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 with . 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 . Here, is the reduced Planck constant, denotes the commutator , and Hρ is the Hamiltonian operator given by
where Vρ is the mean-field potential and is the exchange operator. The mean-field potential is defined as the multiplication operator by the function Vρ(x) = (K*ρ) (x), where 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 ρ, given by
The exchange operator is defined to be the operator with kernel
In this work, we normalize ρ so that
where 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 . In the absence of , 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 L2 and the propagation of higher Hs 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 H2 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 schemes15 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, , where and denotes the weak space on . Our main motivation is to extend the results of the local-in-time regularity obtained in our previous paper19 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, denotes the Schatten norm of order p and 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 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 and the weighted semiclassical Sobolev norms with the operator weight given by
Our main result states the global-in-time propagation of the regularity in terms of these norms.
Note that Theorem 1.1 extends to arbitrarily long times the local-in-time theory studied in Ref. 19, Theorem 3.1 for . 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 are, indeed, of the form (5) with (take p = 2 and q = q1 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 and show that if , 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.
II. PROPAGATION OF MOMENTS
If , then we can still get a short-time estimate when (see Remark 2.3). In particular, an 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 .
The proof of the theorem can be used to get an explicit function Φn(t). It has a polynomial growth in time when and an exponential growth in time when .
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 , a sequence of functions orthonormal in , and a positive summable sequence such that ρ can be written as
For any even integer , we define the moment density of order n by
so that the moment of order n, previously defined, can be rewritten as . Note that we also have . With this notation, inequality (38) in Ref. 10 reads
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 nth order moment density, tells us that for any verifying k ∈ [0, n], we have that
where Θ ≤ 1 when . This leads to the boundedness of moments by a Grönwall-type argument, together with the uniform boundedness of M2 due to the conservation of energy and an induction on n. Moreover, when , 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.
These two inequalities can be merged into a single inequality in terms of the non-homogeneous moments .
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 Mn by a product involving Mn and Mn−2, we directly estimate it by a product involving Mn and M2. 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 in dimension 3.
III. PROOF OF THEOREM 1.1
Note that bounds on the moments imply bounds on the semiclassical Schatten norms for any . More precisely, we have the following proposition.
Summarizing Theorem 2.1 and Propositions 3.1 and 3.3 implies the following result.
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.
Conflict of Interest
The authors have no conflicts to disclose.
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).
The data that support the findings of this study are available within the article.