0-τ)Up0(τ;0)UpI(τ;s)ϕ⟩.

Using that⟨Up0(τ;0)UpI(τ;s)ϕ,Wz-εpτ(k)(t0-τ)Up0(τ;0)UpI(τ;s)ϕ⟩=⟨UpI(τ;s)ϕ,Wz(k)(t0)UpI(τ;s)ϕ⟩,

and recalling (5.3), we get, for all 0≤s,t≤T, by the Gronwall lemma:5.19 ⟨ϕ,UpI(t;s)∗Wz(k)(t0)UpI(t;s)ϕ⟩≤C⟨ϕ,Wz(k)(t0)ϕ⟩,

where the constant C depends on t0, T and n. Going back to (5.12), we have, for all ϕ∈L2(R3):⟨ϕ,Up(t;s)∗Wz(k)(t0)Up(t;s)ϕ⟩=⟨ϕ,Up0(s;0)UpI(t;s)∗Wz+εpt(k)(t0+t)UpI(t;s)Up0(s;0)∗ϕ⟩≤C⟨ϕ,Up0(s;0)Wz+εpt(k)(t0+t)Up0(s;0)∗ϕ⟩=C⟨ϕ,Wz+εp(t-s)(k)(t0+t-s)ϕ⟩,

where the inequality follows from (5.19) with z replaced by z+εpt and t0 replaced by t0+t. This proves the bound in (5.2). □

To conclude this section, we prove Proposition 5.2. The proof is a simple adaptation of the one of Proposition 5.3.

Proof of Proposition 5.2

Let ωN,t=∑j=1N|ϕj,t⟩⟨ϕj,t|, and take 0≤t≤T∗, with T∗≤T1 to be suitably chosen. We start by writing:5.20 trWz(1)ωN,t=∑j=1N⟨ϕj,t,Wz(1)ϕj,t⟩≡∑j=1N⟨ϕj,U(t;0)∗Wz(1)U(t;0)ϕj⟩,

where U(t;0)=Up=0(t;0) is the Hartree dynamics, see (5.1). Let t0∈[0,T∗]. Consider the following quantity:⟨ϕj,UI(t;0)∗Wz(1)(t0)UI(t;0)ϕj⟩.

Recall that, by definition of the evolution in the interaction picture, Eq. (5.10),5.21 ⟨ϕj,UI(t;0)∗Wz(1)(t)UI(t;0)ϕj⟩≡⟨ϕj,U(t;0)∗Wz(1)U(t;0)ϕj⟩.

Next, by the proof of Proposition 5.3, it is not difficult to see that, for a suitable constant K>0:5.22 ⟨ϕj,UI(t;0)∗Wz(1)(t0)UI(t;0)ϕj⟩≤⟨ϕj,Wz(1)(t0)ϕj⟩+∫0tdτKT∗ε3(supz∈R3trWz(1)ωN,τ)⟨ϕj,UI(τ;0)∗Wz(1)(t0)UI(τ;0)ϕj⟩.

Definingα(t;t0):=ε3supz∈R3∑j=1N⟨ϕj,UI(t;0)∗Wz(1)(t0)UI(t;0)ϕj⟩,

Equation (5.22) implies that:5.23 α(t;t0)≤α(0;t0)+KT∗∫0tdτ(ε3supz∈R3trWz(1)ωN,τ)α(τ;t0)=α(0;t0)+KT∗∫0tdτα(τ;τ)α(τ;t0).

Let:f(t):=supt0∈[0,T∗]α(t;t0).

Notice that5.24 trWz(1)ωN,t≤f(t),

see Eqs. (5.20) and (5.21), hence our goal will be to derive an estimate for f(t). Equation (5.23) implies:5.25 f(t)≤f(0)+KT∗∫0tdτf(τ)2.

The quantity f(0) is bounded as follows:f(0)=ε3supt0∈[0,T∗]supz∈R3trWz(1)(t0)ωN≤C,

where the last bound follows from the assumption (2.9) on the initial datum, since T∗≤T1. Next, let us denote by g(t) the r.h.s. of (5.25), so that (5.25) reads f(t)≤g(t). Also,g′(t)=KT∗f(t)2≤KT∗g(t)2.

Equivalently,ddt(-1g(t)-KT∗t)≤0.

Since g(0)=f(0), we have:g(t)≤f(0)1-f(0)KT∗t≤f(0)1-f(0)KT∗2

for 0≤t≤T∗, choosing T∗ so that the denominator is positive. This bound, together with (5.24), proves the final claim with, e.g., T∗=min{12(f(0)K)-1/2,T1}. □

Proof of Theorem 4.1

In this section we shall consider initial data satisfying Assumptions 2.1, and we shall prove the stability of the local bounds in the assumptions (2.6) and (2.8) under the Hartree flow. The proof of (4.3) follows straightforwardly by application of Proposition 5.3 and of Lemma 5.1, in fact for 0≤t≤T:5.26 ‖Wz(n)ωN,t‖tr≤‖Wz(n)U(t;0)ωN‖tr≤C‖Wz(n)(t)ωN‖tr≤Cε-3,

where we used the invariance of the trace norm under unitary conjugation and (2.8).

Our strategy for proving (4.4) also relies on controlling the regularized Hartree evolution of the localization operators by their free evolution. The proof will be divided in a few steps.

Part 1: Setting up the Gronwall estimate. Our goal is to control the following quantity5.27 sups∈[t,T]supq:|q|≤ε-1supz∈R3XΛ(z)1+|q|‖Wz(n)(s-t)[eiq·x^,ωN,t]‖tr

using a Gronwall-type strategy. The claim in the theorem corresponds to the special case s=t. Let us introduce the modified Hartree dynamics Uq(t;s) as in Proposition 5.3:iε∂tUq(t;s)=(-ε2Δ+V∗ρt+iε2q·∇)Uq(t;s),Uq(s;s)=1.

Following [11, Section 5], we have:5.28 iε∂tUq(t;s)∗[eiq·x^,ωN,t]U-q(t;s)=iUq(t;s)∗{eiq·x^,εq[ε∇,ωN,t]}U-q(t;s).

Therefore, taking s=0, we get:Uq(t;0)∗[eiq·x^,ωN,t]U-q(t;0)=[eiq·x^,ωN,0]-iε∫0tdτiε∂τ(Uq(τ;0)∗[eiq·x^,ωN,τ]U-q(τ;0)),

which gives, using (5.28):5.29 [eiq·x^,ωN,t]=Uq(t;0)[eiq·x^,ωN]U-q(t;0)∗+1ε∫0tdτUq(τ;t)∗{eiq·x^,εq[ε∇,ωN,τ]}U-q(τ;t).

We shall now plug this identity into (5.27), and estimate the various terms. Consider the one due to the first term on the r.h.s. of (5.29). Using Proposition 5.3, Eq. (5.2), Lemma 5.1 and the invariance of the trace under unitary conjugation, we have:5.30 ‖Wz(n)(s-t)Uq(t;0)[eiq·x^,ωN]U-q(t;0)∗‖tr≤C‖Wz+εqt(n)(s)[eiq·x^,ωN]‖tr.

We notice that for all t≤TXΛ(z)≤C(1+ε4|q|4)XΛ(z+εqt).

