Pickl’s Proof of the Quantum Mean-Field Limit and Quantum Klimontovich Solutions

. This paper discusses the mean-ﬁeld limit for the quantum dynamics of N identical bosons in R 3 interacting via a binary potential with Coulomb type singularity. Our approach is based on the theory of quantum Klimontovich solutions deﬁned in [F. Golse, T. Paul, Commun. Math. Phys. 369 (2019), 1021–1053]. Our ﬁrst main result is a deﬁnition of the interaction nonlinearity in the equation governing the dynamics of quantum Klimontovich solutions for a class of interaction potentials slightly less general than those considered in [T. Kato, Trans. Amer. Math. Soc. 70 (1951), 195–211]. Our second main result is a new operator inequality satisﬁed by the quantum Klimontovich solution in the case of an interaction potential with Coulomb type singularity. When evaluated on an initial bosonic pure state, this operator inequality reduces to a Gronwall inequality for a functional introduced in [P. Pickl, Lett. Math. Phys. 97 (2011), 151–164], resulting in a convergence rate estimate for the quantum mean-ﬁeld limit leading to the time-dependent Hartree equation.


Introduction and Notation
In classical mechanics, the motion equations for a system of N identical point particles of mass m with positions q j (t) ∈ R 3 and momenta p j (t) ∈ R 3 for all j = 1, . . ., N is ∇V (q j (t) − q k (t)) = −∇ pj H N (p 1 (t), . . ., q N (t)) , where the N -particle classical Hamiltonian is Assuming that V ∈ C 1,1 (R 3 ), this differential system has a unique global solution for all initial data.If V is even 1 , the phase space empirical measure is an exact, weak solution of the Vlasov equation with self-consistent, mean-field potential This remarkable observation is due to Klimontovich, and solutions of the Vlasov equation ( 3) of the form (2) are referred to as "Klimontovich solutions".Thus, if µ N (0) → f in dxdξ weakly in the sense of probability measures as N → ∞, where f in is a probability density on R 3 x × R 3 ξ , one has µ N (t, dxdξ) → f (t, x, ξ)dxdξ weakly in the sense of probability measures for all t ≥ 0 as N → ∞, where f is the solution of the Vlasov equation ( 4) Thus, the mean-field limit in classical mechanics is equivalent to the continuous dependence for the weak topology of probability measures of solutions of the Vlasov equation in terms of their initial data.See [4] for a proof of this result.For instance, the weak convergence of the initial data can be realized by a random choice of (q j (0), p j (0)), independent and identically distributed with distribution f in .
The mean-field limit for bosonic systems in quantum mechanics has been formulated in different settings, by using the so-called BBGKY hierarchy [20,2,1,6], or in the second quantization setting [18].Interestingly, these techniques allow considering singular potentials such as the Coulomb potential, instead of C 1,1 potentials as in the classical case.(The mean-field limit with Coulomb potentials in classical mechanics is still an open problem at the time of this writing; see however [19] in the special case of monokinetic particle distributions.See also [9,10] for potentials less singular than the Coulomb potential).
The quantum mean-field equation analogous to the Vlasov equation ( 4) is the (time-dependent) Hartree equation (5) In [16,14], an original method, close to the second quantization approach in [18], but avoiding the rather heavy formalism of Fock spaces, was proposed and successfully applied to singular potentials including the Coulomb potential.
All these approaches noticeably differ from the classical setting used in [4] for lack of a quantum notion of phase-space empirical measures.However, a quantum analogue of the notion of phase-space empirical measure was recently proposed in [8], along with an equation analogous to (3) governing their evolution.This notion was used in [8] to prove the uniformity of the mean-field limit in the Planck constant ̵ h > 0. However, the discussion in [8] only considers regular potentials (specifically ).Even writing the equation analogous to (3) satisfied by the quantum analogue of the phase-space empirical measure requires V ∈ F L 1 (R d ) in the setting of [8].
The purpose of the present paper is twofold: (a) to extend the formalism of quantum empirical measures considered in [8] to treat the case of singular potentials including the Coulomb potential, which is of particular interest for applications to atomic physics (see Theorem 3.1 in section 3), and (b) to explain how the ideas in [16,14] can be couched in terms of the formalism of quantum empirical measures defined in [8] (see Theorem 4.1 and Corollary 4.2 in section 4).
Specifically, we prove an inequality between operators on the N -particle Hilbert space, of which the key estimates in [16,14] leading to the quantum mean-field limit are straightforward consequences.
The next section briefly recalls only the essential part of [8] used in the sequel.The main results obtained in the present paper are Theorems 3.1 and 4.1 from sections 3 and 4 respectively.The proofs of these results are given in the subsequent sections.

