Abstract
We give a rigorous derivation of the incompressible 2D Euler equation from the von Neumann equation with an external magnetic field. The convergence is with respect to the modulated energy functional, and implies weak convergence in the sense of measures. This is the semi-classical counterpart of theorem 1.5 in (Han-Kwan and Iacobelli in Proc Am Math Soc 149(7):3045–3061, 2021). Our proof is based on a Gronwall estimate for the modulated energy functional, which in turn heavily relies on a recent functional inequality due to (Serfaty in Duke Math J 169:2887–2935, 2020).
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1 Introduction
General background In classical mechanics, Newton’s second law of motion (also known as the N–body problem) is the following system of \(2\times 2\times N\) ODEs
where \(x_{k}(t)\in {\mathbb {R}}^{2}\) (or \(x_{k}(t)\in {\mathbb {T}}^{2}\) where \({\mathbb {T}}^{2}\) is the \(2-\)dimensional torus) and \(\xi _{k}(t)\in {\mathbb {R}}^{2}\) are called the position and momentum respectively. As customary, given a 2D vector \(\xi \in {\mathbb {R}}^{2}\), the vector \(\xi ^{\bot }\) designates the rotation by \(\frac{\pi }{2}\) of \(\xi \), i.e. \(\xi ^{\bot }:=(-\xi ^{2},\xi ^{1})\). From a physical perspective, the system (1.1) describes the dynamics of the positions and momenta of N identical point particles of unit mass in \({\mathbb {R}}^{2}\), interacting via an interaction potential V in the presence of fixed, constant, strong magnetic field. The parameter N represents the number of particles and should be thought of as very large, while \(\frac{1}{\varepsilon }\) represents the the strength of the magnetic field, and the parameter \(\varepsilon >0\) should be thought of as very small. We will be mostly concerned with the magnetic regime, although at times we may refer to the non magnetic regime as well, for which the corresponding dynamics are governed by the system
On the other hand the Vlasov–Poisson system (specified for 2D) with the same strong constant magnetic field on \([0,T]\times {\mathbb {R}}^{2}\times {\mathbb {R}}^{2}\) \(([0,T]\times {\mathbb {T}}^{2}\times {\mathbb {R}}^{2})\) reads:
The unknown is a time dependent probability density function \(f^{\epsilon }(t,x,\xi )\) on \({\mathbb {R}}^{2}\times {\mathbb {R}}^{2}\) \(({\mathbb {T}}^{2}\times {\mathbb {R}}^{2})\) and \(\rho ^{\epsilon }\) is a time dependent probability density function on \({\mathbb {R}}^{2}\) (\({\mathbb {T}}^{2}\)) defined by
Assuming that V is chosen so that the system (1.1) is well posed, denote by
the flow of the system (1.1) and set 
. Unless otherwise stated, we shall focus on the case where the position variable x is in \({\mathbb {R}}^{2}\). Denoting by \({\mathcal {P}}({\mathbb {R}}^{2}\times {\mathbb {R}}^{2})\) the space of probability measures on \({\mathbb {R}}^{2}\times {\mathbb {R}}^{2}\), we have the following classical result due to Klimontovich [15], which enables to construct N-dependent solutions to the Vlasov equation starting from the system (1.2).
