Abstract
This paper discusses the mean-field limit for the quantum dynamics of N identical bosons in \({\textbf{R}}^3\) interacting via a binary potential with Coulomb-type singularity. Our approach is based on the theory of quantum Klimontovich solutions defined in Golse and Paul (Commun Math Phys 369:1021–1053, 2019) . 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 Kato (Trans Am Math Soc 70:195–211, 1951). 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 Pickl (Lett Math Phys 97:151-164, 2011), resulting in a convergence rate estimate for the quantum mean-field limit leading to the time-dependent Hartree equation.
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1 Introduction and notation
In classical mechanics, the motion equations for a system of N identical point particles of mass m with positions \(q_j(t)\in {\textbf{R}}^3\) and momenta \(p_j(t)\in {\textbf{R}}^3\) for all \(j=1,\ldots ,N\) are:
where the N-particle classical Hamiltonian is
Assuming that \(V\in C^{1,1}({\textbf{R}}^3)\), this differential system has a unique global solution for all initial data. If V is even,Footnote 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”. They are discussed in detail in Klimontovich’s own book on the statistical mechanics of plasmas [14]. Thus, if \(\mu _N(0)\rightarrow f^{in}\hbox {d}x\hbox {d}\xi \) weakly in the sense of probability measures as \(N\rightarrow \infty \), where \(f^{in}\) is a probability density on \({\textbf{R}}^3_x\times {\textbf{R}}^3_\xi \), one has
for all \(t\ge 0\) as \(N\rightarrow \infty \), where f is the solution of the Vlasov equation
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 [1, 2, 6, 22], or in the second quantization setting [20]. 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 [21] 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
In [15, 18], an original method, close to the second quantization approach in [20], 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 \(\hbar >0\). However, the discussion in [8] only considers regular potentials (specifically \({\partial }^{\alpha }V\in {\mathcal {F}}L^1({\textbf{R}}^d)\) for \(|{\alpha }|\le 3+[d/2]\)). Even writing the equation analogous to (3) satisfied by the quantum analogue of the phase-space empirical measure requires \(V\in {\mathcal {F}}L^1({\textbf{R}}^d)\) 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 Sect. 3), and
-
(b)
to explain how the ideas in [15, 18] can be couched in terms of the formalism of quantum empirical measures defined in [8] (see Theorem 4.1 and Corollary 4.2 in Sect. 4).
Specifically, we prove an inequality between operators on the N-particle Hilbert space, of which the key estimates in [15, 18] 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 Sects. 3 and 4, respectively. The proofs of these results are given in the subsequent sections.
All the discussions in this paper apply to the mean-field limit in the case of bosons only—in particular, the analysis proposed here is not uniform in \(\hbar \), exactly as in [15, 18], and at variance with [8]. This precludes using the distinguished limit \(\hbar \sim N^{-1/3}\) that is typical of the mean-field limit in the case of fermions.
2 Quantum Klimontovich solutions
Consider the quantum N-body Hamiltonian
on \({\mathfrak {H}}_N:={\mathfrak {H}}^{\otimes N}\simeq L^2({\textbf{R}}^{3N})\), where \({\mathfrak {H}}:=L^2({\textbf{R}}^3)\). Henceforth, it is assumed that V is a real-valued function such that \({\mathcal {H}}_N\) has a (unique) self-adjoint extension to \({\mathfrak {H}}_N\), still denoted by \({\mathcal {H}}_N\). 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
In particular, these conditions include the (repulsive) Coulomb potential in \({\textbf{R}}^3\). In fact, \({\mathcal {H}}_N\) has a self-adjoint extension to \({\mathfrak {H}}_N\) under a condition slightly more general than Kato’s original assumption recalled above:
(see Theorem X.16 and Example 2 in [19], and Theorem V.9 with \(m=1\) in [17]).
In the sequel, we adopt the notation in [8]. In particular, we set
and
The dynamics of the morphism \({\mathcal {M}}_N^{in}\) is defined by conjugation with the N-particle dynamics as follows: for each \(A\in {\mathcal {L}}({\mathfrak {H}})\),
Since \({\mathcal {H}}_N\) is self-adjoint, \(t\mapsto {\mathcal {U}}_N(t)\) is a unitary group by Stone’s theorem. The time-dependent morphism \(t\mapsto {\mathcal {M}}_N(t)\in {\mathcal {L}}({\mathcal {L}}({\mathfrak {H}}),{\mathcal {L}}({\mathfrak {H}}_N))\) is henceforth referred to as the quantum Klimontovich solution.
Assume henceforth that V is even:
The first main result in [8] (Theorem 3.3) is that, if \({{\hat{V}}}\in L^1({\textbf{R}}^d)\), the quantum Klimontovich solution \({\mathcal {M}}_N(t)\) satisfies
where \(K=-\tfrac{1}{2}{\hbar }^2{\Delta }\) is the quantum kinetic energy, and where
for each unbounded self-adjoint operator T on \({\mathfrak {H}}\), each \(A\in {\mathcal {L}}({\mathfrak {H}})\) satisfying the condition \([T,A]\in {\mathcal {L}}({\mathfrak {H}})\), and each \(\Lambda \in {\mathcal {L}}({\mathcal {L}}({\mathfrak {H}}),{\mathcal {L}}({\mathfrak {H}}_N))\). Moreover,
for each \(A\in {\mathcal {L}}({\mathfrak {H}})\) and each \(\Lambda _1,\Lambda _2\in {\mathcal {L}}({\mathcal {L}}({\mathfrak {H}}),{\mathcal {L}}({\mathfrak {H}}_N))\), where \(E_{\omega }\in {\mathcal {L}}({\mathfrak {H}})\) is the operator defined by
Since the integrand of the right-hand side of (15) takes its values in the non-separable space \({\mathcal {L}}({\mathfrak {H}}_N)\), it is worth mentioning that this integral is a weak integral for the ultraweak topology in \({\mathcal {L}}({\mathfrak {H}}_N)\): see footnote 3 on p. 1032 in [8]. (We recall that the ultraweak topology on the algebra \({\mathcal {L}}(H)\) of bounded operators on the Hilbert space H is the topology defined by the family of seminorms
as T runs through the set of trace-class operators on H: see for instance §4.6.10 in chapter 4 of the book [16].)