Quantum Klimontovich Solutions
Consider the quantum N -body Hamiltonian (6) H ). Henceforth it is assumed that V is a real-valued function such that H N has a (unique) self-adjoint extension to H N , still denoted H N .A well-known sufficient condition for this to be true has been found by Kato (see condition (5) in [12]): there exists R > 0 such that (7) z ≤R In particular, these conditions include the (repulsive) Coulomb potential in R 3 .In fact, H N has a self-adjoint extension to H N under a condition slightly more general than Kato's original assumption recalled above: (see Theorem X.16 and Example 2 in [17], and Theorem V.9 with m = 1 in [15]).
In the sequel, we adopt the notation in [8].In particular, we set (9) The dynamics of the morphism M in N is defined by conjugation with the N -particle dynamics as follows: for each A ∈ L(H), ) is henceforth referred to as the quantum Klimontovich solution.
Assume henceforth that V is even: The first main result in [8] where is the quantum kinetic energy, and where ( 14) , where E ω ∈ L(H) is the operator defined by ( 16) Since the integrand of the right-hand side of ( 15) takes its values in the non separable space L(H N ), it is worth mentioning that this integral is a weak integral for the ultraweak topology in L(H N ) (see footnote 3 on p. 1032 in [8]).
At variance with the classical case recalled in (3), the differential equation ( 13) satisfied by the quantum Klimontovich solution t ↦ M N (t) is not formally identical to the mean-field, time-dependent Hartree equation (5).The relation between ( 5) and ( 13) is explained in Theorem 3.5, the second main result in [8], recalled below.
If ψ is a solution of the the time-dependent Hartree equation ( 5) satisfying the normalization condition is a solution of (13).

Extending the Definition of
Our first task is to extend the definition (15) of the term C(V, M N (t), M N (t)) to a more general class of potentials V , including the Coulomb potential in R 3 . Since 2 Throughout this paper, we adopt the Dirac bra-ket notation.Thus a wave function ψ ∈ H viewed as a vector of the linear space H is denoted ψ⟩, whereas ⟨ψ designates the linear functional and ⟨ψ φ⟩ ∶= ⟨ψ I H φ⟩ is the inner product on H.
and to take advantage of the decay of S N [φ] in ω, assuming that φ is regular enough.Our argument does not use any regularity on Φ in N or Ψ in N .This is quite natural, since anyway Kato's condition (8) on the interaction potential V does not entail higher than (Sobolev) H 2 regularity for U N (t)Φ in N or U N (t)Ψ in N , as observed in Note V.10 of [15].
Our first main result in this paper is the following result, leading to a definition of C(V, M N (t), M N (t))( φ⟩⟨φ ) in the case of singular, Coulomb-like potentials V , and for bounded wave functions φ.This theorem can be regarded as an extension to the case of singular, Coulomb like potentials V of the formalism of quantum Klimontovich solutions in [8].
Theorem 3.1.Assume that V is a real-valued measurable function on R 3 satisfying the parity condition (12), and The integral on the right-hand side of the equality above is to be understood as a weak integral and defines as a continuous map from R to L(H N ) endowed with the ultraweak topology, which is moreover bounded on R for the operator norm on L(H N ).
Obviously, condition (17) is stronger than Kato's condition (8).However, the repulsive Coulomb potential z ↦ 1 z in R 3 obviously satisfies (17), since its Fourier transform ζ ↦ C ζ 2 belongs to L 1 (R 3 ) + L 2 (R 3 ).In particular, H N has a selfadjoint extension to H N under condition (17) Observe first that Without loss of generality, consider the term where with the notation We shall prove that where the last inequality is the Cauchy-Schwarz inequality for the inner integral.
On the other hand and and so that for each t ∈ R by polarization, and the function is bounded on R with values in L(H N ) for the norm topology, and continuous on R with values in L(H N ) endowed with the weak operator topology, and therefore for the ultraweak topology (since the weak operator and the ultraweak topologies coincide on norm bounded subsets of L(H N )).
Remark.In the sequel, we shall also need to consider terms of the form (I) ∶= 1 The term (III) is the easiest of all.Indeed, The terms (I) and (II) are slightly more delicate, but can be treated by the same method already used in the proof of the theorem above.First, Thus, we have proved that , which leads to a definition of (I).
The case of (II) is essentially similar.Observe that And (II) ∶= 1 with a similar conclusion for F 3 .On the other hand Therefore the map , thereby leading to a definition of (II).