Proposition 1.1
Let V be the 2D repulsive Coulomb potential, i.e. \(V(x):=-\frac{1}{2\pi }\log (|x|)\), and let \(Z_{N}^{t}\) be the flow of the system (1.2). For each \(N\ge 1\) the map \(t\mapsto \mu _{Z_{N}}(t)\) is continuous on [0, T] with values in \({\mathcal {P}}({\mathbb {R}}^{2}\times {\mathbb {R}}^{2})\) equipped with the weak topology, and is a distributional solution to the Vlasov–Poisson equation
As customary, we refer to the time dependent probability measure \(\mu _{Z_{N}}(t)\) as the empirical measure or as a Klimontovich solution (the latter name is justified thanks to the above proposition). Of course, by the same token the empirical measure associated the the system 1.1 is a distributional solution to the Vlasov-Poisson equation with magnetic field (Eq. 1.3, see also Remark 1.4 in [11]). In what follows, we restrict our attention to the case where V is to the 2D repulsive Coulomb potential \(V(x)=-\frac{1}{2\pi }\log (|x|).\) Despite the fact that in this case V obviously fails to satisfy the conditions of the Cauchy–Lipschitz theorem, existence and uniqueness of solution for the system (1.1) is still ensured provided that the positions of the particles are separated initially (i.e. \(x_{k}^{in}\ne x_{l}^{in}\) for \(k\ne l\)). The Vlasov-Poisson system is also closely linked to a celebrated model from fluid dynamics, namely the incompressible 2D Euler equation in vorticity formulation, which reads
where \(u:[0,T]\times {\mathbb {R}}^{2}\rightarrow {\mathbb {R}}^{2}\), \(\omega \) is a scalar field on \([0,T]\times {\mathbb {R}}^{2}\) and \(v^{\bot }:=(-v_{2},v_{1})\) (if the domain is \({\mathbb {T}}^{2}\), it is obvious how to adjust the formulation). Formal considerations (see e.g., Section 6 in [3]) suggest that Eq. (1.4) (on \({\mathbb {T}}^{2}\)) can be derived from Eq. (1.3) in the limit as \(\epsilon \rightarrow 0\). In the same work [3] (especially theorem 6.1), this derivation has been made rigorous for sufficiently regular/decaying solutions of Eq. (1.3). The same problem has been also dealt within [10], where the authors employ compactness methods. Due to the relation between Eq. (1.3) and the system (1.1) provided by Klimontovich’s observation, it is natural to seek a derivation of Euler from Newton’s system in the presence\absence of a magnetic field. We stress that due to the singular nature of Klimontovich solutions, it is far from straightforward to extend the convergence result of [3] for Klimontovich solutions of (1.3). A recent striking functional inequality due to Serfaty [22] (to be discussed in more detail in the next section) has allowed to overcome the difficulty created due to this singular behavior. This inequality (to which we refer from now on as Serfaty’s inequality) allowed the same authors to derive the pressureless Euler equation from Newton’s system of ODEs in the absence of a magnetic field (system (1.2)) and with monokinetic initial data. The case of non-monokinetic initial data remains a widely open problem. The derivation of the Eq. (1.4) from Newton’s second law in the presence of a magnetic field was established in [11], again through an argument heavily relying on the above mentioned inequality.
Main contribution of the current work In this work, we focus on the semi-classical universe. Therefore, we introduce the von Neumann equation, which is the quantum analogue of Newton’s system of ODEs: The Cauchy problem for the von Neumann equation with a vector potential of the form \(\frac{1}{2\epsilon }x^{\bot }\) reads
where \({\mathscr {H}}_{N}:={\mathscr {H}}_{N,\epsilon ,\hbar }\) is the quantum Hamiltonian defined by the formula
\(V_{pq}\) is the multiplication operator corresponding to the function \(V(x_{p}-x_{q})\), and \(\left[ \cdot ,\cdot \right] \) is the commutator defined for operators A, B by \(\left[ A,B\right] :=AB-BA\). Also, \(x_{i}^{k}\) stands for the \(k-\)th coordinate \((k=1,2)\) of \(x_{i}\). Remark that the scaling in \(\varepsilon \) we use is different from [11], but this is of course doesn’t play any essential role. As will be clarified in Sect. 6, the operator \({\mathscr {H}}_{N}\) can be viewed as a unbounded self-adjoint operator on \(L^{2}({\mathbb {R}}^{2N})\). The Planck constant \(\hbar >0\) should be viewed as a very small parameter, and thus the asymptotics in the semi-classical setting are obtained as a triple limit. For technical reasons we chose to include in the Hamiltonian a quadratic confining potential of the form 
. We elaborate on the reason for this choice in the next section. The unknown \(R_{N,\epsilon ,\hbar }(t)\) is a symmetric density operator i.e. a bounded operator on \(L^{2}({\mathbb {R}}^{2N})\) such that