At variance with the classical case recalled in (3), the differential equation (13) satisfied by the quantum Klimontovich solution \(t\mapsto {\mathcal {M}}_N(t)\) 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 \(\psi \) is a solution of the time-dependent Hartree equation (5) satisfying the normalization condition
the time-dependent morphism \(t\mapsto {\mathcal {R}}(t)\in {\mathcal {L}}({\mathcal {L}}({\mathfrak {H}}),{\mathcal {L}}({\mathfrak {H}}_N))\) defined by the formulaFootnote 2
is a solution of (13).
3 Extending the definition of \({\mathcal {C}}(V,{\mathcal {M}}_N(t),{\mathcal {M}}_N(t))\) when \(V\notin {\mathcal {F}}L^1({\textbf{R}}^3)\)
Our first task is to extend the definition (15) of the term \({\mathcal {C}}(V,{\mathcal {M}}_N(t),{\mathcal {M}}_N(t))\) to a more general class of potentials V, including the Coulomb potential in \({\textbf{R}}^3\).
Since
the idea is to define
for all \(\Phi _N^{in},\Psi _N^{in}\in {\mathfrak {H}}_N\), where
and to take advantage of the decay of \(S_N[\phi ]\) in \({\omega }\), assuming that \(\phi \) is regular enough. Our argument does not use any regularity on \(\Phi _N^{in}\) or \(\Psi _N^{in}\). 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 \({\mathcal {U}}_N(t)\Phi _N^{in}\) or \({\mathcal {U}}_N(t)\Psi _N^{in}\), as observed in Note V.10 of [17].
Our first main result in this paper is the following result, leading to a definition of \({\mathcal {C}}(V,{\mathcal {M}}_N(t),{\mathcal {M}}_N(t))(|\phi \rangle \langle \phi |)\) in the case of singular, Coulomb-like potentials V, and for bounded wave functions \(\phi \). 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 \({\textbf{R}}^3\) satisfying the parity condition (12), and
For each \(\phi \in L^2\cap L^\infty ({\textbf{R}}^3)\) and each \(\Psi _N\in {\mathfrak {H}}_N\), the function
The interaction operator \({\mathcal {C}}(V,{\mathcal {M}}_N(t),{\mathcal {M}}_N(t))(|\phi \rangle \langle \phi |)\) is defined by the formula
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 \({\textbf{R}}\) to \({\mathcal {L}}({\mathfrak {H}}_N)\) endowed with the ultraweak topology, which is moreover bounded on \({\textbf{R}}\) for the operator norm on \({\mathcal {L}}({\mathfrak {H}}_N)\).
Obviously, condition (17) is stronger than Kato’s condition (8). However, the repulsive Coulomb potential \(z\mapsto 1/|z|\) in \({\textbf{R}}^3\) obviously satisfies (17), since its Fourier transform \(\zeta \mapsto {C}/|\zeta |^2\) belongs to \(L^1({\textbf{R}}^3)+L^2({\textbf{R}}^3)\). In particular, \({\mathcal {H}}_N\) has a self-adjoint extension to \({\mathfrak {H}}_N\) under condition (17)
Proof
Assuming that \(\Psi _N^{in}\in {\mathfrak {H}}_N\), one has
Therefore, we henceforth forget the time dependence in \(\Psi _N(t,\cdot )={\mathcal {U}}_N(t)\Psi _N^{in}\), which will be henceforth denoted \(\Psi _N\equiv \Psi _N(x_1,\ldots ,x_N)\).
Observe first that
since
Without loss of generality, consider the term
where
with the notation
We shall prove that \(F\in L^1({\textbf{R}}^3)\cap L^2({\textbf{R}}^3)\), so that \({\hat{F}}\in L^2({\textbf{R}}^3)\cap C_0({\textbf{R}}^3)\).
First
where the last inequality is the Cauchy–Schwarz inequality for the inner integral.
On the other hand,
and
with
so that
and
Hence,
so that
Therefore,
so that \({\omega }\mapsto {\hat{F}}(-{\omega })\) belongs to \(L^2({\textbf{R}}^d)\) by Plancherel’s theorem. Hence, for each \(k\not =l\in \{1,\ldots ,N\}\), one has
Hence,
Since \(S_N[\phi ]({\omega })^*=-S_N[\phi ]({\omega })\in {\mathcal {L}}({\mathfrak {H}}_N)\) for each \({\omega }\in {\textbf{R}}^3\) and \({\hat{V}}\) is even because of (12), the formula
defines
for each \(t\in {\textbf{R}}\) by polarization, and the function
is bounded on \({\textbf{R}}\) with values in \({\mathcal {L}}({\mathfrak {H}}_N)\) for the norm topology, and continuous on \({\textbf{R}}\) with values in \({\mathcal {L}}({\mathfrak {H}}_N)\) 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 \({\mathcal {L}}({\mathfrak {H}}_N)\). This last point is Proposition 4.6.14 in chapter 4 of [16], where the weak operator and the ultraweak topologies are referred to as the weak and the \({\sigma }\)-weak topologies, respectively. \(\square \)
Remark
In the sequel, we shall also need to consider terms of the form
where \(A,B\in {\mathcal {L}}({\mathfrak {H}})\).
