Global-in-time semiclassical regularity for the Hartree–Fock equation

Chong, J. J.; Lafleche, L.; Saffirio, C.

doi:10.1063/5.0089741

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 $V (x) = \pm {|x|}^{- a}$ with $a \in (0, \frac{1}{2})$ ⁠. 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

i ℏ \partial_{t} ρ = [H_{ρ}, ρ]

(1)

describing the evolution of a positive self-adjoint trace class operator ρ = ρ(t) acting on $L^{2} (R^{3})$ ⁠. Here, $ℏ = \frac{h}{2 π}$ is the reduced Planck constant, $[\cdot, \cdot]$ denotes the commutator $[A, B] = A B - B A$ ⁠, and H_ρ is the Hamiltonian operator given by

H_{ρ} = - \frac{ℏ^{2} Δ}{2} + V_{ρ} - h^{3} X_{ρ},

(2)

where V_ρ is the mean-field potential and $X_{ρ}$ is the exchange operator. The mean-field potential is defined as the multiplication operator by the function V_ρ(x) = (K^*ρ) (x), where $K : R^{3} \to R$ 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

ρ (x) = d i a g (ρ) (x) : = h^{3} ρ (x, x) .

The exchange operator is defined to be the operator with kernel

X_{ρ} (x, y) = K (x - y) ρ (x, y) .

In this work, we normalize ρ so that

h^{3} T r (ρ) = 1 and C_{\infty} : = {‖ρ‖}_{\infty},

(3)

where ${‖\cdot‖}_{\infty}$ 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 $\int_{R^{3}} ρ (x) ⅆ x = h^{3} T r (ρ) = 1$ ⁠. In the absence of $X_{ρ}$ ⁠, 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 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 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,

K (x) = \frac{\pm 1}{{|x|}^{a}} with a \in (0, \frac{1}{2}) .

In particular, $K \in L^{b, \infty} (R^{3})$ ⁠, where $b = \frac{3}{a + 1}$ and $L^{b, \infty} (R^{3})$ denotes the weak $L^{b}$ space on $R^{3}$ ⁠. 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

{‖ρ‖}_{L^{p}} : = h^{\frac{3}{p}} {‖ρ‖}_{p} = h^{\frac{3}{p}} T r {({|ρ|}^{p})}^{\frac{1}{p}}

for p ∈ [1, ∞], with the obvious modification for p = ∞. Here, ${‖ρ‖}_{p}$ denotes the Schatten norm of order p and $|A| = \sqrt{A^{*} A}$ denotes the absolute value of the operator A with adjoint A^*. Let p = −iℏ∇ be the momentum operator and

m_{n} : = 1 + {|p|}^{n}, n \in N .

The moment of order n and the weighted semiclassical Lebesgue norms with the operator weight $m_{n}$ are given by

M_{n} : = h^{3} T r ({|p|}^{n} ρ) and {‖ρ‖}_{L^{p} (m_{n})} : = {‖ρ m_{n}‖}_{L^{p}} .

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

\nabla_{x} ρ : = [\nabla, ρ] and \nabla_{ξ} ρ : = [\frac{x}{i ℏ}, ρ] .

Then, the semiclassical homogeneous Sobolev norms are defined by

{‖ρ‖}_{{\dot{W}}^{1, p}} : = \sum_{k = 1}^{3} ({‖\nabla_{x_{k}} ρ‖}_{L^{p}} + {‖\nabla_{ξ_{k}} ρ‖}_{L^{p}}),

with the corresponding inhomogeneous Sobolev norms given by ${‖ρ‖}_{W^{1, p}} : = {‖ρ‖}_{L^{p}} + {‖ρ‖}_{{\dot{W}}^{1, p}}$ and the weighted semiclassical Sobolev norms with the operator weight $m_{n}$ given by

{‖ρ‖}_{W^{1, p} (m_{n})} : = {‖ρ m_{n}‖}_{W^{1, p}} .

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

Theorem 1.1.

Let

a \in (0, \frac{1}{2})

,

n \in 2 N

be an even integer, and ρ be a solution to the Hartree–Fock equation () with initial datum

ρ^{i n} \in L^{\infty} (m_{n})

satisfying () such that

ρ^{i n} \in W^{1, 2} (m_{n}) \cap W^{1, q} (m_{n})

(4)

for

q \in [) 2, \infty ()

and with moments of order strictly larger than

\frac{3}{1 - a} (n + a + 1)

bounded uniformly in ℏ. Then,

ρ \in L_{l o c}^{\infty} (R_{+}, W^{1, 2} (m_{n}) \cap W^{1, q} (m_{n}) \cap L^{\infty} (m_{n}))

(5)

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 $a \in (0, \frac{1}{2})$ ⁠. 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 $a \in (0, \frac{1}{2})$ are, indeed, of the form (5) with $q > \frac{6}{1 + 2 a}$ (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 $a \in () 0, \frac{4}{5} (]$ and show that if $a \in (0, \frac{1}{2})$ ⁠, 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 $a \in (0, \frac{4}{5}]$ 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

Theorem 2.1.