An Operator Inequality. Application to the Mean-Field Limit
First consider the Cauchy problem for the time dependent Hartree equation ( 5).Assuming that the potential V satisfies ( 8) and ( 12), for each φ in ∈ H 2 (R 3 ), there exists a unique solution φ ∈ C(R, H 2 (R 3 )) of ( 5) by Theorems 1.4 and 1.3 of [11].
Pickl's key idea in his proof of the mean-field limit in quantum mechanics is to consider the following functional (see Definition 2.2 and formula (6) in [16], with the choice n(k) ∶= k N , in the notation of [16]): Assuming that ψ ≡ ψ(t, x) is a solution of (5) while Ψ N (t, ⋅) ∶= U N (t)Ψ in N , Pickl studies in section 2.1 of [16] the time-dependent function t ↦ α N (Ψ N (t, ⋅), ψ(t, ⋅)), and proves that it satisfies some Gronwall inequality.
Observe first that Pickl's functional α N (Ψ N (t, ⋅), ψ(t, ⋅)) can be recast in terms of the quantum Klimontovich solution M N (t) as follows (18) . This identity suggests therefore to deduce from ( 13) and ( 5) the expression of in terms of the interaction operator C defined in (15).This is done in the first part of the next theorem, which is our second main result in this paper.
Let ψ in ∈ H 2 (R 3 ) satisfy ψ in H = 1, let ψ be the solution of the Cauchy problem (5) for the time-dependent Hartree equation, and set (2) the operator C(V, M N (t) − R(t), M N (t))(P (t)) is skew-adjoint on H N and satisfies the operator inequality and where C S is the norm of the Sobolev embedding The operator inequality for quantum Klimontovich solutions in the case of potentials with Coulomb type singularity obtained in part (2) of Theorem 4.1 can be thought of as the reformulation of Pickl's argument in terms of the quantum Klimontovich solution M N (t).
Indeed, we deduce from parts (1) and ( 2) in Theorem 4.1 the operator inequality (21) 3 We recall that, if E, F are Banach spaces Then, evaluating both sides of this inequality on the initial N -particle state Ψ in N , and taking into account the identity (18) leads to the Gronwall inequality . This last inequality corresponds to inequality (11) and Lemma 3.2 in [16].
In the sequel, we shall denote by L p (H) for p ≥ 1 the Schatten two-sided ideal of L(H) consisting of operators T such that In with L given by (20).
Let us briefly indicate how one arrives at the operator inequality in part (2) of Theorem 4.1.Let Λ 1 , Λ 2 ∈ L(L(H), L(H N )) be such that 12) and ( 17), define In other words, where the integral on the right hand is to be understood in the ultraweak sense (see footnote 3 on p. 1032 in [8]).
For each A ∈ L(H), denote by Λ j (•A) and Λ j (A•) the linear maps where the first equality follows from the fact that Λ 1 and Λ 2 are *-homomorphisms, while the second equality uses the fact that V is real-valued, since V is real-valued and even.
An easy consequence of (24) and of this lemma is that, for each The key observations leading to Theorem 4.1 are summarized in the two following lemmas.In the first of these two lemmas, the interaction operator is decomposed into a sum of four terms.Lemma 4.4.Under the same assumptions and with the same notations as in Theorem 4.1, the interaction operator satisfies the identity All the terms involved in this decomposition can be defined by the same method already used in the proof of Theorem 3.1.Indeed, one can check that all these terms involve only expressions of the type (I), (II) or (III) in the Remark following Theorem 3.1.This easy verification is left to the reader, and we shall henceforth consider this matter as settled by the detailed explanations concerning (I), (II) and (III) given in the previous section.