and for all \(\sigma \in {\mathfrak {S}}_{N}\)
where \({\mathfrak {S}}_{N}\) is the symmetric group on N elements and where \(U_{\sigma }\) is the operator defined for each \(F_{N}\in L^{2}({\mathbb {R}}^{2N})\) by
In light of the previous discussion, it is natural to seek a derivation of the incompressible Euler or pressureless Euler equation from the von Neumann equation, in the presence/absence of a magnetic field respectively. In order to compare a solution of the von Neumann equation (which is an operator) with the vorticity solution of the Euler equation (which is a time dependent function on the Euclidean space), one attaches to \(R_{N,\epsilon ,\hbar }(t)\) a time dependent probability density called the density of the first marginal of \(R_{N,\epsilon ,\hbar }(t)\). The explicit construction of this density is recalled in the next section. Thus, deriving Euler from von Neumann reduces to proving some kind of weak convergence of the density of the first marginal to the vorticity as the parameters involved grow large or become small (according to their physical interpretation). The derivation of the pressureless Euler equation from the von Neumann equation was achieved in [8] in the limit as \(\frac{1}{N}+\hbar \rightarrow 0\). One of the interesting features of the method of proof of [8] is the observation that Serfaty’s inequality (which was originally applied in the context of a classical mean field limit) can be adapted to the semi-classical regime as well. This observation has also been utilized in the recent work [18], which proves a semi-classical combined mean field quasineutral limit, which is a semi-classical version of Theorem 1.1 in [11]. As already mentioned, the second main result of [11] is a derivation of equation (1.4) from the system (1.1), to which the authors of [11] refer to as a combined mean-field and gyrokinetic limit. Our new contribution—stated precisely in theorem 2.7 of the next section—is a derivation of the incompressible Euler equation (1.4) from the von Neumann equation (1.5), thereby complementing the abovementioned works [8, 11, 18]. Otherwise put, we prove a semi-classical combined mean field and gyrokinetic limit. The recipe for passing from classical to quantum mechanics is summarized neatly in [5]:
1. Functions on phase space are replaced by operators on the Hilbert space of square integrable functions on the underlying configuration space.
2. Integration of functions is replaced by the trace of the corresponding operators.
3. Coordinates q of the configuration space are replaced by multiplication operator \({\widehat{q}}\) by the variable q, while momentum coordinates p are replaced by the operator \({\widehat{p}}=-i\hbar \nabla \) (\({\widehat{p}}=-i\hbar \nabla +\frac{1}{2\epsilon }x^{\bot }\) in case a magnetic field is included).
As typicall in the theory of mean field limits, the argument in [11] rests upon obtaining a Gronwall estimate for a time dependent quantity which is known to control the weak convergence. This quantity is called the modulated energy. Our argument is a modification of the argument leading to a Gronwall estimate on the modulated energy in [11], according to the above mentioned rules 1–3, which is also the central idea in the semi-classical combined mean field quasineutral limit obtained in [18]. Nevertheless, it is important to point out a few points in which the present work differ from [18]:
First, we insist on working on the entire plane \({\mathbb {R}}^{2}\) rather than the torus \({\mathbb {T}}^{2}\), since the magnetic vector potential in question is non-periodic and so it is apriori not obvious how to even make sense of the modulated energy in the \({\mathbb {T}}^{2}\) case. This in turn forces us to include a quadratic confining potential, in order to make sure that the quantum Hamiltonian can be viewed as an essentially self-adjoint operator. Another point which requires some care when working in the plane is the existence and uniqueness theory of solutions to the incompressible 2D Euler—for example, solutions with square summable velocity field cannot have a vorticity with a constant sign (Section 3.1.3 in [16])—and therefore such solutions will be inadequate for the question of interest, in which the vorticity is taken to be a probability density. Finally, the example cooked up in order to witness the initial vanishing of the modulated energy has to be adjusted to these new choices. The utility of each one of the choices we just mentioned will become clearer in the sequel.