Let

a \in (0, \frac{4}{5}]

,

n \in 2 N

⁠, and ρ be a solution of the Hartree–Fock equation with initial condition

ρ^{i n} \in L^{1} \cap L^{\infty}

with moments of order n bounded uniformly in ℏ. Then, there exists a continuous function

Φ_{n} \in C^{0} (R_{+})

independent of ℏ such that for any

t \in R_{+}

,

M_{n} (t) \leq Φ_{n} (t) .

Remark 2.1.

If $a \in (\frac{4}{5}, 2)$ , then we can still get a short-time estimate when $a \leq a_{n} = \frac{2 n}{n + 3}$ (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 $n \geq n_{a} = \frac{3 a}{2 - a}$ .

Remark 2.2.

The proof of the theorem can be used to get an explicit function Φ_n(t). It has a polynomial growth in time when $a < \frac{4}{5}$ and an exponential growth in time when $a = \frac{4}{5}$ .

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 $J \subset N$ ⁠, a sequence of functions ${(ψ_{j})}_{j \in J}$ orthonormal in $L^{2} (R^{3})$ ⁠, and a positive summable sequence ${(λ_{j})}_{j \in J}$ such that ρ can be written as

ρ = \sum_{j \in J} λ_{j} | ψ_{j} 〉 〈 ψ_{j} | .

For any even integer $n \in 2 N$ ⁠, we define the moment density of order n by

ρ_{n} (x) : = h^{3} \sum_{j \in J} λ_{j} {|p^{\frac{n}{2}} ψ_{j} (x)|}^{2} = d i a g (p^{\frac{n}{2}} ρ \cdot p^{\frac{n}{2}}) (x)

(6)

so that the moment of order n, previously defined, can be rewritten as $M_{n} = {‖ρ_{n}‖}_{L^{1}}$ ⁠. Note that we also have $M_{n} = {‖\sqrt{ρ} {|p|}^{\frac{n}{2}}‖}_{L^{2}}^{2}$ ⁠. With this notation, inequality (38) in Ref. 10 reads

|h^{3} T r (\frac{1}{i ℏ} [V_{ρ}, {|p|}^{n}] ρ)| \leq C M_{n}^{\frac{1}{2}} \sup_{\begin{array}{c} (j, k, l) \in N^{3} \\ j + k + l = n / 2 - 1 \end{array}} {‖ρ_{2 j}‖}_{L^{α}}^{\frac{1}{2}} {‖ρ_{2 k}‖}_{L^{β}}^{\frac{1}{2}} {‖ρ_{2 l}‖}_{L^{γ}}^{\frac{1}{2}},

(7)

where

\frac{1}{α^{'}} + \frac{1}{β^{'}} + \frac{1}{γ^{'}} = \frac{1}{b}

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 $(k, n) \in {(2 N)}^{2}$ verifying k ∈ [0, n], we have that

{‖ρ_{k}‖}_{L^{p}} \leq C {‖ρ‖}_{L^{\infty}}^{\frac{1}{p^{'}}} M_{n}^{\frac{1}{p}} with p = p_{n, k} : = \frac{3 + n}{3 + k} .

(8)

Combining this inequality with inequality (7) implies the existence of some positive constants Θ, Θ₀, and Θ₂ such that the following estimate [see Ref. 10, inequality (44)] holds:

|h^{3} T r (\frac{1}{i ℏ} [V_{ρ}, {|p|}^{n}] ρ)| \leq C {‖ρ‖}_{L^{\infty}}^{Θ_{2}} M_{n - 2}^{Θ_{0}} M_{n}^{Θ},

where Θ ≤ 1 when $a \leq \frac{4}{5}$ ⁠. 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 $a < \frac{4}{5}$ ⁠, 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.

Lemma 2.1.

Denote the components of p by

p_{x_{l}}

or simply p_l for l ∈ {1, 2, 3}. Suppose that

n \in N

⁠, and define

Q \subset {[\frac{1}{2}, \infty]}^{4}

⁠, the set of (q₁, q₂, q₃, q₄) verifying

\frac{1}{q_{1}} + \frac{1}{q_{2}} \in (0, 2)

,

\frac{1}{q_{3}} + \frac{1}{q_{4}} \in (0, 2)

⁠, and

\frac{1}{q_{1}^{'}} + \frac{1}{q_{2}^{'}} + \frac{1}{q_{3}^{'}} + \frac{1}{q_{4}^{'}} = \frac{2}{b} .