Each term in this decomposition satisfies an operator inequality involving only the operator norm of the "mean-field squared potential" (V 2 ) R(t) , instead of the "bare" interaction potential V itself. Then Remarks on ℓ(t) in (26) and L(t) in (20). ( , which is the multiplication operator by the function where we recall that C S is the norm of the Sobolev embedding (2) If V satisfies (8), then V (I − ∆) −1 ≤ M for some positive constant M (see the discussion in §5.3 of chapter V in [13], so that In this remark, we shall make a slightly more restrictive assumption, namely that In space dimension d = 3, the Hardy inequality, which can be put in the form 4 1 x 2 ≤ 4(−∆) implies that the Coulomb potential satisfies the assumption above on V .If the potential V satisfies the (operator) inequality ( 27), then . 4 To see that 4 is optimal, minimize in α > 0 the expression (3) A bound on ℓ(t) in terms of ψ(t, ⋅) H 1 (R 3 ) instead of ψ(t, ⋅) H 2 (R 3 ) is advantageous since the former quantity can be controlled rather explicitly by means of the conservation of energy for the Hartree equation ( 5).This explicit control is useful in particular to assess the dependence in ̵ h of the convergence rate for the mean-field limit obtained in Corollary (4.2).
Clearly, the convergence rate for the quantum mean-field limit in Corollary 4.2 is not uniform in the semiclassical regime, in the first place because of the factor 3 ̵ h on the right hand side of the upper bound for F N ∶m (t) − R(t) ⊗m  1 , which comes from the i ̵ h∂ t part of the quantum dynamical equation.However, one should expect that the function ℓ(t), or at least the upper bound for ℓ(t) obtained in ( 2), grows at least as 1 ̵ h, since it involves ∇ x ψ(t, ⋅) L 2 , expected to be of order 1 ̵ h for semiclassical wave functions ψ (think for instance of a WKB wave function, or of a Schrödinger coherent state).
We shall discuss this issue by means of the conservation of energy satisfied by the Hartree solution ψ (see formula (5.2) in [3]): Observe that (where F designates the Fourier transform on R d ), so that the conservation of mass and energy for the Hartree solution implies that Typical states used in the semiclassical regime (WKB or coherent states, for instance) satisfy ̵ h ∇ψ in L 2 = O(1).Thus, in that case Things become worse if the potential energy is a priori of indefinite sign.With (28), the energy conservation implies that and thus Therefore, the exponential amplifying factor in Corollary 4.2 is exp(Kt ̵ h 5 2 ) in the first case, and exp(Kt ̵ h 3 ) in the second.These elementary remarks suggest that Pickl's clever method for proving the quantum mean-field limit with singular potentials including the Coulomb potential (see [16,14]) is not expected to give uniform convergence rates (as in [7,8] in the case of regular interaction potentials) for the mean field limit in the semiclassical regime.

