For arbitrarily large times *T* > 0, we prove the uniform-in-*ℏ* propagation of semiclassical regularity for the solutions to the Hartree–Fock equation with singular interactions of the form $V(x)=\xb1x\u2212a$ with $a\u2208(0,12)$. As a by-product of this result, we extend to arbitrarily long times the derivation of the Hartree–Fock and the Vlasov equations from the many-body dynamics provided in the work of Chong *et al.* [arXiv:2103.10946 (2021)].

## I. INTRODUCTION AND MAIN RESULT

Consider the time-dependent Hartree–Fock equation

describing the evolution of a positive self-adjoint trace class operator ** ρ** =

**(**

*ρ**t*) acting on $L2(R3)$. Here, $\u210f=h2\pi $ is the reduced Planck constant, $\u22c5,\u22c5$ denotes the commutator $A,B=AB\u2212BA$, and

*H*

_{ρ}is the Hamiltonian operator given by

where *V*_{ρ} is the mean-field potential and $X\rho $ is the exchange operator. The mean-field potential is defined as the multiplication operator by the function *V*_{ρ}(*x*) = (*K*^{*}*ρ*) (*x*), where $K:R3\u2192R$ is the potential associated with some two-body interaction and *ρ*(*x*) is the spatial density, defined as the rescaled diagonal of the integral kernel ** ρ**(·, ·) of the operator

**, given by**

*ρ*The exchange operator is defined to be the operator with kernel

In this work, we normalize ** ρ** so that

where $\u22c5\u221e$ is the operator norm. These quantities are preserved by Eq. (1). Furthermore, we assume that the constant in (3) does not depend on *ℏ*. With this scaling, we see that *ρ* satisfies $\u222bR3\rho (x)\u2146x=h3Tr(\rho )=1$. In the absence of $X\rho $, we refer to Eq. (1) as the Hartree equation. All the results presented in this work hold for both the Hartree and the Hartree–Fock equation.

In the case when *ℏ* is fixed, say, *ℏ* = 1, the well-posedness theories for the Hartree and the Hartree–Fock equations are well known. For the case of the Hartree equation with the Coulomb potential, one can find the proof of the global-in-time well-posedness in *L*^{2} and the propagation of higher *H*^{s} regularity for the wave function in Ref. 1, which builds on the earlier works.^{2–4} The case of density operators in Schatten spaces but with an infinite trace was studied in Ref. 5. In the case of the Hartree–Fock equation, well-posedness in *H*^{2} was proved in Ref. 6 for bounded interactions and, then, in Refs. 7 and 8 for more singular potentials, including the case of the Coulomb potential. However, these works do not provide satisfactory estimates when *ℏ* is small and tending toward zero. Obtaining uniform-in-*ℏ* estimates is crucial not only for understanding, of course, the errors in the semiclassical limit *ℏ* → 0, as in Refs. 9–14, but also to create adapted numerical schemes^{15} and to understand the joint mean-field and semiclassical limit.^{16–19}

In this paper, we are interested in proving the global-in-time propagation of regularity uniformly in the semiclassical parameter *ℏ* for solutions to the Hartree–Fock equation (1) when the interaction potential *K* is the inverse power law potential,

In particular, $K\u2208Lb,\u221e(R3)$, where $b=3a+1$ and $Lb,\u221e(R3)$ denotes the weak $Lb$ space on $R3$. Our main motivation is to extend the results of the local-in-time regularity obtained in our previous paper^{19} to global-in-time results, leading to the global-in-time mean-field and semiclassical limits for fermions from the *N* body Schrödinger equation to the Hartree–Fock and Vlasov equation.

Before stating our main result, we introduce the function spaces that we will be working with. First, we define the semiclassical phase space Lebesgue norms by

for *p* ∈ [1, *∞*], with the obvious modification for *p* = *∞*. Here, $\rho p$ denotes the Schatten norm of order *p* and $A=A*A$ denotes the absolute value of the operator *A* with adjoint *A*^{*}. Let ** p** = −

*iℏ*∇ be the momentum operator and

The moment of order *n* and the weighted semiclassical Lebesgue norms with the operator weight $mn$ are given by

In order to consider the quantum analog of Sobolev norms, we introduce the following operators:

Then, the semiclassical homogeneous Sobolev norms are defined by

with the corresponding inhomogeneous Sobolev norms given by $\rho W1,p:=\rho Lp+\rho W\u03071,p$ and the weighted semiclassical Sobolev norms with the operator weight $mn$ given by

Our main result states the global-in-time propagation of the regularity in terms of these norms.