(9)

Then, there exists a constant C independent of ℏ such that

|\frac{h^{3}}{ℏ} T r ([h^{3} X_{ρ}, p_{l}^{n}] ρ)| \leq C \sup_{k_{1}, \dots k_{4}} \inf_{Q} {‖ρ_{k_{1}}‖}_{L^{q_{1}}}^{\frac{1}{2}} {‖ρ_{k_{2}}‖}_{L^{q_{2}}}^{\frac{1}{2}} {‖ρ_{k_{3}}‖}_{L^{q_{3}}}^{\frac{1}{2}} {‖ρ_{k_{4}}‖}_{L^{q_{4}}}^{\frac{1}{2}},

(10)

where the supremum is taken over all the integers

(k_{1}, \dots, k_{4}) \in N^{4}

such that k₁ + ⋯ + k₄ = 2(n − 1).

Proof.

Let us begin by making the observation that

[X_{ρ}, p_{l}^{n}] ρ

is trace class. By the Leibniz formula, the kernel of

p_{l}^{n} X_{ρ} ρ

is given by

\begin{matrix} p_{l}^{n} X_{ρ} ρ (x, y) & = \sum_{k = 0}^{n} (\binom{n}{k}) \int_{R^{3}} p_{x_{l}}^{n - k} K (x - z) p_{x_{l}}^{k} ρ (x, z) ρ (z, y) ⅆ z \\ = \sum_{k = 0}^{n} \sum_{ℓ = 0}^{n - k} (\binom{n}{k}) (\binom{n - k}{ℓ}) {(- 1)}^{n - k} \int_{R^{3}} K (x - z) (p_{x_{l}}^{k} p_{z_{l}}^{ℓ} ρ (x, z)) p_{z_{l}}^{n - k - ℓ} ρ (z, y) ⅆ z \\ = : \sum_{k = 0}^{n} \sum_{ℓ = 0}^{n - k} (\binom{n}{k}) (\binom{n - k}{ℓ}) {(- 1)}^{n - k + 1} A_{n, k, l} (x, y) . \end{matrix}

It is clear that each A_n,k,l is a product of two Hilbert–Schmidt operators, and so is trace class, provided that ρ is sufficiently regular. Since

X_{ρ} p_{l}^{n} ρ = A_{n, 0, 0}

is also trace class, it follows that

[X_{ρ}, p_{l}^{n}] ρ

is, indeed, trace class.

Let

I_{n} : = T r ([X_{ρ}, p_{l}^{n}] ρ)

⁠. Then, by Ref. 20, VI.7, Theorem 17, we may express the trace of

[X_{ρ}, p_{l}^{n}] ρ

in terms of its kernel as follows:

I_{n} = \iint_{R^{6}} K (x - y) ρ (y, x) p_{x_{l}}^{n} (ρ) (x, y) - p_{x_{l}}^{n} (K (x - y) ρ (x, y)) ρ (y, x) ⅆ x ⅆ y .

Therefore, using the Leibniz formula and, then, diagonalizing the self-adjoint compact operator ρ yield

\begin{matrix} I_{n} & = - \sum_{k = 0}^{n - 1} (\binom{n}{k}) \iint_{R^{6}} (p_{x_{l}}^{n - k} K) (x - y) (p_{x_{l}}^{k} ρ (x, y)) ρ (y, x) ⅆ x ⅆ y \\ = - \sum_{(i, j) \in J^{2}} \sum_{k = 0}^{n - 1} (\binom{n}{k}) λ_{i} λ_{j} \int_{R^{3}} (p_{l}^{n - k} K) * (\bar{ψ_{i}} ψ_{j}) p_{l}^{k} (ψ_{i}) \bar{ψ_{j}}, \end{matrix}

and so

\frac{I_{n}}{i ℏ} = \sum_{(i, j) \in J^{2}} \sum_{k = 0}^{n - 1} (\binom{n}{k}) λ_{i} λ_{j} \int_{R^{3}} (\partial_{l} K) * p_{l}^{n - 1 - k} (\bar{ψ_{i}} ψ_{j}) p_{l}^{k} (ψ_{i}) \bar{ψ_{j}} .