Accordingly, we get:5.31 supz∈R3XΛ(z)‖Wz+εqt(n)(s)[eiq·x^,ωN]‖tr≤C(1+ε4|q|4)supz∈R3XΛ(z)‖Wz(n)(s)[eiq·x^,ωN]‖tr≤C(1+ε4|q|4)(1+|q|)ε-2,

where the last step follows from assumption (2.6), since t≤s≤T. This bound, combined with (5.30), shows that:5.32 supz∈R3XΛ(z)‖Wz(n)(s-t)Uq(t;0)[eiq·x^,ωN]U-q(t;0)∗‖tr≤C(1+|q|)ε-2.

Next, let us consider the contribution due to the second term in (5.29), namely:∫0tdτ‖Wz(n)(s-t)Uq(τ;t)∗{eiq·x^,q·[ε∇,ωN,τ]}U-q(τ;t)‖tr.

By Proposition 5.3 and Lemma 5.1, we have:5.33 ‖Wz(n)(s-t)Uq(τ;t)∗{eiq·x^,q·[ε∇,ωN,τ]}U-q(τ;t)‖tr≤C‖Wz+εq(t-τ)(n)(s-τ){eiq·x^,q·[ε∇,ωN,τ]}‖tr

The two contributions to the anticommutator are estimated in the same way. For instance, consider:‖Wz+εq(t-τ)(n)(s-τ)eiq·x^q·[ε∇,ωN,τ]‖tr≤‖Wz+εq(t-τ)-2qε(s-τ)(n)(s-τ)q·[ε∇,ωN,τ]‖tr,

where in the localization operator we used that e-iq·x^x^(s-τ)eiq·x^=x^(s-τ)+2εq(s-τ). Since for 0≤τ≤t≤s≤T5.34 XΛ(z)≤C(1+ε4|q|4)XΛ(z+εq(t-τ)-2qε(s-τ)),

we have:5.35 supz∈R3XΛ(z)‖Wz+εq(t-τ)(n)(s-τ)eiq·x^q·[ε∇,ωN,τ]‖tr≤C(1+ε4|q|4)supz∈R3XΛ(z)‖Wz(n)(s-τ)q·[ε∇,ωN,τ]‖tr.

A similar bound holds for the second contribution to the anticommutator in (5.33). Hence, combining (5.29), (5.32), (5.35), we obtain:5.36 supz∈R3XΛ(z)‖Wz(n)(s-t)[eiq·x^,ωN,t]‖tr≤C(1+|q|)ε-2+C(1+ε4|q|4)|q|∫0tdτsupz∈R3XΛ(z)‖Wz(n)(s-τ)[ε∇,ωN,τ]‖tr.

Part 2: Control of the convolution. Our last task is to estimate the argument of the integral in Eq. (5.36). As in [11], we start by writing:iε∂τU(τ;s)∗[ε∇,ωN,τ]U(τ;s)=U(τ;s)∗[ωN,τ,[V∗ρτ,ε∇]]U(τ;s),

which gives the identity:5.37 [ε∇,ωN,τ]=U(τ;0)[ε∇,ωN]U(τ;0)∗-iε∫0τdηU(η;τ)∗[ωN,η,[V∗ρη,ε∇]]U(η;τ).

We plug this identity in the integrand in (5.36). We get:5.38 ‖Wz(n)(s-τ)[ε∇,ωN,τ]‖tr≤‖Wz(n)(s-τ)U(τ;0)[ε∇,ωN]U(τ;0)∗‖tr+1ε∫0τdη‖Wz(n)(s-τ)U(η;τ)∗[ωN,η,[V∗ρη,ε∇]]U(η;τ)‖tr.

Consider the first term on the r.h.s. of (5.38). We have, by Proposition 5.3, Lemma 5.1 and the invariance of the trace under unitary conjugation:‖Wz(n)(s-τ)U(τ;0)[ε∇,ωN]U(τ;0)∗‖tr≤C‖Wz(n)(s)[ε∇,ωN]‖tr.

Hence, the contribution to the integrand in (5.36) due to the first term on the r.h.s. of (5.38) is bounded by, for t≤s≤T:supz∈R3XΛ(z)‖Wz(n)(s-τ)U(τ;0)[ε∇,ωN]U(τ;0)∗‖tr≤Csupz∈R3XΛ(z)‖Wz(n)(s)[ε∇,ωN]‖tr≤Kε-2,

where in the last step we used the assumption (2.7). Let us consider now the second term in (5.38). Again using Proposition 5.3 and Lemma 5.1:‖Wz(n)(s-τ)U(η;τ)∗[ωN,η,[V∗ρη,ε∇]]U(η;τ)‖tr≤C‖Wz(n)(s-η)[ωN,η,[V∗ρη,ε∇]]‖tr≡Cε‖Wz(n)(s-η)[ωN,η,∇V∗ρη]‖tr,

Therefore, the integrand in (5.36) is bounded as:5.39 supz∈R3XΛ(z)‖Wz(n)(s-τ)[ε∇,ωN,τ]‖tr≤Kε-2+C∫0τdηsupz∈R3XΛ(z)‖Wz(n)(s-η)[ωN,η,∇V∗ρη]‖tr.

To estimate the latter integrand, we write:5.40 ‖Wz(n)(s-η)[ωN,η,∇V∗ρη]‖tr=‖∫dyρη(y)Wz(n)(s-η)[ωN,η,∇V(x^-y)]‖tr≤‖∫dyρη(y)Wz(n)(s-η)Wy(1)[ωN,η,Fy(x^)]‖tr+‖∫dyρη(y)Wz(n)(s-η)[ωN,η,Wy(1)]Fy(x^)‖tr,

where we let Fy(x^):=(Wy(1))-1∇V(x^-y). We estimate the first term on the right-hand side as:5.41 ‖∫dyρη(y)Wz(n)(s-η)Wy(1)[ωN,η,Fy(x^)]‖tr≤∫dp|F^(p)|‖∫dyρη(y)e-ip·yWz(n)(s-η)Wy(1)[ωN,η,eip·x^]‖tr=∫dp|F^(p)|‖Wz(n)(s-η)(W(1)∗ρp,η)[ωN,η,eip·x^]‖tr≤∫dp|F^(p)|‖Wz(n)(s-η)(W(1)∗ρp,η)[Wz(n)(s-η)]-1‖op×‖Wz(n)(s-η)[ωN,η,eip·x^]‖tr≤(supp∈R3‖Wz(n)(s-η)(W(1)∗ρp,η)[Wz(n)(s-η)]-1‖op)×∫dp|F^(p)|‖Wz(n)(s-η)[ωN,η,eip·x^]‖tr

where we set ρp,η(y):=ρη(y)e-ip·y. To control the operator norm, we write5.42 ‖Wz(n)(s-η)(W(1)∗ρp,η)[Wz(n)(s-η)]-1‖op≤‖W(1)∗ρp,η‖op+‖Wz(n)(s-η)[(W(1)∗ρp,η),|x^z(s-η)|4n]‖op