The term (III) is the easiest of all. Indeed,
and since \(V\in L^2({\textbf{R}}^3)+C_b({\textbf{R}}^3)\) while \(\phi \in L^1\cap L^\infty ({\textbf{R}}^3)\), one has \(V\star |\phi |^2\in C_b({\textbf{R}}^3)\), so that \(A(V\star |\phi |^2)B\in {\mathcal {L}}({\mathfrak {H}})\).
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,
with
so that
Then,
Besides
so that
On the other hand,
where
so that
Thus, we have proved that \(F_1\in L^1\cap L^2({\textbf{R}}^d)\), and since \({\hat{V}}\in L^2({\textbf{R}}^d)+L^1({\textbf{R}}^d)\), the product \({\hat{V}}{\hat{F}}\in L^1({\textbf{R}}^d)\), which leads to a definition of (I).
The case of (II) is essentially similar. Observe that
where
And
Observe that
so that
while
with a similar conclusion for \(F_3\). On the other hand
so that
Therefore, the map
belongs to \(L^2\cap L^\infty ({\textbf{R}}^3)\). Since \({\hat{V}}\in L^2({\textbf{R}}^3)+L^1({\textbf{R}}^3)\), this implies that
belongs to \(L^1({\textbf{R}}^3)\), thereby leading to a definition of (II).
4 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 \(\phi ^{in}\in H^2({\textbf{R}}^3)\), there exists a unique solution \(\phi \in C({\textbf{R}},H^2({\textbf{R}}^3))\) 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 [18], with the choice \(n(k):=k/N\), in the notation of [18]):
for all \(\Psi _N\in {\mathfrak {H}}_N\) and \(\psi \in {\mathfrak {H}}\).
Assuming that \(\psi \equiv \psi (t,x)\) is a solution of (5) while \(\Psi _N(t,\cdot ):={\mathcal {U}}_N(t)\Psi _N^{in}\), Pickl studies in section 2.1 of [18] the time-dependent function \(t\mapsto {\alpha }_N(\Psi _N(t,\cdot ),\psi (t,\cdot ))\), and proves that it satisfies some Gronwall inequality.
Observe first that Pickl’s functional \({\alpha }_N(\Psi _N(t,\cdot ),\psi (t,\cdot ))\) can be recast in terms of the quantum Klimontovich solution \({\mathcal {M}}_N(t)\) as follows
This identity suggests therefore to deduce from (13) and (5) the expression of
in terms of the interaction operator \({\mathcal {C}}\) defined in (15).
This is done in the first part of the next theorem, which is our second main result in this paper.
Theorem 4.1
Assume that the (real-valued) interaction potential V, viewed as an (unbounded) multiplication operator acting on \({\mathfrak {H}}:=L^2({\textbf{R}}^3)\), satisfies the parity condition (12) and (17).
Let \(\psi ^{in}\in H^2({\textbf{R}}^3)\) satisfy \(\Vert \psi ^{in}\Vert _{{\mathfrak {H}}}=1\), let \(\psi \) be the solution of the Cauchy problem (5) for the time-dependent Hartree equation, and set
Then,
-
(1)
the N-body quantum Klimontovich solution \(t\mapsto {\mathcal {M}}_N(t)\) satisfies
$$\begin{aligned} i{\hbar }{\partial }_t({\mathcal {M}}_N(t)(P(t)))={\mathcal {C}}(V,{\mathcal {M}}_N(t)-{\mathcal {R}}(t),{\mathcal {M}}_N(t))(R(t)), \end{aligned}$$where
$$\begin{aligned} {\mathcal {R}}(t)A:=\langle \psi (t,\cdot )|A|\psi (t,\cdot )\rangle I_{{\mathfrak {H}}_N}={\text {trace}}_{{\mathfrak {H}}}(R(t)A)I_{{\mathfrak {H}}_N}; \end{aligned}$$ -
(2)
the operator \({\mathcal {C}}(V,{\mathcal {M}}_N(t)-{\mathcal {R}}(t),{\mathcal {M}}_N(t))(P(t))\) is skew-adjoint on \({\mathfrak {H}}_N\) and satisfies the operator inequality
$$\begin{aligned} \pm i{\mathcal {C}}(V,{\mathcal {M}}_N(t)-{\mathcal {R}}(t),{\mathcal {M}}_N(t))(R(t))\le 6L(t)\left( {\mathcal {M}}_N(t)(P(t))+\tfrac{2}{N}I_{{\mathfrak {H}}_N}\right) , \end{aligned}$$
whereFootnote 3
and where \(C_S\) is the norm of the Sobolev embedding \(H^2({\textbf{R}}^3)\subset L^\infty ({\textbf{R}}^3)\).
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 \({\mathcal {M}}_N(t)\).