Let 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 = 2 \tilde{n} \in 2 N

⁠. The case when n is odd is similar. Then, if

k \leq \tilde{n}

⁠, we have

I_{n, k} = \int_{R^{3}} (\partial_{l} K) * p_{l}^{\tilde{n} - 1} (\bar{ψ_{i}} ψ_{j}) p_{l}^{\tilde{n} - k} (p_{l}^{k} (ψ_{i}) \bar{ψ_{j}}) = \sum_{ℓ = 0}^{\tilde{n} - k} \sum_{\tilde{ℓ} = 0}^{\tilde{n} - 1} (\binom{\tilde{n} - k}{ℓ}) (\binom{\tilde{n} - 1}{\tilde{ℓ}}) I_{n, k, ℓ, \tilde{ℓ}},

where

I_{n, k, ℓ, \tilde{ℓ}} = \int_{R^{3}} (\partial_{l} K) * (p_{l}^{\tilde{ℓ}} (\bar{ψ_{i}}) p_{l}^{\tilde{n} - 1 - \tilde{ℓ}} (ψ_{j})) p_{l}^{\tilde{n} - ℓ} (ψ_{i}) p_{l}^{ℓ} (\bar{ψ_{j}}) .

Now, note that by the Cauchy–Schwarz inequality for sums and Definition () for the moment density, we have that

\begin{array}{l} h^{2 d} \sum_{(i, j) \in J^{2}} λ_{i} λ_{j} I_{n, k, ℓ, \tilde{ℓ}} & \leq \int_{R^{3}} |\partial_{l} K| * (ρ_{2 \tilde{ℓ}}^{\frac{1}{2}} ρ_{2 (\tilde{n} - 1 - \tilde{ℓ})}^{\frac{1}{2}}) ρ_{2 (\tilde{n} - ℓ)}^{\frac{1}{2}} ρ_{2 ℓ}^{\frac{1}{2}} \\ \leq C_{K} {‖ρ_{2 \tilde{ℓ}}‖}_{L^{q_{1}}}^{\frac{1}{2}} {‖ρ_{2 (\tilde{n} - 1 - \tilde{ℓ})}‖}_{L^{q_{2}}}^{\frac{1}{2}} {‖ρ_{2 (\tilde{n} - ℓ)}‖}_{L^{q_{3}}}^{\frac{1}{2}} {‖ρ_{2 ℓ}‖}_{L^{q_{4}}}^{\frac{1}{2}}, \end{array}

where the last inequality follows from the Hardy–Littlewood–Sobolev inequality and the Hölder inequality. Similarly, when

k > \tilde{n}

⁠, we have that

\begin{array}{l} I_{n, k} : = & \int_{R^{3}} p_{l}^{k - \tilde{n}} ((\partial_{l} K) * p_{l}^{n - k - 1} (\bar{ψ_{i}} ψ_{j}) \bar{ψ_{j}}) p_{l}^{\tilde{n}} (ψ_{i}) \\ = & \sum_{ℓ = 0}^{k - \tilde{n}} \sum_{\tilde{ℓ} = 0}^{\tilde{n} - 1 - ℓ} (\binom{k - \tilde{n}}{ℓ}) (\binom{\tilde{n} - 1 - ℓ}{\tilde{ℓ}}) I_{n, k, ℓ, \tilde{ℓ}}, \end{array}

where

I_{n, k, ℓ, \tilde{ℓ}} = \int_{R^{3}} (\partial_{l} K) * (p_{l}^{\tilde{ℓ}} (\bar{ψ_{i}}) p_{l}^{\tilde{n} - 1 - ℓ - \tilde{ℓ}} (ψ_{j})) p_{l}^{ℓ} (\bar{ψ_{j}}) p_{l}^{\tilde{n}} (ψ_{i}) .

Mimicking the estimates we obtained above for the case

k \leq \tilde{n}

completes the proof of the lemma.□

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

Lemma 2.2.

Let 0 ≤ k ≤ n and

p_{n, k}^{'} = {(\frac{n}{k})}^{'} (1 + \frac{3}{n}) = \frac{3 + n}{n - k}

. Then, for any p ∈ [1, p_n,k], if k ≥ 2 or if

p \geq p_{2, 0} = \frac{5}{3}

, it holds that

{‖ρ_{k}‖}_{L^{p}} \leq C_{n, k}^{\frac{1}{p^{'}}} {‖ρ‖}_{L^{\infty}}^{\frac{1}{p^{'}}} M_{2}^{θ_{2}} M_{n}^{θ_{n}}

with

θ_{2} = \frac{n - k}{n - 2} - \frac{3 + n}{n - 2} \frac{1}{p^{'}}

and

θ_{n} = \frac{k - 2}{n - 2} + \frac{5}{n - 2} \frac{1}{p^{'}}

. If k = 0 and p ≤ p_2,0, then

{‖ρ‖}_{L^{p}} \leq C^{\frac{1}{p^{'}}} {‖ρ‖}_{L^{\infty}}^{\frac{1}{p^{'}}} M_{0}^{1 - \frac{5}{2 p^{'}}} M_{2}^{\frac{3}{2 p^{'}}} .