where we set x^z(s-η):=x^(s-η)-z. We write the commutator by moving the powers of x^z(s-η) to the left:5.43 ‖Wz(n)(s-η)[(W(1)∗ρp,η),|x^z(s-η)|4n]‖op≤∑α:|α|=4n∑α′:|α′|≥1,α′≤ααα′‖Wz(n)(s-η)(x^z(s-η))α-α′adx^z(s-η)(α′)(W(1)∗ρp,η)‖op≤∑α:|α|=4n∑α′:|α′|≥1,α′≤ααα′‖adx^z(s-η)(α′)(W(1)∗ρp,η)‖op,

where adx^z(s-η)(α)(O) denotes the α-folded commutator of O with x^z(s-η), that is, αi commutators of O with the i-th component x^z(s-η), for i=1,2,3. We then estimate the latter operator norm as follows, for 0≤|α′|≤4n:5.44 ‖adx^z(s-η)(α′)(W(1)∗ρp,η)‖op=(2|s-η|)|α′|supz∈R3|∫dy∂zα′W(1)(z-y)ρη(y)e-ip·y|≤C‖(1+|·|4)∂α′W(1)‖∞‖W(1)∗ρη‖∞≤C,

where we used Proposition 5.2 and that W(1) and ρη are positive. All in all, putting together the bounds (5.41), (5.42), (5.43) and (5.44), we have:5.45 supz∈R3XΛ(z)‖∫dyρη(y)Wz(n)(s-η)Wy(1)[ωN,η,Fy(x^)]‖tr≤C∫dp|F^(p)|supz∈R3XΛ(z)‖Wz(n)(s-η)[ωN,η,eip·x^]‖tr.

To control this integral, we split it into small and large momenta:5.46 ∫dp|F^(p)|supz∈R3XΛ(z)‖Wz(n)(s-η)[ωN,η,eip·x^]‖tr=∫|p|≤ε-1dp|F^(p)|supz∈R3XΛ(z)‖Wz(n)(s-η)[ωN,η,eip·x^]‖tr+∫|p|>ε-1dp|F^(p)|supz∈R3XΛ(z)(‖Wz(n)(s-η)ωN,η‖tr+‖Wz(n)(s-η)eip·x^ωN,η‖tr).

The first term on the right-hand side is estimated as follows:5.47 ∫|p|≤ε-1dp|F^(p)|supz∈R3XΛ(z)‖Wz(n)(s-η)[ωN,η,eip·x^]‖tr≤‖(1+|·|)F^‖1supp:|p|≤ε-1XΛ(z)1+|p|‖Wz(n)(s-η)[ωN,η,eip·x^]‖tr≤Csupp:|p|≤ε-1XΛ(z)1+|p|‖Wz(n)(s-η)[ωN,η,eip·x^]‖tr,

where we used the assumptions on the interaction potential (2.10). To estimate the second term on the r.h.s. of (5.46), we follow the computations in (5.34) and (5.35), to write5.48 supz∈R3XΛ(z)‖Wz(n)(s-η)eip·x^ωN,η‖tr≤C(1+ε4|p|4)supz∈R3XΛ(z)‖Wz(n)(s-η)ωN,η‖tr.

Then, by (5.26) and by the assumption on the potential (2.10), we obtain:5.49 ∫|p|>ε-1dp|F^(p)|supz∈R3XΛ(z)(‖Wz(n)(s-η)ωN,η‖tr+‖Wz(n)(s-η)eip·x^ωN,η‖tr)≤C∫|p|>ε-1dpε|p|(1+ε4|p|4)|F^(p)|supz∈R3XΛ(z)‖Wz(n)(s-η)ωN,η‖tr≤Cε‖(1+|·|5)F^‖1supz∈R3XΛ(z)‖Wz(n)(s-η)ωN,η‖tr≤Cε-2.

Therefore, putting together (5.41), (5.45), (5.46), (5.47), (5.49), we have:5.50 supz∈R3XΛ(z)‖∫dyρη(y)Wz(n)(s-η)Wy(1)[ωN,η,Fy(x^)]‖tr≤Cε-2+Csupp:|p|≤ε-1XΛ(z)1+|p|‖Wz(n)(s-η)[ωN,η,eip·x^]‖tr.

To estimate the second term on the r.h.s. of (5.40), we proceed in a similar way:‖∫dyρη(y)Wz(n)(s-η)[ωN,η,Wy(1)]Fy(x^)‖tr≤∫dp|W(1)^(p)|‖∫dyρη(y)e-ip·yWz(n)(s-η)[ωN,η,eip·x^]Fy(x^)‖tr=∫dp|W(1)^(p)|‖Wz(n)(s-η)[ωN,η,eip·x^](F∗ρp,η)‖tr≤(supp∈R3‖F∗ρp,η‖op)∫dp|W^(1)(p)|‖Wz(n)(s-η)[ωN,η,eip·x^]‖tr,

then estimate the operator norm as in (5.44) and the integral as in (5.46), using that W(1)^ decays fast enough.

Part 3: Conclusion. The estimate (5.40) together with the bounds (5.41), (5.50) imply:5.51 supz∈R3XΛ(z)‖Wz(n)(s-η)[ωN,η,∇V∗ρη]‖tr≤Cε-2+Csupp:|p|≤ε-1XΛ(z)1+|p|‖Wz(n)(s-η)[ωN,η,eip·x^]‖tr.

Combining (5.39) with the estimate (5.51), we have:supz∈R3XΛ(z)‖Wz(n)(s-τ)[ε∇,ωN,τ]‖tr≤Kε-2+C∫0τdηsupp:|p|≤ε-1supz∈R3XΛ(z)1+|p|‖Wz(n)(s-η)[ωN,η,eip·x^]‖tr.

Plugging this estimate into (5.36) we find:sups∈[t,T]supq:|q|≤ε-1supz∈R3XΛ(z)1+|q|‖Wz(n)(s-t)[eiq·x^,ωN,t]‖tr≤Cε-2+C∫0tdτ∫0τdηsups∈[η,T]supp:|p|≤ε-1supz∈R3XΛ(z)1+|p|‖Wz(n)(s-η)[ωN,η,eip·x^]‖tr,

where we used that 0≤η≤τ≤t. Hence, by the Gronwall lemma we finally get, for 0≤t≤T:sups∈[t,T]supp:|p|≤ε-1supz∈R3XΛ(z)1+|p|‖Wz(n)(s-t)[eip·x^,ωN,t]‖tr≤Cε-2.

This concludes the proof of Theorem 4.1. □

Propagation of the Semiclassical Structure: Pseudo-Relativistic Case

In this section, we show how to propagate the local semiclassical structure along the flow of the pseudo-relativistic Hartree equationiε∂tωN,t=[hrel(t),ωN,t]

with hrel(t):=1-ε2Δ+V∗ρt, and ρt(x):=ε3ωN,t(x;x). With respect to the non-relativistic case, here we will be able to propagate the local semiclassical structure for all times. This allows us to prove the convergence of the many-body pseudo-relativistic dynamics to the pseudo-relativistic Hartree dynamics, Theorem 2.5, following the strategy of the non-relativistic case. The following theorem is the analogue of Theorem 4.1.