*Let*$a\u2208(0,12)$

*,*$n\u22082N$

*be an even integer, and*

*ρ**be a solution to the Hartree–Fock equation (*

*1*

*) with initial datum*$\rho in\u2208L\u221e(mn)$

*satisfying (*

*3*

*) such that*

*for*$q\u22082,\u221e$

*and with moments of order strictly larger than*$31\u2212a(n+a+1)$

*bounded uniformly in*

*ℏ*

*. Then,*

*uniformly in*

*ℏ*∈ (0, 1)

*.*

Note that Theorem 1.1 extends to arbitrarily long times the local-in-time theory studied in Ref. 19, Theorem 3.1 for $a\u2208(0,12)$. As a corollary, Theorem 1.1 entails the global-in-time derivation of the Hartree–Fock and the Vlasov equations from the many-body Schrödinger equation in the mean-field regime for mixed states, thus extending from local to global-in-time results (Theorems 3.2 and 3.3) in Ref. 19. The crucial regularity conditions needed to perform the joint mean-field and semiclassical limit in Ref. 19 when $a\u2208(0,12)$ are, indeed, of the form (5) with $q>61+2a$ (take *p* = 2 and *q* = *q*_{1} in the beginning of Ref. 19, Sec. 10.1).

As in Refs. 10 and 13, the key ingredient to getting long-time estimates in this work is the usage of quantum moments, used to bound the semiclassical weighted Lebesgue norms. In this paper, we prove the global-in-time propagation of quantum moments for the solution of the Hartree–Fock equation (1) when $a\u22080,45$ and show that if $a\u2208(0,12)$, then the global-in-time bound on the moments, combined with a Grönwall-type argument, proves the uniform-in-*ℏ* propagation of regularity in weighted semiclassical Sobolev spaces.

This paper is structured as follows: in Sec. II, we recall the result obtained in Ref. 10 about the global-in-time propagation of quantum moments for the Hartree equation when $a\u2208(0,45]$ and extend it to solutions of the Hartree–Fock equation (1), while Sec. III is devoted to the Proof of Theorem 1.1.

## II. PROPAGATION OF MOMENTS

*Let*$a\u2208(0,45]$

*,*$n\u22082N$,

*and*

*ρ**be a solution of the Hartree–Fock equation with initial condition*$\rho in\u2208L1\u2229L\u221e$

*with moments of order*

*n*

*bounded uniformly in*

*ℏ*

*. Then, there exists a continuous function*$\Phi n\u2208C0(R+)$

*independent of*

*ℏ*

*such that for any*$t\u2208R+$

*,*

*If* $a\u2208(45,2)$*, then we can still get a short-time estimate when* $a\u2264an=2nn+3$ *(see Remark 2.3). In particular,* *a*_{n} *is larger than* 1* as soon as* *n* ≥ 3*. More generally, for any* *a* ∈ (0, 2)*, the propagation of moment of order* *n* *holds for any even* $n\u2265na=3a2\u2212a$*.*

*The proof of the theorem can be used to get an explicit function* Φ_{n}(*t*)*. It has a polynomial growth in time when* $a<45$ *and an exponential growth in time when* $a=45$*.*

### A. The Hartree equation

In this section, we recall the main ingredients of the proof of the propagation of moments for the Hartree equation obtained in Ref. 10, Theorem 3. This provides us a guide for the extension to the case of the Hartree–Fock equation addressed in Theorem 2.1.

Recall that for any density operator ** ρ** (i.e., any positive trace class operator with trace one), there exists $J\u2282N$, a sequence of functions $(\psi j)j\u2208J$ orthonormal in $L2(R3)$, and a positive summable sequence $(\lambda j)j\u2208J$ such that

**can be written as**

*ρ*For any even integer $n\u22082N$, we define the moment density of order *n* by

so that the moment of order *n*, previously defined, can be rewritten as $Mn=\rho nL1$. Note that we also have $Mn=\rho pn2L22$. With this notation, inequality (38) in Ref. 10 reads

where

and *α*′, *β*′, *γ*′ are the Hölder conjugates of *α*, *β*, *γ*, respectively. The semiclassical kinetic inequality (Ref. 10, Theorem 6), which is a generalization of the Lieb–Thirring inequality for the *n*th order moment density, tells us that for any $(k,n)\u2208(2N)2$ verifying *k* ∈ [0, *n*], we have that

Combining this inequality with inequality (7) implies the existence of some positive constants Θ, Θ_{0}, and Θ_{2} such that the following estimate [see Ref. 10, inequality (44)] holds:

where Θ ≤ 1 when $a\u226445$. This leads to the boundedness of moments by a Grönwall-type argument, together with the uniform boundedness of *M*_{2} due to the conservation of energy and an induction on *n*. Moreover, when $a<45$, then Θ < 1, thus proving a polynomial growth in time in this latter case.