The paper is organized as follows: in Sect. 2 a semi-classical version of the modulated energy is introduced along with other preliminaries, and in Sect. 3 a Gronwall estimate is established for this quantity. The calculations which lead to this estimate are more tedious in comparison to the classical setting, partially due to the fact that quantization gives rise to commutators of a differential operator with a multiplication operator—which contributes non zero terms of course. As in [8, 11, 18] this estimate heavily relies on Serfaty’s inequality. Section 4 aims to explain how weak convergence is implied from this estimate—which is a simple consequence of a different (yet intimately related) inequality of Serfaty. In Sect. 5 we construct an explicit example witnessing the asymptotic vanishing of the initial modulated energy. Finally, Sect. 6 elaborates on the self-adjointness of the Hamiltonian and related functional analytic material.
2 Preliminaries and Main Result
The equation which is to be derived (to which it is customary to refer to as the “target equation”) is (1.4). We will work with an equivalent formulation which reads
Taking the divergence of both sides of (2.1) gives
We elaborate more on the objects related to the von Neumann equation (1.5). To this aim, let us fix some notations from Hilbert space theory, adapting mainly the terminology introduced in [8]. Set \({\mathfrak {H}}:=L^{2}({\mathbb {R}}^{2})\) and for each \(N\ge 1\) denote \({\mathfrak {H}}_{N}:={\mathfrak {H}}^{\otimes N}\simeq L^{2}({\mathbb {R}}^{2N})\). As customary, \({\mathcal {L}}({\mathfrak {H}})\) stands for the normed space of bounded linear operators on \({\mathfrak {H}}\), \({\mathcal {L}}^{1}({\mathfrak {H}})\) stands for the normed space of trace class operators on \({\mathfrak {H}}\) and \({\mathcal {L}}^{2}({\mathfrak {H}})\) stands for the normed space of Hilbert Schmidt operators on \({\mathfrak {H}}\). For each \(\sigma \in {\mathfrak {S}}_{N}\) (where \({\mathfrak {S}}_{N}\) is the group of permutations on \(\{1,\ldots ,N\}\)) we define the operator \(U_{\sigma }\in {\mathcal {L}}({\mathfrak {H}}_{N})\) by
With this notation, we denote by \({\mathcal {L}}_{s}({\mathfrak {H}}_{N})\) (\({\mathcal {L}}_{s}^{1}({\mathfrak {H}}_{N})\)) the set of bounded (trace class) operators \(F_{N}\in {\mathcal {L}}({\mathfrak {H}}_{N})({\mathcal {L}}^{1}({\mathfrak {H}}_{N}))\) such that \(U_{\sigma }F_{N}U_{\sigma }^{*}=F_{N}\) for all \(\sigma \in {\mathfrak {S}}_{N}\).We denote by \({\mathcal {D}}({\mathfrak {H}})\) the set of density operators on \({\mathfrak {H}}\), i.e. operators \(R\in {\mathcal {L}}({\mathfrak {H}})\) such that \(R=R^{*}\ge 0,\textrm{trace}_{{\mathfrak {H}}}(R)=1\). In addition set \({\mathcal {D}}_{s}({\mathfrak {H}}_{N}):={\mathcal {D}}({\mathfrak {H}}_{N})\cap {\mathcal {L}}_{s}({\mathfrak {H}}_{N})\). Let us recall the notion of marginal
Definition 2.1
Let \(N\ge 1\) and \(F_{N}\in {\mathcal {D}}_{s}({\mathfrak {H}}_{N})\). For each \(1\le k\le N\) we denote by \(F_{N:k}\in {\mathcal {D}}_{s}({\mathfrak {H}}_{k})\) the k-th marginal, which is by definition unique element of \({\mathcal {D}}_{s}({\mathfrak {H}}_{k})\) such that
for all \(A_{k}\in {\mathcal {L}}({\mathfrak {H}}_{k})\).