Indeed, we deduce from parts (1) and (2) in Theorem 4.1 the operator inequality
Then, evaluating both sides of this inequality on the initial N-particle state \(\Psi _N^{in}\) and taking into account the identity (18) lead to the Gronwall inequality
satisfied by Pickl’s functional \({\alpha }_N({\mathcal {U}}_N(t)\Psi ^{in}_N,\psi (t,\cdot ))\). This last inequality corresponds to inequality (11) and Lemma 3.2 in [18].
In the sequel, we shall denote by \({\mathcal {L}}^p({\mathfrak {H}})\) for \(p\ge 1\) the Schatten two-sided ideal of \({\mathcal {L}}({\mathfrak {H}})\) consisting of operators T such that
In particular, \({\mathcal {L}}^1({\mathfrak {H}})\) is the set of trace-class operators on \({\mathfrak {H}}\) and \(\Vert \cdot \Vert _1\) the trace norm, while \({\mathcal {L}}^2({\mathfrak {H}})\) is the set of Hilbert–Schmidt operators on \({\mathfrak {H}}\) and \(\Vert \cdot \Vert _2\) the Hilbert–Schmidt norm.
Corollary 4.2
Under the same assumptions and with the same notations as in Theorem 4.1, consider the N-body wave function \(\Psi _N(t,\cdot ):={\mathcal {U}}_N(t)(\psi ^{in})^{\otimes N}\), and the N-body density operator \(F_N(t):=|\Psi _N(t,\cdot )\rangle \langle \Psi _N(t,\cdot )|\). For each \(m=1,\ldots ,N\), the m-particle reduced density operator \(F_{N:m}(t)\), defined by the identity
for all \(A_1,\ldots ,A_m\in {\mathcal {L}}({\mathfrak {H}})\), satisfies
with L given by (20).
As already mentioned at the end of the introduction (Sect. 1), the results discussed here apply to the case of bosons, and one reason for this is that the analysis in the present paper is not uniform in \(\hbar \) (exactly as in [18]). In particular, one cannot consider the distinguished limit \(h\sim N^{-1/3}\) which is typical of the mean-field limit for fermions. Another reason is that the initial condition for \(\Psi _N\) is of the form \(\Psi _N(0,\cdot )=(\psi ^{in})^{\otimes N}\), which is an example of pure state for bosons.
Let us briefly indicate how one arrives at the operator inequality in part (2) of Theorem 4.1. Let \(\Lambda _1,\Lambda _2\in {\mathcal {L}}({\mathcal {L}}({\mathfrak {H}}),{\mathcal {L}}({\mathfrak {H}}_N))\) be such that
for all \(\Psi _N\in {\mathfrak {H}}_N\). For all V satisfying (12) and (17), define \({\mathcal {T}}(V,{\Lambda }_1,{\Lambda }_2)\in {\mathcal {L}}({\mathfrak {H}}_N)\) by polarization of the formula
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 \(A\in {\mathcal {L}}({\mathfrak {H}})\), denote by \(\Lambda _j({\bullet }A)\) and \(\Lambda _j(A{\bullet })\) the linear maps
respectively. If \(A\in {\mathcal {L}}({\mathfrak {H}})\) is such that \(\Lambda _1,\Lambda _2({\bullet }A)\) and \(\Lambda _2(A{\bullet }),\Lambda _1\) satisfy (22), then one has
Lemma 4.3
Let \(\Lambda _1,\Lambda _2\in {\mathcal {L}}({\mathcal {L}}({\mathfrak {H}}),{\mathcal {L}}({\mathfrak {H}}_N))\) be \(*\)-homomorphisms, in other words
for all \(A\in {\mathcal {L}}({\mathfrak {H}})\). Assume that \(\Lambda _1,\Lambda _2\) satisfy (22). Then
Proof
Indeed
where the first equality follows from the fact that \(\Lambda _1\) and \(\Lambda _2\) are *-homomorphisms, while the second equality uses the fact that \({\hat{V}}\) is real-valued, since V is real-valued and even. \(\square \)
An easy consequence of (24) and of this lemma is that, for each \(A=A^*\in {\mathcal {L}}({\mathfrak {H}})\) such that \(\Lambda _1,\Lambda _2\in {\mathcal {L}}({\mathcal {L}}({\mathfrak {H}}),{\mathcal {L}}({\mathfrak {H}}_N))\) are \(*\)-homomorphisms such that \(\Lambda _1,\Lambda _2({\bullet }A)\) satisfy (22), then
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
with
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.
Lemma 4.5
Under the same assumptions and with the same notations as in Theorem 4.1, set
Then
Remarks on \(\ell (t)\) in (26) and L(t) in (20).