### B. The Hartree–Fock equation

We now consider the Hartree–Fock equation. By estimating the exchange (operator) term with a similar strategy, we get the analog of Ref. 10, Theorem 3. The analog of inequality (7) for the exchange term is given by the following lemma.

*Denote the components of*

*p**by*$pxl$

*or simply*

*p*_{l}

*for*

*l*∈ {1, 2, 3}

*. Suppose that*$n\u2208N$,

*and define*$Q\u2282[12,\u221e]4$,

*the set of*(

*q*

_{1},

*q*

_{2},

*q*

_{3},

*q*

_{4})

*verifying*$1q1+1q2\u2208(0,2)$

*,*$1q3+1q4\u2208(0,2)$,

*and*

*Then, there exists a constant*

*C*

*independent of*

*ℏ*

*such that*

*where the supremum is taken over all the integers*$(k1,\u2026,k4)\u2208N4$

*such that*

*k*

_{1}+ ⋯ +

*k*

_{4}= 2(

*n*− 1)

*.*

**A**

_{n,k,l}is a product of two Hilbert–Schmidt operators, and so is trace class, provided that

**is sufficiently regular. Since $X\rho pln\rho =An,0,0$ is also trace class, it follows that $[X\rho ,pln]\rho $ is, indeed, trace class.**

*ρ***yield**

*ρ**I*

_{n,k}denote the integral in the above formula. We need to balance the powers of

*p*_{l}in

*I*

_{n,k}. For simplicity of notation, we assume that $n=2n\u0303\u22082N$. The case when

*n*is odd is similar. Then, if $k\u2264n\u0303$, we have

Now, combining the semiclassical kinetic interpolation inequality (8) for *n* = 2 and another $n\u2208N$, we get the following inequalities (see Ref. 13, Proposition 3.2).

*Let*0 ≤

*k*≤

*n*

*and*$pn,k\u2032=(nk)\u2032(1+3n)=3+nn\u2212k$

*. Then, for any*

*p*∈ [1,

*p*

_{n,k}]

*, if*

*k*≥ 2

*or if*$p\u2265p2,0=53$

*, it holds that*

*with*$\theta 2=n\u2212kn\u22122\u22123+nn\u221221p\u2032$

*and*$\theta n=k\u22122n\u22122+5n\u221221p\u2032$

*. If*

*k*= 0

*and*

*p*≤

*p*

_{2,0},

*then*

These two inequalities can be merged into a single inequality in terms of the non-homogeneous moments $1+Mn=h3Tr(\rho mn)$.

*For any*

*n*≥ 2

*,*

*k*∈ [0,

*n*],

*and*

*p*∈ [1,

*p*

_{n,k}]

*, there exists a constant*

*C*> 0

*such that for any compact operator*

**,**

*ρ**we have the following estimate:*

*with*$\theta n=k\u22122n\u22122+5n\u221221p\u2032$

*.*

*If*$(qj)j=1,\u2026,4$

*verify*$qj\u2208[1,pn,kj]$

*and*$\u2211j=14kj=2(n\u22121)$

*, then*$qj\u2032\u2208[pn,kj\u2032,\u221e]$,

*so*

*Hence, we can find such a family verifying (*

*9*

*) as soon as*$2(n+1)n+3\u22652b$

*or, equivalently, as soon as*$a\u22642nn+3=:an$

*.*

We can now complete the Proof of Theorem 2.1 using a similar strategy as to the one explained in Sec. II A for the Hartree equation. Instead of proceeding by induction and bounding the time derivative of *M*_{n} by a product involving *M*_{n} and *M*_{n−2}, we directly estimate it by a product involving *M*_{n} and *M*_{2}. This method allows us to slightly improve the result of Ref. 10, Theorem 3, even in the case of the Hartree equation, as it allows us to propagate moments of high order locally in time for any *a* ∈ (0, 2), while Ref. 10, Theorem 3 only covers the case $a\u2208(0,87)$ in dimension 3.

**|**

*p*^{n}, and inequality (10), we deduce that

*M*

_{n}can be further bounded by

*C*> 0. In particular,

## III. PROOF OF THEOREM 1.1

Note that bounds on the moments imply bounds on the semiclassical Schatten norms for any $p\u22082,\u221e$. More precisely, we have the following proposition.

*Let*

*ρ**be a positive trace class operator. Then, for any*$p\u22082,\u221e$

*, there exists a constant*

*C*> 0

*such that*

**is a bounded operator, we get that**

*ρ*For *p* = *∞*, we control $\rho mnL\u221e$ by means of a Grönwall argument. We will need the following commutator estimates in the spirit of Refs. 10 and 19, but improved to lead to large time estimates.