Theorem 6.1

(Propagation of the local semiclassical structure). Under the same assumptions of Theorem 2.5, we have for any t∈R6.1 supz∈R3XΛ(z)‖Wz(n)ωN,t‖tr≤Cexp(C|t|)ε-3supq:|q|≤ε-1supz∈R3XΛ(z)1+|q|‖Wz(n)[eiq·x^,ωN,t]‖tr≤Cexp(CexpC|t|)ε-2.

The main improvement with respect to the non-relativistic case is that in the pseudo-relativistic case we are able to rule out excessive concentration of the density globally in time, thanks to the boundedness of the velocity of the particles. We start by adapting the propagation estimate for the localization operators, Proposition 5.3.

Proposition 6.2

(Bounds for the evolution of the localization operator). Under the same assumptions of Theorem 2.5, consider the pseudo-relativistic Hartree evolution generator Urel(t;s) defined byiε∂tUrel(t;s)=hrel(t)Urel(t;s),Urel(t;t)=1.

Then, for all k≥1, there exists a constant C such that for all z∈R3, 0≤s≤tUrel(t;s)∗Wz(k)(x^)Urel(t;s)≤eC(t-s)Wz(k)(x^).

The reader should compare the result with Eq. (5.2). The reason why we are able to control Urel(t)∗Wz(x^)Urel(t) with Wz(x^), for all times, is the fact that the velocity operator [x^,1-ε2Δ] is bounded.

Proof

We compute the derivativeiε∂tUrel(t;s)∗Wz(k)Urel(t;s)=Urel(t;s)∗[Wz(k),hrel(t)]Urel(t;s)=Urel(t;s)∗[Wz(k),1-ε2Δ]Urel(t;s),

so that, for any ϕ∈L2(R3) we have⟨ϕ,Urel(t;s)∗Wz(k)Urel(t;s)ϕ⟩≤⟨ϕ,Wz(k)ϕ⟩+Cε‖(Wz(k/2))-1[Wz(k),1-ε2Δ](Wz(k/2))-1‖op×∫stdτ⟨ϕ,Urel(τ;s)∗Wz(k)Urel(τ;s)ϕ⟩

where we used that (Wz(k/2))2≤CWz(k). To control the operator norm, we write‖(Wz(k/2))-1[Wz(k),1-ε2Δ](Wz(k/2))-1‖op≤‖(Wz(k/2))-1Wz(k)[|x^-z|2k,1-ε2Δ]Wz(k)(Wz(k/2))-1‖op≤C‖Wz(k/2)[|x^-z|2k,1-ε2Δ]‖op≤C∑α:|α|=2k∑α′:|α′|≥1α′≤α‖Wz(k/2)(x^-z)α-α′adx^(α′)(1-ε2Δ)‖op.

Since adx^(α′)(1-ε2Δ)≤Cε|α′| for 1≤|α′|≤2k, we obtain⟨ϕ,Urel(t;s)∗Wz(k)Urel(t;s)ϕ⟩≤⟨ϕ,Wz(k)ϕ⟩+C∫stdτ⟨ϕ,Urel(τ;s)∗Wz(k)Urel(τ;s)ϕ⟩

which implies the claim by the Gronwall lemma. □

As a corollary, Proposition 6.2 immediately implies absence of excessive concentration for the density, for all times.

Corollary 6.3

(No-concentration bound). Under the same assumption of Theorem 2.5, we have:supz∈R3trWz(1)ωN,t≤CeCtε-3.

Proof

We have:trWz(1)ωN,t=trUrel(t;0)∗Wz(1)Urel(t;0)ωN≤eCttrWz(1)ωN≤CeCtε-3,

where the first inequality follows from Proposition 6.2 and from the positivity of ωN, while the last inequality follows from the assumption of Eq. (2.9). □

Proof of Theorem 6.1

The first bound is an immediate consequence of Proposition 6.2 and of the assumption (2.8) on the initial datum. Let us now prove the second inequality. By using the Jacobi identity, we write:iε∂t[eiq·x^,ωN,t]=[hrel(t),[eiq·x^,ωN,t]]+[ωN,t,[hrel(t),eiq·x^]].

Consider the second term on the right-hand side. It can be rewritten as[ωN,t,[hrel(t),eiq·x^]]=[ωN,t,[1-ε2Δ,eiq·x^]]=[ωN,t,eiq·x^(1+ε2(-i∇+q)2-1-ε2Δ)]=[ωN,t,eiq·x^]εA(q)+eiq·x^[ωN,t,εA(q)]

where we have introduced the operatorA(q):=∫01dsε(-i∇+sq)·q1+ε2(-i∇+sq)2.

Let us introduce the modified dynamicsiε∂tUrel;q(t;s)=(hrel(t)+εA(q))Urel;q(t;s),Urel;q(s;s)=1.

This allows us to writeiε∂tUrel(t;s)∗[eiq·x^,ωN,t]Urel;q(t;s)=Urel(t;s)∗eiq·x^[ωN,t,εA(q)]Urel;q(t;s).

Writing this equation in integral form we get:6.2 [eiq·x^,ωN,t]=Urel(t;0)[eiq·x^,ωN]Urel;q(t;0)∗-i∫0tdτUrel(τ;t)∗eiq·x^[ωN,τ,A(q)]Urel;q(τ;t).

We shall now plug this identity into ‖Wz(n)[eiq·x^,ωN,t]‖tr, and estimate the various terms. The first term gives the contribution,‖Wz(n)Urel(t;0)[eiq·x^,ωN]Urel;q(t;0)∗‖tr≤Ct‖Wz(n)[eiq·x^,ωN]‖tr,

where we used Proposition 6.2, Lemma 5.1 and the invariance of the trace under unitary conjugation. We then bound the term due to the integrand in (6.2) as follows:6.3 ‖Wz(n)Urel(τ;t)∗eiq·x^[ωN,τ,A(q)]Urel;q(τ;t)‖tr≤Ct-τ‖Wz(n)[ωN,τ,A(q)]‖tr≤Ct-τ∫01ds‖Wz(n)[ωN,τ,ε(-i∇+sq)·q1+ε2(-i∇+sq)2]‖tr≤Ct-τ∫01ds‖Wz(n)[ωN,τ,ε(-i∇)·q]11+ε2(-i∇+sq)2‖tr+Ct-τ∫01ds‖Wz(n)ε(-i∇+sq)·q[ωN,τ,11+ε2(-i∇+sq)2]‖tr,

where Cτ=Cexp(Cτ). Since ‖(1+ε2(-i∇+sq)2)-1/2‖op≤1, the first term on the last line of (6.3) is estimated by6.4 ∫01ds‖Wz(n)[ωN,τ,ε(-i∇)·q]11+ε2(-i∇+sq)2‖tr≤|q|‖Wz(n)[ωN,τ,ε∇]‖tr.

To bound the second term, we use the integral representation1B=1π∫0∞dλλ1B+λ,

valid for any self-adjoint B>0. Accordingly:6.5 ‖Wz(n)ε(-i∇+sq)·q[ωN,τ,11+ε2(-i∇+sq)2]‖tr≤1π∫0∞dλλ‖Wz(n)ε(-i∇+sq)·q1+ε2(-i∇+sq)2+λ×[ωN,τ,ε2(-i∇+sq)2]11+ε2(-i∇+sq)2+λ‖tr≤∑j1π∫0∞dλλ‖Wz(n)ε(-i∇+sq)·qε(-i∇+sq)j1+ε2(-i∇+sq)2+λ×[ωN,τ,ε∇j]11+ε2(-i∇+sq)2+λ‖tr+∑j1π∫0∞dλλ‖Wz(n)ε(-i∇+sq)·q1+ε2(-i∇+sq)2+λ×[ωN,τ,ε∇j]ε(-i∇+sq)j1+ε2(-i∇+sq)2+λ‖tr.