Proof of part (1) in Theorem 4.1
For each σ ∈ S N and each ] is a bounded operator on H.According to formula (25) in [8], denoting by V kl the multiplication operator The core result in the proof of Theorem 3.1 is that the function , and more generally, using a spectral decomposition of the trace-class operator With the definition of C in Theorem 3.1, we conclude that the operator for each operator F N ∈ L(H N ) satisfying (31).One easily checks that for all D N ∈ L(H N ) satisfying (32).Since any trace-class operator on H N is a linear combination of 4 density operators, we conclude that On the other hand Finally, by condition ( 17) on V , one has so that, returning to (34), one arrives at the equality which proves part (1) in Theorem 4.1.Remark.In [8], the equality This argument cannot be used here since V ∉ F L 1 (R 3 ).Besides, the definition of the operator C(V, M N (t), M N (t))(R(t)) in Theorem 3.1 makes critical use of the fact that R(t) = ψ(t, ⋅)⟩⟨ψ(t, ⋅) with ψ(t, ⋅) ∈ L 2 ∩ L ∞ (R 3 ).This is the reason for the rather lengthy justification of (33) in this section.

Proof of Lemma 4.4
In the sequel, we seek to "simplify" the expression of the interaction operator This will lead to rather involved computations which do not seem much of a simplification.However, we shall see that the final result of these computations, reported in Lemma 4.4, although algebraically more cumbersome, has better analytical properties.
and observe that ]dω .All the terms in the right hand side of the equality above are either similar to the one considered in Theorem 3.1, or of the type denoted (III) in the Remark following Theorem 3.1.
An elementary computation shows that, for all ) -see formula before (41) on p. 1041 in [8].On the other hand (12).With the formula (36), we conclude that (39) . By (35), one can further simplify the term Observe again that all the integrals in the right hand side of the equalities defining T 1 and T 3 are of the form defined in Theorem 3.1, or of the form (I), (II) or (III), or their adjoint, in the Remark following Theorem 3.1. That follows from (12) and the definition (23).This concludes the proof of Lemma 4.4.

Proof of Lemma 4.5
In the sequel, we shall estimate these four terms in increasing order of technical difficulty.-see the formula following (41) on p. 1041 in [8].Thus , where the equality follows from the fact that R(t) = R(t) * , which implies that (41) On the other hand, by Jensen's inequality for each A ∈ L(H), so that (44) ) .Then (41) and (44) imply that so that T * 4 = −T 4 .Hence ±iT 4 are self-adjoint operators on H N , so that (45) One has By cyclicity of the trace, for each F in N satisfying (31), denoting .
By (44), Next we use the following elementary observation.Proof.Indeed, we seek to prove that ⟨Ψ T Ψ⟩ ≥ 0 for each Ψ ∈ H N .
For each Ψ ∈ H N such that Ψ N H N = 1, set Then F satisfies (31), so that Therefore, by cyclicity of the trace, for each and we study the quantity ) . where so that, by the Cauchy-Schwarz inequality, .
First, one has (the second equality follows from the fact that J k (P (t)) commutes with Π N (t) and Π N (t) −1 ), so that The inequality above follows from the fact that trace H N (Π N (t) −1 (J k P (t)) 2 F N (t)) = trace H N (F N (t) For instance, if F in N = ψ in ⟩⟨ψ in ⊗N with ψ in ∈ H and ψ in H = 1, one has α N (0) = trace H N (R(0) ⊗N M in N (P (0))) = trace H (R(0)P (0)) = 0 , so that 8.2.2.Pickl's functional and the trace norm.How the inequality above implies the mean-field limit is explained by the following lemma, which recaps the results stated as Lemmas 2.1 and 2.2 in [14], and whose proof is given below for the sake of keeping the present paper self-contained.If F in N ∈ L(H N ) satisfies (31), for each m = 1, . . ., N , we denote by F N ∶m (t) the m-particle reduced density operator deduced from F N (t) = U N (t)F in N U N (t) * , i.e. trace Hm (F N ∶m (t)A 1 ⊗ . . .⊗ A m ) = trace H N (F N (t)(J 1 A 1 ) . . .(J m A m )) for all A 1 , . . ., A m ∈ L(H).This completes the proof of Corollary 4.2.

Lemma 4 . 5 .
Under the same assumptions and with the same notations as in Theorem 4.1, set

7. 1 .
Bound for T 4 .The easiest term to treat is obviously T 4 .We first recall that (40) M N (t)(A) ≤ A for each A ∈ L(H)