*Let*$a\u2208(0,12)$

*,*$n\u2208N$,

*and*

*q*∈ [1,

*∞*]

*. Then, for any*

*ɛ*∈ (0, 1)

*and*

*n*

_{1}>

*n*+

*a*+ 1

*, there exists a constant*

*C*> 0

*such that for every compact self-adjoint operators*

*ρ**and*

**,**

*μ**where*$r=31\u2212a$

*and*$\u22c5Lr\xb1\epsilon $

*stands for the norm*

*g*

_{ℓ}is the function defined by $g\u2113(x)=(pj\u2113(\u2202j\u2207V\rho ))(x)$ and $C\u2113=\u2211k=\u2113n\u22121k\u2113$. Noting that

_{A}(

*X*) := [

*A*,

*X*], and using the fact that $\u2202j\u2207K\u2208L3a+2+\u03f5+L3a+2\u2212\u03f5$ for any

*ϵ*> 0 sufficiently small, then Young’s inequality yields

*ɛ*∈ (0, 1). By Proposition 6.4 and Lemma 6.5 of Ref. 19, it implies that

*n*

_{0}> 3/

*r*′. This proves inequality (12). When ∇

*V*

_{ρ}is replaced by

*V*

_{ρ}, then we just replace ∇

*K*by

*K*, and so we need to find a

*L*

^{∞}bound for the function

*ℓ*> 0, we just write $g\u2113=i\u210f\u2202j2K*(pj\u2113\u22121\rho )$ and, then, use the same estimates as for ∇

*V*

_{ρ}. If

*ℓ*= 0, then

*g*

_{ℓ}= −

*∂*

_{j}

*K**

*ρ*is bounded using Young’s inequality by

*n*

_{0}> 3/

*r*′ =

*a*+ 2.□

*Let*$a\u2208(0,12)$

*and*

*ρ**be a solution to the Hartree–Fock equation (*

*1*

*). Assume that*$\rho in\u2208L\u221e(mn)$

*for some even*$n\u22082N$

*with moments of order*$(n+a+1)31\u2212a+\epsilon $

*bounded uniformly in*

*ℏ*

*for some*$\epsilon \u22080,1$

*. Then,*

*q*→

*∞*, we have

*x*≤

*y*to denote

*x*≤

*Cy*for some constant

*C*≤ 0, which is independent of

*x*,

*y*, and

*h*. Since

*ɛ*≤ 1 and

*r*≥ 3, we see that

*r*−

*ɛ*≥ 2, which allows us to apply Proposition 3.1 to get

*M*

_{2(n+1)}and obtain the following inequality for the left-hand side of Eq. (14):

Summarizing Theorem 2.1 and Propositions 3.1 and 3.3 implies the following result.

*Let*$a\u2208(0,12)$

*and*

*ρ**be a solution to the Hartree–Fock equation (*

*1*

*). For*$n\u22082N$,

*let*$\rho in\u2208L\u221e(mn)$

*with the moment of order*$(n+a+1)(31\u2212a+\epsilon )$

*bounded uniformly in*

*ℏ*

*for some*

*ɛ*∈ (0, 1)

*. Then, for any*

*q*∈ [2,

*∞*]

*, we have that*

*n*

_{1}>

*n*+

*a*+ 1. Note that for $a\u2208(0,12)$, the quantity $1q+12\u22121b$ is positive for every $q\u22082,\u221e$. Hence, for

*h*∈ (0, 1), we have that

*q*is due to the fact that we do not want to have any ∇

*ρ*in the right-hand side of inequality (16). However, now that we obtained the boundedness of

*N*

_{2}, we can propagate higher norms of the form $Nq2$ when $q2>61+2a$. Indeed, Ref. 19, Proposition 6.5 also yields, for instance,

*N*

_{2}. All

*q*

_{2}∈ (2,

*∞*) follow in the same way, finishing the proof of the theorem.□

## ACKNOWLEDGMENTS

J.J.C. was supported by the NSF through the RTG under Grant No. DMS- RTG 184031. C.S. acknowledges the NCCR SwissMAP and the support of the SNSF through the Eccellenza project under Grant No. PCEFP2_181153.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Laurent Lafleche**: Writing – original draft (equal); Writing – review & editing (equal). **Jacky J. Chong**: Writing – original draft (equal); Writing – review & editing (equal). **Chiara Saffirio**: Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

## REFERENCES

^{2}solutions to the Schrödinger–Poisson system: Existence, uniqueness, time behaviour, and smoothing effects