Using the following bounds, for |α|≤4n:‖adx^(α)(ε(-i∇+sq)j1+ε2(-i∇+sq)2+λ)‖op≤C(1+λ)-1+|α|2,‖adx^(α)(ε(-i∇+sq)jε(-i∇+sq)k1+ε2(-i∇+sq)2+λ)‖op≤C(1+λ)-|α|2,

we obtain6.6 ‖Wz(n)ε(-i∇+sq)j1+ε2(-i∇+sq)2+λ(Wz(n))-1‖op≤1+∑α:|α|=4n∑α′≤α‖Wz(n)(x^-z)α-α′adx^(α′)(ε(-i∇+sq)j1+ε2(-i∇+sq)2+λ)‖op≤C(1+λ)-1/2

and also6.7 ‖Wz(n)ε(-i∇+sq)jε(-i∇+sq)k1+ε2(-i∇+sq)2+λ(Wz(n))-1‖op≤C

Putting together (6.5), (6.6) and (6.7) and using ‖(1+ε2(-i∇+sq)2+λ)-1‖op≤(1+λ)-1, we obtain the estimate∫01ds‖Wz(n)ε(-i∇+sq)·q[ωN,τ,11+ε2(-i∇+sq)2]‖tr≤C|q|‖Wz(n)[ωN,τ,ε∇]‖tr,

which, combined with (6.4) implies‖Wz(n)Urel(τ;t)∗eiq·x^[ωN,τ,A(q)]Urel;q(τ;t)‖tr≤Ct-τ|q|‖Wz(n)[ωN,τ,ε∇]‖tr.

All in all, we have thus proven that6.8 supz∈R3XΛ(z)‖Wz(n)[eiq·x^,ωN,t]‖tr≤Ctε-2+|q|∫0tdτCt-τsupz∈R3XΛ(z)‖Wz(n)[ωN,τ,ε∇]‖tr.

Proceeding as in the proof of Theorem 4.1, we find6.9 ‖Wz(n)[ωN,τ,ε∇]‖tr≤Cτ‖Wz(n)[ωN,ε∇]‖tr+∫0τdηCτ-η‖Wz(n)[ωN,η,∇V∗ρη]‖tr,

where we used Proposition 6.2, Lemma 5.1 and the invariance of the trace under unitary conjugation. With respect to the non-relativistic case, notice the simplification introduced by the fact that the localization operator is not time-evolved. Letting Fy(x^) be as below (5.40), we have:6.10 ‖Wz(n)[ωN,η,∇V∗ρη]‖tr≤‖∫dyρη(y)Wz(n)Wy(1)[ωN,η,Fy(x^)]‖tr+‖∫dyρη(y)Wz(n)[ωN,η,Wy(1)]Fy(x^)‖tr≤supp∈R3(‖W(1)∗ρη,p‖op+‖F∗ρη,p‖op)∫dp(|F^(p)|+|W(1)^(p)|)‖Wz(n)[ωN,η,eip·x^]‖tr,

compare with (5.40) and (5.41). By Corollary 6.3 we have that ‖W(1)∗ρη,p‖op,‖F∗ρη,p‖op≤Cη uniformly in p, whereas the integral on the last line in (6.10) is controlled by splitting in |p|≤ε-1 and |p|>ε-1 as was done in (5.46). Accordingly, by using the assumptions on the potential we obtain6.11 supz∈R3XΛ(z)‖Wz(n)[ωN,η,∇V∗ρη]‖tr≤Cηε-2+Cηsupp:|p|≤ε-1XΛ(z)1+|p|‖Wz(n)[ωN,η,eip·x^]‖tr.

Putting together the bounds (6.8), (6.9) and (6.11), we finally get:supq:|q|≤ε-1XΛ(z)1+|q|‖Wz(n)[ωN,η,eiq·x^]‖tr≤Ctε-2+Ct∫0tdτ∫0τdηsupp:|p|≤ε-1XΛ(z)1+|p|‖Wz(n)[ωN,η,eip·x^]‖tr.

The final claim, Eq. (6.1), follows by the application of the Gronwall lemma. □

Appendix A: Check of Assumption 2.1

Here we shall discuss examples of fermionic states satisfying Assumptions 2.1. Specifically, we shall consider the case of the free Fermi gas on R3 at positive density, and of coherent states. We expect these assumptions to hold true for a wide class of fermionic equilibrium states.

Appendix A.1: The free Fermi gas

Here we prove the assumptions for the free Fermi on R3, at positive density. Instead of carrying out the computations in a large but finite periodic box Λ⊂R3, we work directly in infinite volume, for the sake of simplicity. Let ω be the operator on L2(R3) with integral kernel:ω(x;y)=∫R3dq(2π)31|q|≤ε-1eiq·(x-y),

This operator describes the homogeneous free Fermi gas in R3 at density ω(x;x)=(1/6π2)ε-3. Clearly, ω=ω2=ω∗, and trω=∞.

Check of (2.6). Being the state defined in Λ=R3, here we shall replace XΛ(z) with 1. Simple computations show that the kernel of the operator [eip·x^,ω] is given by[eip·x^,ω](x,y)=eip·x+y2∫R3dq(2π)3(1|q-p/2|≤ε-1-1|q+p/2|≤ε-1)eiq·(x-y).

Furthermore, |[eip·x^,ω]|2=|[eip·x^,ω]| and|[eip·x^,ω]|2(x;y)=∫R3dq(2π)31Speiq·(x-y),

with Sp the following set:Sp:={q∈R3||q-p|≤ε-1}⊖{q∈R3||q|≤ε-1}

where ⊖ denotes the symmetric difference. Clearly,|Sp|≤C|p|ε-2.

Let Fp(q) be a C∞ smoothing of the characteristic function χSp(q), such that:A.1 χSp(q)≤Fp(q),Fp(q)↾Sp=1,Fp(q)=0ifdist(q,Sp)≥1.

One can check that the following holds:A.2 ‖Fp‖1≤lC(1+|p|)ε-2,‖DkFp‖1≤Ck(1+|p|)ε-2,∀k>0.

Let Op be the operator with integral kernel:Op(x;y)=∫R3dq(2π)3Fp(q)eiq·(x-y).

Then, |[eip·x^,ω]|2≤Op. In particular, by Lemma 5.1A.3 ‖Wz(n)(t)[eip·x^,ω]‖tr=‖Wz(n)(t)|[eip·x^,ω]|2‖tr≤‖Wz(n)(t)Op‖tr.

Since Op is invariant under free time evolution we then have, by the invariance under conjugation with unitary transformations:‖Wz(n)(t)Op‖tr=‖Wz(n)Op‖tr.