In the sequel we will take V to be as in Proposition 1.1. The unknown \(R_{N,\epsilon ,\hbar }(t)\) in equation (1.5) is an element of \({\mathcal {D}}_{s}({\mathfrak {H}}_{N})\). For brevity we put
The Hamiltonian can be decomposed to the kinetic energy \({\mathscr {K}}_{N}:={\mathscr {K}}_{N,\epsilon ,\hbar }\) and the interaction part \({\mathscr {V}}_{N}:={\mathscr {V}}_{N,\epsilon }\) which are defined by
and
The operator \({\mathscr {K}}_{N}\) may be viewed as an unbounded essentially self-adjoint operator in \({\mathfrak {H}}_{N}\) with domain \(C_{0}^{\infty }({\mathbb {R}}^{2N})\). This can be seen through the machinery of quadratic forms methods (see Sect. 6 of the present work or Section 2.2 in [26] for details). In Sect. 6 we will prove the estimate
for some \(0<\alpha <1,\beta >0\) and all \(\varphi \in C_{0}^{\infty }({\mathbb {R}}^{2N})\). Due to the perturbation theory developed by Kato-Rellich or Kato in [12,13,14] this implies that \({\mathscr {H}}_{N}\) is essentially self-adjoint, which by Stone’s theorem implies that \(e^{-it{\mathscr {H}}_{N}}\) is a unitary operator commuting with \({\mathscr {H}}_{N}\)- a fact which is of utter importance. We stress again that some special care is needed in the case of the 2D Coulomb potential, since originaly Kato proved that self-adjointness of the kinetic energy is invariant under perturbations which belong to \(L^{\infty }+L^{2}\), whereas \(\log (|x|)\) obviously fails to obey this condition. The external potential 
is responsible for handeling this obstcale: as clarified in Sect.
6, it turns out that this term is helpful while establishing the self-adjointness of \({\mathscr {H}}_{N}\), since it compensate the divergence of \(\log (|x|)\) as \(|x|\rightarrow \infty \). This is one technical difference in comparison to the work of [11]. A second related technical difference is that contrary to [11], we do not treat the case \({\mathbb {T}}^{2}\) and confine the discussion to \({\mathbb {R}}^{2}\), since the vector potential \(x^{\bot }\) is non-periodic. Possibly, a gauge invariance principle can be applied in order to overcome this problem, but this direction remains to be pursued. It is well known that the generalized solution to equation (1.5) is given by
If \(R_{N}\) is a density operator then both \(R_{N},\sqrt{R_{N}}\) are Hilbert–Schmidt operators on \({\mathfrak {H}}_{N}\), and denote their integral kernels by \(k(X_{N},Y_{N}),\kappa (X_{N},Y_{N})\in {\mathfrak {H}}_{N}^{\otimes 2}\) respectively. Using that \(R_{N}=\sqrt{R_{N}}\sqrt{R_{N}}\) we see that
and therefore \((X_{N},\Xi _{N})\mapsto k(X_{N},X_{N}+\Xi _{N})\) defines an element of \(C({\mathbb {R}}_{\Xi _{N}}^{2N};L^{1}({\mathbb {R}}_{X_{N}}^{2N}))\), which implies that \(\rho [R_{N}](X_{N}):=X_{N}\mapsto k(X_{N},X_{N})\in L^{1}({\mathbb {R}}^{2N})\), and furthermore that \(\rho [R_{N}]\ge 0\) and \(\underset{{\mathbb {R}}^{2N}}{\int }\rho [R_{N}](X_{N})dX_{N}=1\). Hence \(\rho [R_{N}]\) is a probability density on \({\mathbb {R}}^{dN}\) which is moreover symmetric provided \(R_{N}\) is symmetric.We will denote by \(\rho _{N:k,\epsilon ,\hbar }(t,\cdot )\) the density associated to \(R_{N:k,\epsilon ,\hbar }\). Inspired by [11] we introduce the time dependent quantity