-
(1)
If V satisfies condition (17) in Theorem 3.1, then \(V\!\in L^2({\textbf{R}}^3)\!+\!L^\infty ({\textbf{R}}^3)\), so that \(V^2\!\in L^1({\textbf{R}}^d)+L^\infty ({\textbf{R}}^d)\). Thus \((V^2)_{R(t)}\), which is the multiplication operator by the function \(V^2\star |\psi (t,\cdot )|^2\), satisfies
$$\begin{aligned} \begin{aligned} \ell (t)^2:=\Vert V^2\star |\psi (t,\cdot )|^2\Vert _{L^\infty ({\textbf{R}}^3)}\le&\Vert V^2\Vert _{L^1({\textbf{R}}^3)+L^\infty ({\textbf{R}}^3)}\Vert \psi (t,\cdot )\Vert ^2_{L^1\cap L^\infty ({\textbf{R}}^3)} \\ \le&2\Vert V\Vert ^2_{L^1({\textbf{R}}^3)+L^\infty ({\textbf{R}}^3)}\max (1,\Vert \psi (t,\cdot )\Vert _{L^\infty ({\textbf{R}}^3)})^2 \\ \le&2C^2_S\Vert V\Vert ^2_{L^1({\textbf{R}}^3)+L^\infty ({\textbf{R}}^3)}\Vert \psi (t,\cdot )\Vert _{H^2({\textbf{R}}^3)}^2 \end{aligned} \end{aligned}$$where we recall that \(C_S\) is the norm of the Sobolev embedding \(H^2({\textbf{R}}^3)\subset L^\infty ({\textbf{R}}^3)\).
-
(2)
If V satisfies (8), then \(\Vert V(I-{\Delta })^{-1}\Vert \le M\) for some positive constant M (see the discussion in §5.3 of chapter V in [13], so that
$$\begin{aligned} V^2\le M^2(I-{\Delta })^2. \end{aligned}$$In this remark, we shall make a slightly more restrictive assumption, namely that \(V^2\) satisfies
$$\begin{aligned} V^2\le C(I-{\Delta })\,. \end{aligned}$$(27)In space dimension \(d=3\), the Hardy inequality, which can be put in the formFootnote 4
$$\begin{aligned} \frac{1}{|x|^2}\le 4(-{\Delta }) \end{aligned}$$implies that the Coulomb potential satisfies the assumption above on V. If the potential V satisfies the (operator) inequality (27), then
$$\begin{aligned} \begin{aligned} 0\le (V^2)_{R(t)}(x)&=\int _{{\textbf{R}}^d}V^2(y)|\psi (t,x-y)|^2\hbox {d}y=\langle \psi (t,x-\cdot )|V^2|\psi (t,x-\cdot )\rangle \\&\le C\langle \psi (t,x-\cdot )|(I-{\Delta })|\psi (t,x-\cdot )\rangle =C\Vert \psi (t,x-\cdot )\Vert _{L^2}^2\\&\quad +C\Vert {\nabla }\psi (t,x-\cdot )\Vert _{L^2}^2 \\&=C\Vert \psi (t,\cdot )\Vert _{L^2}^2+C\Vert {\nabla }\psi (t,\cdot )\Vert _{L^2}^2. \end{aligned} \end{aligned}$$Thus, if \(\psi \in C({\textbf{R}};H^1({\textbf{R}}^d))\) is a solution of the Hartree equation,
$$\begin{aligned} \ell (t)\le \sqrt{C}\Vert \psi (t,\cdot )\Vert _{H^1({\textbf{R}}^3)}. \end{aligned}$$ -
(3)
A bound on \(\ell (t)\) in terms of \(\Vert \psi (t,\cdot )\Vert _{H^1({\textbf{R}}^3)}\) instead of \(\Vert \psi (t,\cdot )\Vert _{H^2({\textbf{R}}^3)}\) 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 \(\hbar \) 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 \(3/{\hbar }\) on the right-hand side of the upper bound for \(\Vert F_{N:m}(t)-R(t)^{\otimes m}\Vert _1\), which comes from the \(i{\hbar }{\partial }_t\) part of the quantum dynamical equation.
However, one should expect that the function \(\ell (t)\), or at least the upper bound for \(\ell (t)\) obtained in (2), grows at least as \(1/{\hbar }\), since it involves \(\Vert {\nabla }_x\psi (t,\cdot )\Vert _{L^2}\), expected to be of order \(1/{\hbar }\) for semiclassical wave functions \(\psi \) (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 \(\psi \) (see formula (5.2) in [3]):
Observe that
so that
Thus, if \(V\ge 0\), or if \({\hat{V}}\ge 0\), one has
(where \({\mathcal {F}}\) designates the Fourier transform on \({\textbf{R}}^d\)), so that the conservation of mass and energy for the Hartree solution implies that
In that case
Typical states used in the semiclassical regime (WKB or coherent states, for instance) satisfy \({\hbar }\Vert {\nabla }\psi ^{in}\Vert _{L^2}=O(1)\). Thus, in that case
Things become worse if the potential energy is a priori of indefinite sign. With (28), the energy conservation implies that
so that
and thus
Therefore, the exponential amplifying factor in Corollary 4.2 is \(\exp (Kt/{\hbar }^{5/2})\) in the first case, and \(\exp (Kt/{\hbar }^3)\) 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 [15, 18]) 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.
5 Proof of part (1) in Theorem 4.1
For each \({\sigma }\in {\mathfrak {S}}_N\) and each \(\Psi _N\in {\mathfrak {H}}_N\), set
Since \(\psi (t,\cdot )\in H^2({\textbf{R}}^3)\), the commutator \([{\Delta },R(t)]\) is a bounded operator on \({\mathfrak {H}}\). According to formula (25) in [8], denoting by \(V_{kl}\) the multiplication operator
one has
for all \(F_N\in {\mathcal {L}}({\mathfrak {H}}_N)\) such that
The core result in the proof of Theorem 3.1 is that the function
for each \(k\not =l\in \{1,\ldots ,N\}\). Since \({\hat{V}}\in L^1({\textbf{R}}^3)+L^2({\textbf{R}}^3)\), this has led us to define
and more generally, using a spectral decomposition of the trace-class operator \(F_N\),
with
Since \(U_{\sigma }F_NU_{\sigma }^*=F_N\) for all \({\sigma }\in {\mathfrak {S}}_N\), for each \(m\not =n\in \{1,\ldots ,N\}\), one has
With the definition of \({\mathcal {C}}\) in Theorem 3.1, we conclude that the operator
satisfies
for each operator \(F_N\in {\mathcal {L}}({\mathfrak {H}}_N)\) satisfying (31). One easily checks that
Let \(D_N\in {\mathcal {L}}({\mathfrak {H}}_N)\) be a density operator on \({\mathfrak {H}}_N\), i.e.