To bound the right-hand side, it is enough to consider n=1. From the inequality Wz(1)≤C(1+|x^-z|2)-2, we have by Lemma 5.1A.4 ‖Wz(1)Op‖tr≤C‖1(1+|x^-z|2)2Op‖tr≤C‖11+|x^-z|2Op11+|x^-z|2‖tr+C‖11+|x^-z|2[11+|x^-z|2,Op]‖tr≡I+II.

Consider the first term. Let:gz,p(x)=eip·x(1+|x-z|2)2.

Clearly, gz,p∈L2(R3). Moreover, by the first bound in (A.2):A.5 |I|≤C∫dqF(q)‖|gz,p⟩⟨gz,p|‖tr≤K(1+|p|)ε-2.

Consider now the term II. We have:[11+|x^-z|2,Op]=-11+|x^-z|2[|x^-z|2,Op]11+|x^-z|2=-∑i=1311+|x^-z|2[(x^i-zi)2,Op]11+|x^-z|2.

We then have:‖11+|x^-z|2[11+|x^-z|2,Op]‖tr≤∑i=13‖1(1+|x^-z|2)2[(x^i-zi)2,Op]11+|x^-z|2‖tr≤∑i=13‖(x^i-zi)(1+|x^-z|2)2[x^i,Op]11+|x^-z|2‖tr+∑i=13‖1(1+|x^-z|2)2[x^i,Op](x^i-zi)1+|x^-z|2‖tr≡II1+II2.

Consider II1. We have:|II1|≤C∑i=13‖11+|x^-z|2[x^i,Op]11+|x^-z|2‖tr.

Using that:[x^i,Op]=i∫R3dq(2π)3∂qiFp(q)|eiq·x⟩⟨eiq·x|,

and recalling the second bound in (A.2), we have:A.6 |II1|≤Csupi=1,2,3∫dq|∂qiFp(q)|≤K(1+|p|)ε-2.

Consider now the term II2. We have:∑i=13‖1(1+|x^-z|2)2[x^i,Op](x^i-zi)1+|x^-z|2‖tr≤∑i=13‖11+|x^-z|2[x^i,Op]11+|x^-z|2(x^i-zi)1+|x^-z|2‖tr+∑i=13‖11+|x^-z|2[11+|x^-z|2,[x^i,Op]](x^i-zi)1+|x^-z|2]‖tr≡II2;1+II2;2.

The first term is bounded as II1:A.7 |II2;1|≤∑i=13‖11+|x^-z|2[x^i,Op]11+|x^-z|2‖tr≤C(1+|p|)ε-2.

Finally, consider II2;2. Writing:-[11+|x^-z|2,[x^i,Op]]=11+|x^-z|2[|x^-z|2,[x^i,Op]]11+|x^-z|2=∑i=13(x^i-zi)1+|x^-z|2[x^i,[x^i,Op]]11+|x^-z|2+∑i=1311+|x^-z|2[x^i,[x^i,Op]](x^i-zi)1+|x^-z|2,

it is clear that II2;2 can be estimated in terms of a sum of terms bounded by:A.8 ‖11+|x^-z|2[x^i,[x^i,Op]]11+|x^-z|2‖tr≤C∫dq|∂qi2Fp(q)|≤K(1+|p|)ε-2,

where the last inequality follows from (A.2). Hence, (A.6), (A.7), (A.8) imply:|II|≤C(1+|p|)ε-2.

Combined with (A.5) and with (A.4), we have:‖Wz(1)Op‖tr≤C(1+|p|)ε-2.

Recalling (A.3), this concludes the check of the assumption (2.6) for the free Fermi gas.

Check of (2.7). This assumption is trivially true for the free Fermi gas, since [ω,∇]=0.

Check of (2.8). By stationarity of the free Fermi gas:A.9 ‖ωWz(n)(t)‖tr=‖ωWz(n)‖tr≤Cε-3,

where the last bound is proven as we did with the assumption (2.6), replacing [eip·x^,ω] with ω. Finally, since we allow for the value n=1 in the localizer, assumption (2.9) in Proposition 5.2 immediately follows.

Appendix A.2: Coherent states

Let ρ∈L1(R3)∩L∞(R3), ρ(r)≥0, such thatA.10 ∫R3drρ(r)=N,|ρ(r)|≤Cε-3,XΛ(r)ρ(r)2/3≤Cε-2,|∇rρ(r)1/3|≤Cε-1,

where recall that XΛ(r):=1+dist(r,Λ)4. The function ρ(r) plays the role of density for the fermionic state that we are going to introduce. The second inequality in (A.10) introduces a form of localization of ρ(r) in the domain Λ, while the last one allows us to bound derivatives of the local Fermi momentum, to be defined below. Here we shall consider coherent states, corresponding to the following reduced one-particle density matrix:A.11 ωN=1(2π)3∫R3×R3dqdrM(q,r)πq,r,πq,r=|fq,r⟩⟨fq,r|,

with fq,r(x)=eiq·xg(x-r), where the function g is even, ‖g‖2=1, smooth and fast decaying; for definiteness, we choose g(x)=1(2πδ2)3/4e-|x|22δ2 with δ>0 to be chosen later. We also set:M(q,r)=1|q|≤kF(r),

where we choose the local Fermi momentum kF(r)=κρ(r)1/3, with κ=(6π2)1/3. With this choice:trωN=1(2π)3∫dqdrM(q,r)=∫drρ(r)=N.

Closeness to a projection. The state ωN is not an orthogonal projection. However, it can be viewed as an approximate projection, in the following sense. Consider the quantity trωN(1-ωN). Since 0≤ωN≤1, it satisfies the trivial bound trωN(1-ωN)≤N. We claim that, for ωN given by (A.11), for δ=ε,A.12 trωN(1-ωN)≤CεN.

To prove this estimate, we proceed as follows. We write:A.13 tr(ωN-ωN2)=1(2π)6∫dqdq′drdr′M(q,r)(M(q,r)-M(q′,r′))|⟨fq,r,fq′,r′⟩|2

where we used the completeness of coherent states, and the fact that M(q,r)=M(q,r)2. Next, notice that:A.14 M(q,r)(M(q,r)-M(q′,r′))=χ(|q|≤kF(r))(χ(|q|≤kF(r))-χ(|q′|≤kF(r′))),

which implies:A.15 ∫dqdq′drdr′M(q,r)(M(q,r)-M(q′,r′))|⟨fq,r,fq′,r′⟩|2=∫dqdq′drdr′χ(|q|≤kF(r))χ(|q′|>kF(r′))|⟨fq,r,fq′,r′⟩|2.

We compute:A.16 |⟨fq,r,fq′,r′⟩|2=e-(r-r′)2/2δ2-(q-q′)2δ2/2,

and we consider the integral:A.17 ∫dqdq′χ(|q|≤kF(r))χ(|q′|>kF(r′))e-(q-q′)2δ2/2.

By the regularity properties of the Fermi momentum (A.10), we have:A.18 kF(r′)=kF(r)+kF(r′)-kF(r)≥kF(r)-Cε-1|r-r′|.