These two inequalities can be merged into a single inequality in terms of the non-homogeneous moments $1 + M_{n} = h^{3} T r (ρ m_{n})$ ⁠.

Corollary 2.1.

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:

{‖ρ_{k}‖}_{L^{p}} \leq C {‖ρ‖}_{L^{\infty}}^{\frac{1}{p^{'}}} {(1 + M_{2})}^{θ_{2}} {(1 + M_{n})}^{θ_{n}}

(11)

with

θ_{n} = \frac{k - 2}{n - 2} + \frac{5}{n - 2} \frac{1}{p^{'}}

.

Remark 2.3.

If

{(q_{j})}_{j = 1, \dots, 4}

verify

q_{j} \in [1, p_{n, k_{j}}]

and

\sum_{j = 1}^{4} k_{j} = 2 (n - 1)

, then

q_{j}^{'} \in [p_{n, k_{j}}^{'}, \infty]

⁠, so

0 \leq \sum_{j = 1}^{4} \frac{1}{q_{j}^{'}} \leq \sum_{j = 1}^{4} \frac{n - k_{j}}{n + 3} = \frac{2 (n + 1)}{n + 3} .

Hence, we can find such a family verifying () as soon as

\frac{2 (n + 1)}{n + 3} \geq \frac{2}{b}

or, equivalently, as soon as

a \leq \frac{2 n}{n + 3} = : a_{n}

.

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 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 $a \in (0, \frac{8}{7})$ in dimension 3.

Proof of Theorem 2.1.

Taking the time derivative of moments and using the cyclicity of the trace yield

i ℏ \frac{d}{d t} T r (ρ p_{l}^{n}) = - T r ([V_{ρ}, p_{l}^{n}] ρ) - T r ([X_{ρ}, p_{l}^{n}] ρ) .

Then, by inequality (), which also holds for

p_{l}^{n}

in place of |p|ⁿ, and inequality (), we deduce that

|h^{3} \frac{d}{d t} T r (ρ p_{l}^{n})| \leq C_{n, a} \sup_{k_{1} + \dots + k_{4} = 2 (n - 1)} \inf_{Q} {‖ρ_{k_{1}}‖}_{L^{q_{1}}}^{\frac{1}{2}} {‖ρ_{k_{2}}‖}_{L^{q_{2}}}^{\frac{1}{2}} {‖ρ_{k_{3}}‖}_{L^{q_{3}}}^{\frac{1}{2}} {‖ρ_{k_{4}}‖}_{L^{q_{4}}}^{\frac{1}{2}}

with the notations of Lemma 2.1. Using the interpolation formula () for each of the terms in the right-hand side of the above inequality, we get

|\frac{h^{3}}{i ℏ} T r ([h^{3} X_{ρ}, p_{l}^{n}] ρ)| \leq C_{n, a} {‖ρ‖}_{L^{\infty}}^{\frac{1}{b}} {(1 + M_{2})}^{Θ_{2}} {(1 + M_{n})}^{Θ_{n}}

with

Θ_{2} = \frac{n + 1}{n - 2} - \frac{n + 3}{n - 2} \frac{1}{b}

and

Θ_{n} = \frac{2 n - 10}{2 (n - 2)} + \frac{5}{n - 2} \frac{1}{b} = 1 + \frac{1}{n - 2} (\frac{5}{b} - 3) .

Moreover, by Ref. 19, Lemma 6.3, M_n can be further bounded by

C^{- 1} \sum_{l = 1}^{3} T r (ρ (1 + p_{l}^{n})) \leq M_{n} \leq C \sum_{l = 1}^{3} T r (ρ (1 + p_{l}^{n}))

for some C > 0. In particular,

Θ_{n} \leq 1 ⟺ b \geq \frac{5}{3} ⟺ a \leq \frac{4}{5},

which yields the result by Grönwall’s lemma.□

III. PROOF OF THEOREM 1.1

Note that bounds on the moments imply bounds on the semiclassical Schatten norms for any $p \in [) 2, \infty ()$ ⁠. More precisely, we have the following proposition.

Proposition 3.1.

Let ρ be a positive trace class operator. Then, for any

p \in [) 2, \infty ()

, there exists a constant C > 0 such that

{‖ρ m_{n}‖}_{L^{p}} \leq C {‖ρ‖}_{L^{\infty}}^{\frac{1}{p^{'}}} {(1 + M_{n p})}^{\frac{1}{p}} .

Proof.