Obviously
satisfies (31), so that
for all \(D_N\in {\mathcal {L}}({\mathfrak {H}}_N)\) satisfying (32). Since any trace-class operator on \({\mathfrak {H}}_N\) is a linear combination of 4 density operators, we conclude that
so that
On the other hand,
so that
Finally, by condition (17) on V, one has
so that
Hence
so that, returning to (34), one arrives at the equality
which proves part (1) in Theorem 4.1.
Remark
In [8], the equality
is proved for all \(A\in {\mathcal {L}}({\mathfrak {H}})\) such that \([{\Delta },A]\in {\mathcal {L}}({\mathfrak {H}})\), assuming that \(V\in {\mathcal {F}}L^1({\textbf{R}}^3)\). This argument cannot be used here since \(V\notin {\mathcal {F}}L^1({\textbf{R}}^3)\). Besides, the definition of the operator \({\mathcal {C}}(V,{\mathcal {M}}_N(t),{\mathcal {M}}_N(t))(R(t))\) in Theorem 3.1 makes critical use of the fact that \(R(t)=|\psi (t,\cdot )\rangle \langle \psi (t,\cdot )|\) with \(\psi (t,\cdot )\in L^2\cap L^\infty ({\textbf{R}}^3)\). This is the reason for the rather lengthy justification of (33) in this section.
6 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.
6.1 A first simplification
First we decompose \(E_{\omega }R(t)\) and \(R(t)E_{\omega }\) in the terms \({\mathcal {M}}_N(t)(E_{\omega }R(t))\) and \({\mathcal {M}}_N(t)(R(t)E_{\omega })\) as
and observe that
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 \({\omega }\in {\textbf{R}}^d\),
Recall indeed that, for each \(A,B\in {\mathcal {L}}({\mathfrak {H}})\), one has
— see formula before (41) on p. 1041 in [8]. On the other hand,
so that
Besides
Indeed,
where we recall that \({\mathcal {U}}_N(t):=e^{-it{\mathcal {H}}_N/{\hbar }}\), while \({\mathcal {H}}_N\) is the N-body Hamiltonian.
Therefore,
in view of (12). With the formula (36), we conclude that
6.2 A second simplification
Next we decompose \(E_{\omega }^*\) in \({\mathcal {M}}_N(t)(E_{\omega }^*)\) as
The identity (37) shows that
and hence
Therefore,
since \(R(t)P(t)=P(t)R(t)=0\). Thus,
Using again (12) implies that
so that
By (35), one can further simplify the term
Finally
with
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.
7 Proof of Lemma 4.5
In the sequel, we shall estimate these four terms in increasing order of technical difficulty.
7.1 Bound for \(T_4\)
The easiest term to treat is obviously \(T_4\). We first recall that
— 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
On the other hand, by Jensen’s inequality
so that
and therefore
Finally, we recall that
for each \(A\in {\mathcal {L}}({\mathfrak {H}})\), so that
Then, (41) and (44) imply that
so that \(T_4^*=-T_4\). Hence, \(\pm iT_4\) are self-adjoint operators on \({\mathfrak {H}}_N\), so that
7.2 Bound for \(T_2\)
Set
One has
Then
so that
By cyclicity of the trace, for each \(F_N^{in}\) satisfying (31), denoting
one has
so that
By (44),
so that
Thus,
so that
by (42).
Next we use the following elementary observation.
Lemma 7.1
Let \(T=T^*\in {\mathcal {L}}({\mathfrak {H}}_N)\) satisfy
for each \(F\in {\mathcal {L}}({\mathfrak {H}}_N)\) satisfying (31). Then \(T\ge 0\).
Proof
Indeed, we seek to prove that
For each \(\Psi \in {\mathfrak {H}}_N\) such that \(\Vert \Psi _N\Vert _{{\mathfrak {H}}_N}=1\), set
Then F satisfies (31), so that
since \(U_{\sigma }^*TU_{\sigma }=T\) for each \({\sigma }\in {\mathfrak {S}}_N\). Thus, \(\langle \Psi |T|\Psi \rangle \ge 0\) for each \(\Psi \in {\mathfrak {H}}_N\) such that \(\Vert \Psi _N\Vert _{{\mathfrak {H}}_N}=1\), and thus for each \(\Psi \in {\mathfrak {H}}_N\setminus \{0\}\) by normalization. \(\square \)
The inequality (48) implies that
and we conclude from Lemma 7.1 that
7.3 Bound for \(T_1\)
Next we estimate
Observe that
since \(P(t)=P(t)^2\), so that \(J_kP(t)=(J_kP(t))^2\). Then
so that
Hence, (12) implies that
On the other hand,
since \(J_k(P(t)E_{\omega }R(t))J_k(P(t))=0\), with \(V_{kl}\) defined as in (29).