Therefore, the expression in (A.17) is bounded above by:A.19 ∫dqdq′χ(|q|≤kF(r))χ(|q′|>kF(r)-Cε-1|r-r′|)e-(q-q′)2δ2/2,

which we further decompose as:A.20 ∫dqdq′χ(|q|≤kF(r))χ(kF(r)+δ-1>|q′|>kF(r)-Cε-1|r-r′|)e-(q-q′)2δ2/2+∫dqdq′χ(|q|≤kF(r))χ(kF(r)+δ-1≤|q′|)e-(q-q′)2δ2/2.

The second term is easily estimated as:A.21 ∫dqdq′χ(|q|≤kF(r))χ(kF(r)+δ-1≤|q′|)e-(q-q′)2δ2/2≤CkF(r)3,

which contributes to trωN(1-ωN) with a term bounded by:A.22 C∫drdr′kF(r)3e-(r-r′)2/2δ2≤CNδ3.

Consider now the first term in (A.20). We estimate it as:A.23 ∫dqdq′χ(|q|≤kF(r))χ(kF(r)+δ-1>|q′|>kF(r)-Cε-1|r-r′|)e-(q-q′)2δ2/2≤Cδ-3kF(r)2(δ-1+Cε-1|r-r′|);

this contributes to trωN(1-ωN) with a term bounded by:A.24 C∫drdr′δ-3kF(r)2(δ-1+Cε-1|r-r′|)e-(r-r′)2/2δ2≤K∫drkF(r)2(δ-1+Cε-1δ)=K∫dr1XΛ(r)XΛ(r)kF(r)2(δ-1+Cε-1δ)≤C|Λ|ε-2(δ-1+Cε-1δ),

where we used the assumptions (A.10). Putting everything together, and choosing δ=ε we find:A.25 trωN(1-ωN)≤CNε

as claimed.

Check of (2.6). We write:[eip·x^,ωN]=1(2π)3∫dqdr[M(q-p/2,r)-M(q+p/2,r)]|fq+p/2,r⟩⟨fq-p/2,r|,

and notice that|M(q-p/2,r)-M(q+p/2,r)|=1Sp(r)(q)

the set Sp(r) being the symmetric difference of two Fermi balls of radius kF(r), shifted by p, i.e.,A.26 Sp(r):={q∈R3||q-p/2|≤κρ(r)1/3}⊖{q∈R3||q+p/2|≤κρ(r)1/3}

with measureA.27 |Sp(r)|≤C|p|ρ(r)2/3.

We compute:A.28 ‖Wz(n)(t)[eip·x^,ωN]‖tr≤1(2π)3∫dqdr1Sp(r)(q)‖Wz(n)(t)|fq+p/2,r⟩⟨fq-p/2,r|‖tr=1(2π)3∫dqdr1Sp(r)(q)‖Wz(n)(t)fq+p/2,r‖2=1(2π)3∫dqdr1Sp(r)(q)‖Wz(n)eiεΔtfq+p/2,r‖2=1(2π)3∫dqdr1Sp(r)(q)1+|z-r|4n×‖(1+|z-r|4n)Wz(n)eiεΔtfq+p/2,r‖2.

We estimate:A.29 ‖(1+|z-r|4n)Wz(n)eiεΔtfq+p/2,r‖2≤C‖(1+|z-x^|4n)Wz(n)eiεΔtfq+p/2,r‖2+C‖(1+|x^-r|4n)Wz(n)eiεΔtfq+p/2,r‖2≤C+C‖(1+|x^-r|4n)Wz(n)eiεΔtfq+p/2,r‖2.

We estimate the second term as, using the unitarity of the free dynamics:A.30 ‖(1+|x^-r|4n)Wz(n)eiεΔtfq+p/2,r‖2≤‖(1+|x^(t)-r|4n)fq+p/2,r‖2≤‖(1+|x^(t)-r-2ε(q+p/2)t|4n)g(·-r)‖2≤C(1+ε4n|q+p/2|4nt4n+(εδ-1t)4n).

The last inequality follows from the smoothness of g, and from its fast decay at infinity. Therefore, going back to (A.28), for ε|p|≤1, using that εδ-1≤C:A.31 ‖Wz(n)(t)[eip·x^,ωN]‖tr≤∫dqdr1Sp(r)(q)1+|z-r|4nC(1+ε4n|q+p/2|4nt4n)≤C(1+t4n)∫dqdr1Sp(r)(q)1+|z-r|4n,

where in the last step we used that, by the compact support properties of the integral and by (A.10) and (A.26), 1Sp(r)(q)=0, if |q|≥Cε-1. Recalling the bound (A.27):‖Wz(n)(t)[eip·x^,ωN]‖tr≤C(1+t4n)∫dr|p|ρ(r)2/31+|z-r|4n;

hence,supp:|p|≤ε-1supz∈R3XΛ(z)1+|p|‖Wz(n)(t)[eip·x^,ωN]‖tr≤K(1+t4n)supz∈R3XΛ(z)∫drρ(r)2/31+|z-r|4n.

To estimate the supremum, we proceed as follows, using that by the triangle inequality XΛ(z)≤CXΛ(r)(1+|z-r|4):XΛ(z)∫drρ(r)2/31+|z-r|4n=∫drXΛ(z)XΛ(r)XΛ(r)ρ(r)2/31+|z-r|4n≤∫drXΛ(r)ρ(r)2/31+|z-r|4(n-1)≤Cε-2,

where the last bound follows from the last assumption in (A.10). This concludes the check of (2.6) for n≥2.

Check of (2.7). We start by writing:[∇,ωN](x;y)=(∇x+∇y)ωN(x;y)=1(2π)3∫R3×R3dqdrM(q,r)eiq(x-y)(∇xg(x-r)g(y-r)¯+g(x-r)∇yg(y-r)¯)≡-1(2π)3∫R3×R3dqdrM(q,r)eiq(x-y)∇rg(x-r)g(y-r)¯.

Integrating by parts, we get:A.32 [∇,ωN](x;y)=1(2π)3∫R3×R3dqdrδ(|q|-kF(r))(∇rkF(r))eiq(x-y)g(x-r)g(y-r)¯.

We then have, proceeding as in (A.28)–(A.31):‖Wz(n)(t)[ε∇,ωN]‖tr≤Cε∫|q|=kF(r)dqdr|∇rkF(r)|1+|z-r|4n‖(1+|z-r|4n)Wz(n)eiεΔteiq·x^g(·-r)‖2≤Cε(1+t4n)∫drkF(r)2|∇rkF(r)|1+|z-r|4n.

Hence:supz∈R3XΛ(z)‖Wz(n)(t)[ε∇,ωN]‖tr≤Cε(1+t4n)supz∈R3XΛ(z)∫drkF(r)2|∇rkF(r)|1+|z-r|4n=Cε(1+t4n)supz∈R3∫drXΛ(z)XΛ(r)XΛ(r)kF(r)2|∇rkF(r)|1+|z-r|4n≤Cε(1+t4n)supz∈R3∫drXΛ(r)kF(r)2|∇rkF(r)|1+|z-r|4(n-1)≤K(1+t4n)ε-2,

where the last step follows from the assumptions in (A.10), recalling that kF(r)=κρ(r)1/3. This concludes the check of (2.7) for n≥2.