We recall that in the above definition \(u,\omega \) are the solutions of Eq. 1.4. The quantity \({\mathcal {E}}(t)\) is a semi-classical rescaled version of the distance introduced in [11] formula 1.10. A few remarks are in order after this definition. First, let us justify that the 2 last integrals on the second line are well defined. Since \(V\star {\mathfrak {U}}(t,\cdot )=\textrm{div}(V\star (u\cdot \nabla u)(t,\cdot ))\), we see that \(V\star {\mathfrak {U}}(t,\cdot )\in L^{\infty }({\mathbb {R}}^{2})\) provided \(u\cdot \nabla u(t,\cdot )\in L^{\infty }\cap L^{q}({\mathbb {R}}^{2})\) for some \(1<q<2\). In addition \({\mathfrak {U}}(t,\cdot )\in L^{1}({\mathbb {R}}^{2})\) assuming \(\nabla u(t,\cdot )\in L^{2}({\mathbb {R}}^{2})\). Both of these integrability assumptions (\(\nabla u(t,\cdot )\in L^{2}({\mathbb {R}}^{2})\) and \(u\cdot \nabla u(t,\cdot )\in L^{\infty }\cap L^{q}({\mathbb {R}}^{2})\)) are obviously implied by the assumption on u stated in our main theorem (2.7) below, and explain why the product \(V\star {\mathfrak {U}}(\omega +\epsilon {\mathfrak {U}})(t,\cdot )\in L^{1}({\mathbb {R}}^{2})\). The convolution of the logarithm with a summable function is known to be a function of bounded mean oscillation, or in brief \(\textrm{BMO}\) (see 6.3, (i) in [25]). In addition, we recall that a \(\textrm{BMO}\) function is in the weighted space \(L^{1}((1+|x|)^{-3})\) ([25], 1.1.4). In light of this reminder we see that in order to make sense of the above integrals it suffices to require \((1+|x|)^{3}\omega (t,\cdot )\in L^{\infty }({\mathbb {R}}^{2})\). If we require this assumption to hold initially (\(t=0\)), we can ensure it is propagated in time for compactly supported initial data. We elaborate on this last claim in appendix A. As for the first integral in the definition of \({\mathcal {E}}(t)\) as well as \({\mathcal {E}}_{1}(t)\), we shall impose the following technical assumption which will be needed in order to justify the fact that both are well defined for all times
Assumption (A).
The usefulness of assumption A will become clear while establishing conservation of energy, as stated in the following
Lemma 2.2
Let \(R_{N,\epsilon ,\hbar }^{in}\in {\mathcal {D}}_{s}({\mathfrak {H}}_{N})\) satisfy assumption (A). Let \(R_{N}(t):=e^{-\frac{it{\mathscr {H}}_{N}}{\hbar }}R_{N,\epsilon ,\hbar }^{in}e^{\frac{it{\mathscr {H}}_{N}}{\hbar }}\). Then
We remark that the proof of energy conservation in [8] does not consider the presence of a magnetic field. The proof of lemma (2.2) is postponed to Sect. 6. In particular the conservation of energy includes the (non obvious at first sight) statement that both terms on the right hand side of (2.4) are separately well defined for all times \(t\in [0,T]\). Assumption (A) also implies in particular
which in turn makes the Riesz representation theorem available, thereby allowing to define a notion of a current, which is the quantum analog of the first moment of the solution of Vlasov-Poisson system. Denote by \(\vee \) the anticommutator.