Hence,
Therefore, by cyclicity of the trace, for each \(F_N^{in}\in {\mathcal {L}}({\mathfrak {H}}_N)\) satisfying (31), denoting \(F_N(t):={\mathcal {U}}_N(t)F_N^{in}{\mathcal {U}}_N(t)^*\), one has
so that
Finally
because of Lemma 4.3, so that
Since R(s) is a rank-one orthogonal projection
Thus,
In particular,
and since this inequality holds for each \(F_N^{in}\in {\mathcal {L}}_s({\mathfrak {H}}_N)\) such that \(F_N^{in}=(F_N^{in})^*\ge 0\), we conclude from Lemma 7.1 that
7.4 The operator \(\Pi _N\)
In order to treat the last term \(T_3\), we need the following auxiliary lemma—see the formula preceding (13) in [18].
Lemma 7.2
Let \(R=R^*\) be a rank-one projection on \({\mathfrak {H}}\) and let \(P:=I-R\). Set \(\Pi _N:={\mathcal {M}}_N^{in}P\). For each \(N>1\),
so that
In particular, there exists a pseudo-inverse \(\Pi _N^{-1}:\,({\text {Ker}}\Pi _N)^\perp \rightarrow ({\text {Ker}}\Pi _N)^\perp \), with extension by 0 on \({\text {Ker}}\Pi _N\) also (abusively) denoted \(\Pi _N\), such that
In [18], the definition of the pseudo-inverse of \(\Pi _N\) immediately follows from formula (6), which can be viewed as the spectral decomposition of \(\Pi _N\). The proof below is quite straightforward and avoids using the clever argument leading to formula (6) in [18], which is not entirely obvious unless one already knows the result.
Proof
That \(\Pi _N\) is self-adjoint is obvious by definition of \({\mathcal {M}}_N^{in}\). Then,
If \(X\in {\text {Ker}}\Pi _N\), one has, for each \(k=1,\ldots ,N\),
Hence,
so that
Thus, \({\text {Ker}}\Pi _N={\text {Ker}}(I-R^{\otimes N})\). Finally,
Therefore, for each \(X\in {\mathfrak {H}}_N\), one has
Since \(\Pi _N=\Pi _N^*\), one has
(see for instance Corollary 2.18 (iv) in chapter 2 of [5]). Since
and since one has obviously \(\Vert \Pi _N\Vert \le 1\), a straightforward density argument shows that
Hence
The existence of the pseudo-inverse \(\Pi _N^{-1}\) follows from this inequality. \(\square \)
7.5 Bound for \(T_3\)
Finally, we treat the term \(T_3\). Set
One easily checks that
At this point, we set \(\Pi _N(t):={\mathcal {M}}_N^{in}P(t)\) and use Lemma 7.2 to define the pseudo-inverse \(\Pi _N(t)^{-1}\). One has \(\Pi _N(t)=\Pi _N(t)^*\ge 0\), thus \(\Pi _N(t)^{-1}=(\Pi _N(t)^{-1})^*\ge 0\) on \({\text {Ker}}(I-R(t)^{\otimes N})\). Abusing the notation \(\Pi _N(t)^{-1/2}\) to designate the linear map \((\Pi _N(t)^{-1})^{1/2}\), we deduce from (54) that
so that
Hence,
and we study the quantity
where \(F_N(t)={\mathcal {U}}_N(t)F_N^{in}{\mathcal {U}}_N(t)^*\), for each \(F_N^{in}\in {\mathcal {L}}({\mathfrak {H}}_N)\) satisfying (31). Observe that
so that, by the Cauchy–Schwarz inequality,
First, one has
(the second equality follows from the fact that \(J_k(P(t))\) commutes with \(\Pi _N(t)\) and \(\Pi _N(t)^{-1}\)), so that
The inequality above follows from the fact that
On the other hand,
Now, \(m\notin \{k,l\}\) implies that \(J_mP(t)\) commutes with \(V_{kl}\), \(J_kR(t)\) and \(J_lR(t)\), so that
Therefore,
Now
according to Lemma 4.3. Thus, (55) implies that
According to (51)
so that
In particular,
Since this last inequality holds for each \(F_N^{in}\in {\mathcal {L}}({\mathfrak {H}}_N)\) satisfying (31), we deduce from Lemma 7.1 that
8 Proofs of part (2) in Theorem 4.1 and Corollary 4.2
8.1 Proof of part (2) in Theorem 4.1
Applying Lemma 4.4 shows that
With Lemma 4.5, this shows that
and that
It remains to bound the function
Since
one has
Minimizing \(\Vert V_1\Vert _{L^\infty ({\textbf{R}}^3)}+\Vert V_2\Vert _{L^2({\textbf{R}}^3)}\) over all possible decompositions of \(V=V_1+V_2\) as above, one has
8.2 Proof of Corollary 4.2
Pickl’s functional defined in [18] and recalled in formula (18) can be recast as
(see Definition 2.2 and formula (6) in [18]), where \(F_{N:1}(t)\) is the single-body reduced density operator deduced from
where \(F_N^{in}\in {\mathcal {L}}({\mathfrak {H}}_N)\) satisfies (31). Specifically, \(F_{N:1}(t)\) is defined by the formula
Since \(F_N(t)\) satisfies (31), it holds
This is Lemma 2.3 in [8], and the raison d’être of \({\mathcal {M}}_N(t)\). Thus, formula (18) and (58) are indeed equivalent.