Using the fact that ρ is a bounded operator, we get that

{‖ρ m_{n}‖}_{L^{p}} \leq {‖ρ‖}_{L^{\infty}}^{\frac{1}{p^{'}}} {‖ρ^{\frac{1}{p}} m_{n}‖}_{L^{p}} \leq {‖ρ‖}_{L^{\infty}}^{\frac{1}{p^{'}}} {‖\sqrt{ρ} m_{n}^{\frac{p}{2}}‖}_{L^{2}}^{\frac{2}{p}} = {‖ρ‖}_{L^{\infty}}^{\frac{1}{p^{'}}} h^{3} T r {(ρ m_{n}^{p})}^{\frac{1}{p}},

where the second inequality follows from the Araki–Lieb–Thirring inequality.□

For p = ∞, we control ${‖ρ m_{n}‖}_{L^{\infty}}$ 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.

Proposition 3.2

(weighted commutator estimate). Let

a \in (0, \frac{1}{2})

,

n \in N

⁠, and q ∈ [1, ∞]. Then, for any ɛ ∈ (0, 1) and n₁ > n + a + 1, there exists a constant C > 0 such that for every compact self-adjoint operators ρ and μ,

\frac{1}{ℏ} {‖[\nabla V_{ρ}, p_{j}^{n}] μ‖}_{L^{q}} \leq C {‖ρ m_{n_{1}}‖}_{L^{r \pm ε}} {‖μ m_{n}‖}_{L^{q}},

(12)

\frac{1}{ℏ} {‖[V_{ρ}, p_{j}^{n}] μ‖}_{L^{q}} \leq C ({‖ρ m_{n_{1}}‖}_{L^{r \pm ε}} + M_{0}) {‖μ m_{n}‖}_{L^{q}},

(13)

where

r = \frac{3}{1 - a}

and

{‖\cdot‖}_{L^{r \pm ε}}

stands for the norm

{‖\cdot‖}_{L^{r \pm ε}} : = {‖\cdot‖}_{L^{r + ε} \cap L^{r - ε}} = {‖\cdot‖}_{L^{r + ε}} + {‖\cdot‖}_{L^{r - ε}} .

Proof of Proposition 3.2.

We first proceed as in Refs. 10 and 19 and write

\frac{1}{ℏ} {‖[\nabla V_{ρ}, p_{j}^{n}] μ‖}_{L^{q}} \leq \sum_{ℓ = 0}^{n - 1} C_{ℓ} {‖g_{ℓ} p_{j}^{n - 1 - ℓ} μ‖}_{L^{q}} \leq \sum_{ℓ = 0}^{n - 1} C_{ℓ} {‖g_{ℓ}‖}_{L^{\infty}} {‖μ m_{n}‖}_{L^{q}},

where g_ℓ is the function defined by

g_{ℓ} (x) = (p_{j}^{ℓ} (\partial_{j} \nabla V_{ρ})) (x)

and

C_{ℓ} = \sum_{k = ℓ}^{n - 1} (\binom{k}{ℓ})

⁠. Noting that

g_{ℓ} = - \partial_{j} \nabla K * (p_{j}^{ℓ} ρ) = - \partial_{j} \nabla K * d i a g ({a d}_{p_{j}}^{ℓ} (ρ)),

where ad_A(X) := [A, X], and using the fact that

\partial_{j} \nabla K \in L^{\frac{3}{a + 2} + ϵ} + L^{\frac{3}{a + 2} - ϵ}

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

{‖g_{ℓ}‖}_{L^{\infty}} \leq C_{K} {‖d i a g ({a d}_{p_{j}}^{ℓ} (ρ))‖}_{L^{r \pm ε}}

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

{‖g_{ℓ}‖}_{L^{\infty}} \leq C_{K} {‖{a d}_{p_{j}}^{ℓ} (ρ) m_{n_{0}}‖}_{L^{r \pm ε}} \leq 2^{ℓ + 1} C_{K} {‖ρ m_{n_{0} + n - 1}‖}_{L^{r \pm ε}}

with n₀ > 3/r′. This proves inequality (). 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

g_{ℓ} = - \partial_{j} K * (p_{j}^{ℓ} ρ) .

When ℓ > 0, we just write

g_{ℓ} = i ℏ \partial_{j}^{2} K * (p_{j}^{ℓ - 1} ρ)

and, then, use the same estimates as for ∇V_ρ. If ℓ = 0, then g_ℓ = −∂_jK*ρ is bounded using Young’s inequality by

{‖g_{ℓ}‖}_{L^{\infty}} \leq {‖ρ‖}_{L^{b^{'} \pm ε}} \leq {‖ρ‖}_{L^{r}} + {‖ρ‖}_{L^{1}}

since

b^{'} \leq r

⁠. By Ref. 19, Proposition 6.4,

{‖ρ‖}_{L^{r}} \leq {‖ρ m_{n_{0}}‖}_{L^{r}}

with n₀ > 3/r′ = a + 2.□

Proposition 3.3.