Check of (2.8). To begin, we estimate:‖Wz(n)(t)ωN‖tr≤1(2π)3∫dqdrM(q,r)‖Wz(n)(t)fq,r‖2≤1(2π)3∫dqdrM(q,r)‖Wz(n)eiεΔtfq,r‖2=1(2π)3∫dqdrM(q,r)1+|z-r|4n‖(1+|z-r|4n)Wz(n)eiεΔtfq,r‖2,

where in the second last step we used the unitarity of the free dynamics. Next, following (A.29) and (A.30), we have:‖(1+|z-r|4n)Wz(n)eiεΔtfq,r‖2≤C(1+ε4n|q|4nt4n+(εδ-1t)4n).

To conclude, using that the integral is supported on |q|≤kF(r)≤Cε-1:supz∈R3XΛ(z)‖Wz(n)(t)ωN‖tr≤Csupz∈R3XΛ(z)∫dqdrM(q,r)(1+ε4n|q|4nt4n+(εδ-1t)4n)1+|z-r|4n≤C(1+t4n)supz∈R3∫drXΛ(z)XΛ(r)XΛ(r)ρ(r)1+|z-r|4n≤C(1+t4n)supz∈R3∫drXΛ(r)ρ(r)1+|z-r|4(n-1)≤C(1+t4n)ε-3,

where the last step follows from the assumptions (A.10). This concludes the check of (2.8), and establishes the validity of the local semiclassical structure for coherent states for n≥2.

Check of (2.9). To conclude, we check the validity of the assumption (2.9). This follows from the previous computations, in fact:supz∈R3trWz(1)(t)ωN≤C(1+t4)supz∈R3∫drρ(r)1+|z-r|4≤C(1+t4)ε-3.

Appendix B: Comparison of Hartree and Hartree–Fock Dynamics

In this section we prove that the solutions of the Hartree and the Hartree–Fock dynamics are close. Specifically, we show that the distance between the evolutions under the two dynamics of initial data enjoying the local semiclassical structure is much smaller than the estimate for the distance between the many-body and Hartree evolution, stated in Theorem 2.3. In the mean-field setting, a similar result has been proved in e.g. [11, Appendix A]. We will prove the statement in the non-relativistic setting, for short times. The analogous result in the pseudo-relativistic case can be proved in the same way, for all times.

Proposition B.1

(Comparison of Hartree and Hartree–Fock dynamics). Under the same assumptions of Theorem 2.3, the following is true. Let ωN,t, ω~N,t be the solutions of the Hartree and Hartree–Fock equations, respectively, with initial datum ωN. Then, for t∈[0,T], with T>0 as in Theorem 2.3:B.1 ‖ωN,t-ω~N,t‖tr≤CNε2.

Remark B.2

This bound is smaller than the trace-norm estimate in (2.13). Also, using that ωN,t≤1, ω~N,t≤1, the estimate (B.1) implies‖ωN,t-ω~N,t‖HS≤CN12ε,

which is smaller than the Hilbert–Schmidt estimate in (2.13).

Proof

Let ω~N,t be the solution of the time-dependent Hartree–Fock equation:B.2 iε∂tω~N,t=[-ε2Δ+ρ~t∗V-Xt,ω~N,t],ω~N,0=ωN,

with ρ~t(x)=ε3ω~N,t(x;x) and Xt(x;y)=ε3V(x-y)ω~N,t(x;y). Let U~(t;s) be the unitary operator generating the Hartree–Fock dynamics:iε∂tU~(t;s)=(-ε2Δ+ρ~t∗V-Xt)U~(t;s),U~(s;s)=1,

which allows us to rewrite the solution of (B.2) as ω~N,t=U~(t;0)ωNU~(t;0)∗. Let ωN,t be the solution of the time-dependent Hartree equation, with initial datum ωN. By unitarity,B.3 ‖ωN,t-ω~N,t‖tr=‖U~(t;0)∗ωN,tU~(t;0)-ωN‖tr.

Next, we have:U~(t;0)∗ωN,tU~(t;0)-ωN=1iε∫0tdsiε∂sU~(s;0)∗ωN,sU~(s;0)=1iε∫0tdsU~(s;0)∗[V∗(ρs-ρ~s)+Xs,ωN,s]U~(s;0).

Taking the trace norm, we have:B.4 ‖U~(t;0)∗ωN,tU~(t;0)-ωN‖tr≤1ε∫0tds‖[V∗(ρs-ρ~s),ωN,s]‖tr+1ε∫0tds‖[Xs,ωN,s]‖tr≡I+II.

Consider the term II. We have, using that Xt=ε3∫dpV^(p)eip·x^ω~N,te-ip·x^:B.5 II≤ε3ε∫0tds∫dp|V^(p)|‖[eip·x^ω~N,se-ip·x^,ωN,s]‖tr≤CtNε2,

where we used ‖[eip·x^ω~N,se-ip·x^,ωN,s]‖tr≤2‖ωN,s‖tr=2N. Consider now the term I. We have:B.6 I=1ε∫0tds‖[V∗(ρs-ρ~s),ωN,s]‖tr=1ε∫0tds‖∫dy(ρs(y)-ρ~s(y))[V(x^-y),ωN,s]‖tr.

We estimate the right-hand side as:B.7 I≤1ε∫0tds∫dy|ρs(y)-ρ~s(y)|‖[V(x^-y),ωN,s]‖tr≤1ε∫0tds‖ρs-ρ~s‖1supy∈R3‖[V(x^-y),ωN,s]‖tr.

The L1 norm can be estimated as, by duality:B.8 ‖ρs-ρ~s‖1=ε3∫dyJ(y)(ωN,s(y;y)-ω~N,s(y;y))=ε3trJ(x^)(ωN,s-ω~N,s)≤ε3‖ωN,s-ω~N,s‖tr,

where we have introduced the function J(y)=sign(ωN,s(y;y)-ω~N,s(y;y)). Next, the trace norm of the commutator in (B.7) can be estimated using Corollary 4.3. We get, for all |s|≤T:B.9 supy∈R3‖[V(x^-y),ωN,s]‖tr≤Cε-2.

Thus, the estimates (B.7), (B.8), (B.9) imply:B.10 I≤C∫0tds‖ωN,s-ω~N,s‖tr.

All in all, from Eqs. (B.3), (B.4), (B.5), (B.10) we obtain, for 0≤t≤T:‖ωN,t-ω~N,t‖tr≤C∫0tds‖ωN,s-ω~N,s‖tr+CtNε2.

The final claim, Eq. (B.1), follows from Gronwall’s lemma. □

Acknowledgements

We thank Alessandro Giuliani for suggesting the relation of our result with the Kac limit. The work of L.F. has been supported by the European Research Council through the ERC-AdG CLaQS and by the Swiss National Science Foundation grant number 200160. The work of M.P. has been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program ERC StG MaMBoQ, n.80290. The work of M.P. has been carried out under the auspices of the GNFM of INdAM. B.S. acknowledges partial support from the NCCR SwissMAP, from the Swiss National Science Foundation through the Grant “Dynamical and energetic properties of Bose-Einstein condensates” and from the European Research Council through the ERC-AdG CLaQS.

Funding Information

Open Access funding enabled and organized by Projekt DEAL.

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Declarations

Conflict of interest

All authors declare that they have no conflicts of interest to disclose.

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