8.2.1 The Gronwall inequality for Pickl’s functional
One deduces from part (2) in Theorem 4.1 that
This inequality implies that
Now, by cyclicity of the trace and (59),
so that, by Gronwall’s inequality,
For instance, if \(F_N^{in}=|\psi ^{in}\rangle \langle \psi ^{in}|^{\otimes N}\) with \(\psi ^{in}\in {\mathfrak {H}}\) and \(\Vert \psi ^{in}\Vert _{\mathfrak {H}}=1\), one has
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 [15], and whose proof is given below for the sake of keeping the present paper self-contained.
If \(F_N^{in}\in {\mathcal {L}}({\mathfrak {H}}_N)\) satisfies (31), for each \(m=1,\ldots ,N\), we denote by \(F_{N:m}(t)\) the m-particle reduced density operator deduced from \(F_N(t)={\mathcal {U}}_N(t)F_N^{in}{\mathcal {U}}_N(t)^*\), i.e.
for all \(A_1,\ldots ,A_m\in {\mathcal {L}}({\mathfrak {H}})\).
Lemma 8.1
The Pickl functional satisfies the inequality
Proof
Call \({\mathcal {P}}_-\) the spectral projection on the direct sums of eigenspaces of the trace-class operator \(F_{N:m}(t)-R(t)^{\otimes m}\) corresponding to negative eigenvalues. Then, the self-adjoint operator
must have only negative eigenvalues by definition of \({\mathcal {P}}_-\) and is obviously nonnegative on the orthogonal complement of \({\mathcal {P}}_-\psi (t,\cdot )^{\otimes m}\) in the range of \({\mathcal {P}}_-\). By definition of \({\mathcal {P}}_-\), this orthogonal complement must be \(\{0\}\). Hence, \({\mathcal {P}}_-\) is a rank-one projection, so that \(F_{N:m}(t)-R(t)^{\otimes m}\) has only one negative eigenvalue \(\lambda _0\), with all its other eigenvalues \({\lambda }_1,{\lambda }_2,\ldots \) being nonnegative. Since
one hasFootnote 5
Now \(F_{N:m}(t)\) is self-adjoint, and therefore
Hence,
Since \(R(t)=|\psi (t,\cdot )\rangle \langle \psi (t,\cdot )|\) is a self-adjoint projection
so that
Since \(F_{N}^{in}\) satisfies (31), the reduced m-particle operator \(F_{N:m}(t)\in {\mathcal {L}}({\mathfrak {H}}_m)\) also satisfies (31) (with N replaced by m), and hence,
by induction, which implies the inequality in the lemma. \(\square \)
With this lemma, the consequence of the Gronwall inequality above implies that, under the assumptions of Corollary 4.2,
This completes the proof of Corollary 4.2.
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Notes
In this case, the exclusion \(j\not =k\) in the right-hand side of Newton’s second law for \(\dot{p}_j(t)\) is useless since V even \(\implies {\nabla }V(0)=0\).
Throughout this paper, we adopt the Dirac bra–ket notation. Thus, a wave function \(\psi \in {\mathfrak {H}}\) viewed as a vector of the linear space \({\mathfrak {H}}\) is denoted \(|\psi \rangle \), whereas \(\langle \psi |\) designates the linear functional
$$\begin{aligned} \langle \psi |:\,\,{\mathfrak {H}}\ni \phi \mapsto \int _{{\textbf{R}}^d}\overline{\psi (x)}\phi (x)\hbox {d}x\in {\textbf{C}}. \end{aligned}$$If \(A\in {\mathcal {L}}({\mathfrak {H}})\), we denote
$$\begin{aligned} \langle \psi |A|\phi \rangle :=\int _{{\textbf{R}}^d}\overline{\psi (x)}(A\phi )(x)\hbox {d}x \end{aligned}$$and \(\langle \psi |\phi \rangle :=\langle \psi | I_{\mathfrak {H}}|\phi \rangle \) is the inner product on \({\mathfrak {H}}\).
We recall that, if E, F are Banach spaces
$$\begin{aligned} \Vert v\Vert _{E\cap F}:=\max (\Vert v\Vert _E,\Vert v\Vert _F), \end{aligned}$$and
$$\begin{aligned} \Vert f\Vert _{L^p({\textbf{R}}^d)+L^q({\textbf{R}}^d)}=\inf \{\Vert f_1\Vert _{L^p({\textbf{R}}^d)}\!+\!\Vert f_2\Vert _{L^q({\textbf{R}}^d)}\text { s.t. }f\!=\!f_1\!+\!f_2\text { with }f_1\!\in \! L^p({\textbf{R}}^d),\,f_2\!\in \! L^q({\textbf{R}}^d)\}. \end{aligned}$$To see that 4 is optimal, minimize in \({\alpha }>0\) the expression
$$\begin{aligned} \int _{{\textbf{R}}^3}\left| {\nabla }u+{\alpha }\frac{x}{|x|^2}u\right| ^2\hbox {d}x. \end{aligned}$$This observation is attributed to Seiringer on p. 35 in [20].
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Porat, I.B., Golse, F. Pickl’s proof of the quantum mean-field limit and quantum Klimontovich solutions. Lett Math Phys 114, 51 (2024). https://doi.org/10.1007/s11005-023-01768-7
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DOI: https://doi.org/10.1007/s11005-023-01768-7