Let

a \in (0, \frac{1}{2})

and ρ be a solution to the Hartree–Fock equation (). Assume that

ρ^{i n} \in L^{\infty} (m_{n})

for some even

n \in 2 N

with moments of order

(n + a + 1) (\frac{3}{1 - a} + ε)

bounded uniformly in ℏ for some

ε \in (0, 1)

. Then,

ρ \in L_{l o c}^{\infty} (R_{+}, L^{\infty} (m_{n})) .

Proof.

By Ref. 19, Lemma 6.2 with q → ∞, we have

\frac{d}{d t} {‖ρ m_{n}‖}_{L^{\infty}} \leq \frac{1}{ℏ} {‖[V_{ρ}, m_{n}] ρ‖}_{L^{\infty}} + \frac{1}{ℏ} {‖[h^{3} X_{ρ}, m_{n}] ρ‖}_{L^{\infty}} .

(14)

Next, we use Proposition 3.2 to bound the first term on the right-hand side of inequality (),

\frac{1}{ℏ} {‖[V_{ρ}, m_{n}] ρ‖}_{L^{\infty}} ≲ (1 + {‖ρ m_{n_{1}}‖}_{L^{r \pm ε}}) {‖ρ m_{n}‖}_{L^{\infty}},

where

r = \frac{3}{1 - a}

and where we use the standard notation 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

\frac{1}{ℏ} {‖[V_{ρ}, m_{n}] ρ‖}_{L^{\infty}} ≲ (1 + {‖ρ‖}_{L^{\infty}} + M_{n_{1} (r + ε)}) {‖ρ m_{n}‖}_{L^{\infty}} .

(15)

As for the second term on the right-hand side of inequality (), we first observe that by Ref. 19, Lemma 6.4,

ℏ^{\frac{3}{2} - a} {‖\nabla_{x} ρ m_{n}‖}_{L^{p}} = ℏ^{\frac{1}{2} - a} {‖[p, ρ] m_{n}‖}_{L^{p}} \leq 2 ℏ^{\frac{1}{2} - a} {‖ρ m_{n + 1}‖}_{L^{p}},

and then we apply Ref. 19, Proposition 6.8 with

m_{n} = 1 + | p |^{n}

to get

\frac{1}{ℏ} {‖[h^{3} X_{ρ}, m_{n}] ρ‖}_{L^{\infty}} ≲ ℏ^{\frac{1}{2} - a} {‖ρ m_{n + 1}‖}_{L^{2}} {‖ρ m_{n}‖}_{L^{\infty}} .

Proposition 3.1 allows us to bound

{‖ρ m_{n + 1}‖}_{L^{2}}

in terms of M_2(n+1) and obtain the following inequality for the left-hand side of Eq. ():

\frac{d}{d t} {‖ρ m_{n}‖}_{L^{\infty}} ≲ (1 + {‖ρ‖}_{L^{\infty}} + M_{n_{1} (r + ε)} + M_{2 (n + 1)}) {‖ρ m_{n}‖}_{L^{\infty}},

which gives a bound on

{‖ρ m_{n}‖}_{L^{\infty}}

by means of Grönwall’s lemma and the control on moments established in Theorem 2.1.□

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

Corollary 3.1.

Let

a \in (0, \frac{1}{2})

and ρ be a solution to the Hartree–Fock equation (). For

n \in 2 N

⁠, let

ρ^{i n} \in L^{\infty} (m_{n})

with the moment of order

(n + a + 1) (\frac{3}{1 - a} + ε)

bounded uniformly in ℏ for some ɛ ∈ (0, 1). Then, for any q ∈ [2, ∞], we have that

ρ \in L_{l o c}^{\infty} (R_{+}, L^{q} (m_{n})) .

Proof of Theorem 1.1.

To propagate quantum Sobolev norms, we proceed as in Ref. 19, Sec. 6 and consider the inequalities

\begin{matrix} \frac{d}{d t} {‖\nabla_{x} ρ m_{n}‖}_{L^{q}} \leq \frac{1}{ℏ} ({‖[V_{ρ}, m_{n}] \nabla_{x} ρ‖}_{L^{q}} + {‖[\nabla V_{ρ}, m_{n}] ρ‖}_{L^{q}}) + \frac{1}{ℏ} {‖[\nabla V_{ρ}, ρ m_{n}]‖}_{L^{q}} \\ + \frac{1}{ℏ} {‖[h^{3} X_{ρ}, m_{n}] \nabla_{x} ρ‖}_{L^{q}} + \frac{1}{ℏ} {‖[h^{3} X_{\nabla_{x} ρ}, ρ] m_{n}‖}_{L^{q}} \end{matrix}

and

\begin{matrix} \frac{d}{d t} {‖\nabla_{ξ} ρ m_{n}‖}_{L^{q}} \leq \frac{1}{ℏ} {‖[V_{ρ}, m_{n}] \nabla_{ξ} ρ‖}_{L^{q}} + {‖\nabla_{x} ρ m_{n}‖}_{L^{q}} \\ + \frac{1}{ℏ} {‖[h^{3} X_{ρ}, m_{n}] \nabla_{ξ} ρ‖}_{L^{q}} + \frac{1}{ℏ} {‖[h^{3} X_{\nabla_{ξ} ρ}, ρ] m_{n}‖}_{L^{q}}, \end{matrix}

and we define

N_{q} = N_{q} (t) : = {‖ρ‖}_{W^{1, q} (m_{n})}

⁠. Now, when

q \leq \frac{6}{1 + 2 a}

⁠, Ref. 19, Proposition 6.5 yields an estimate of the form

\frac{1}{ℏ} {‖[\nabla V_{ρ}, ρ m_{n}]‖}_{L^{q}} \leq {‖ρ‖}_{L^{r \pm ε} \cap L^{2}} {‖\nabla_{x} (ρ m_{n})‖}_{L^{q}}

(16)

with

r = \frac{3}{1 - a}

⁠. Then, using the above inequality, Proposition 3.2, and, similarly as in the proof of the previous proposition, Ref. 19, Propositions 6.8 and 6.9 to bound the terms involving the exchange term, we obtain for any

q \leq \frac{6}{1 + 2 a}

⁠,

\begin{matrix} \frac{d}{d t} N_{q} ≲ (1 + {‖ρ m_{n_{1}}‖}_{L^{r \pm ε}}) N_{q} + {‖ρ‖}_{L^{r \pm ε} \cap L^{2}} N_{q} \\ + h^{\frac{1}{2} - a} {‖ρ m_{n + 1}‖}_{L^{2}} N_{q} + h^{3 (\frac{1}{q} + \frac{1}{2} - \frac{1}{b})} {‖ρ m_{n}‖}_{L^{\infty}} N_{2} \end{matrix}

with n₁ > n + a + 1. Note that for

a \in (0, \frac{1}{2})

⁠, the quantity

\frac{1}{q} + \frac{1}{2} - \frac{1}{b}

is positive for every

q \in [) 2, \infty ()

⁠. Hence, for h ∈ (0, 1), we have that

\frac{d}{d t} (N_{2} + N_{q}) ≲ (1 + {‖ρ m_{n_{1}}‖}_{L^{r \pm ε}} + {‖ρ m_{n}‖}_{L^{\infty}} + {‖ρ m_{n + 1}‖}_{L^{2}}) (N_{2} + N_{q}),

which proves Theorem 1.1 for

q \leq \frac{6}{1 + 2 a}

by Corollary 3.1 and Grönwall’s lemma. The limitation on q is due to the fact that we do not want to have any ∇ρ in the right-hand side of inequality (). However, now that we obtained the boundedness of N₂, we can propagate higher norms of the form

N_{q_{2}}

when

q_{2} > \frac{6}{1 + 2 a}

⁠. Indeed, Ref. 19, Proposition 6.5 also yields, for instance,

\frac{1}{ℏ} {‖[\nabla V_{ρ}, ρ m_{n}]‖}_{L^{\infty}} \leq {‖ρ‖}_{L^{2}}^{1 - θ} {‖ρ‖}_{H^{1}}^{θ} {‖\nabla_{x} (ρ m_{n})‖}_{L^{\infty}},

with

θ = \frac{3}{b} - \frac{1}{2} = a + \frac{1}{2} \in (0, 1)

and

{‖ρ‖}_{L^{2}}

being controlled by N₂. All q₂ ∈ (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.

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Author(s)

Global-in-time semiclassical regularity for the Hartree–Fock equation

I. INTRODUCTION AND MAIN RESULT

II. PROPAGATION OF MOMENTS

A. The Hartree equation

B. The Hartree–Fock equation

III. PROOF OF THEOREM 1.1

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Global-in-time semiclassical regularity for the Hartree–Fock equation

I. INTRODUCTION AND MAIN RESULT

II. PROPAGATION OF MOMENTS

A. The Hartree equation

B. The Hartree–Fock equation

III. PROOF OF THEOREM 1.1

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

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Related Content

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