Definition 2.3
Let \(R\in {\mathcal {L}}^{1}({\mathfrak {H}})\) such that \(R=R^{*}\ge 0\) and \(\textrm{trace}_{{\mathfrak {H}}}(\sqrt{R}((-i\hbar \partial _{x^{k}}+\frac{1}{2\epsilon }(x^{\bot })^{k})^{2})\sqrt{R})<\infty \). For each \(a\in C_{b}({\mathbb {R}}^{2})\) and \(k=1,2\) we denote by \(J_{N:1}^{k}\) the unique signed Radon measure on \({\mathbb {R}}^{2}\) such that
The current of R is the signed measure valued vector field \((J_{N:1}^{1},J_{N:1}^{2})\).
We now state Serfaty’s remarkable functional inequality, whose considerable importance while obtaining a Gronwall estimate on the functional \({\mathcal {E}}(t)\) has already been mentioned. For \(X_{N}=(x_{1},\ldots ,x_{N})\in {\mathbb {R}}^{2N}\) denote 
, and denote by \(\triangle \) the diagonal of \({\mathbb {R}}^{2N}\), i.e. \(\triangle :=\left\{ \left. X_{N}\in {\mathbb {R}}^{2N}\right| \exists 1\le i,j\le N:x_{i}=x_{j}\right\} \).
Theorem 2.4
([22], Proposition 1.1) There is a constant \(C>0\) with the following property. Assume that \(\mu \in L^{\infty }({\mathbb {R}}{}^{2})\) is a probability density. Then for any \(X_{N}\in ({\mathbb {R}}^{2})^{N}\) s.t. \(\forall i\ne j:x_{i}\ne x_{j}\) and any Lipschitz map \(\psi :{\mathbb {R}}^{2}\rightarrow {\mathbb {R}}^{2}\) we have
where
To fully appreciate the vital importance of this estimate in the context of singular mean field limits, let us examine for instance the system 1.2 in arbitrary dimension d with V taken to be the Coulomb potential \((V(x)\sim \frac{1}{\left| x\right| ^{d-2}}\)). In this case the expected target equation is the Euler-Poisson system which reads
It turns out that the modulated energy corresponding to this regime is
The modulated energy \({\mathcal {H}}_{N}\) is a reminiscent of a time dependent quantity used in classical stability estimates for the Euler-Poisson equation, and can be viewed as a renormalized version of this quantity (see lemma A.2 in [22]). After some tedious algebra (see appendix A in [22]) one finds that \(\dot{{\mathcal {H}}}_{N}(t)\) can be recast as
The first term in the above expression is clearly bounded by means of \(\mathcal {{\mathcal {H}}}_{N}^{1}(t)\), while theorem 2.4 enables to control the second term by means of \(\mathcal {{\mathcal {H}}}_{N}^{2}(t)\), simply upon taking \(\psi =u\). This type of argument is in fact quite robust, and variants of it have been applied in a few other regimes as well- in addition to the references already mentioned, we refer also to [4, 20].
Remark 2.5
After completing this work, Matthew Rosenzweig drew our attention to corollary (4.3) in [23] which provides an improved (and sharp) version of theorem 2.4, as well as to proposition (3.9) in [19] which provides a simpler proof of this improvement.
Originally, Serfaty’s inequality was used in order to derive the pressureless Euler system from Newton’s second order system of ODEs with monokinetic initial data. This is a classical mechanics mean field limit type result. The following lemma serves as a bridge between theorem (2.4) and the quantum settings which are relevant for us
Lemma 2.6
([8], Lemma 3.5) Set \(\mu (t,\cdot ):=\omega (t,\cdot )+\epsilon {\mathfrak {U}}(t,\cdot )\) and
and
Then
and
where
In its essence, Lemma 2.6 tells us how to represent the quantum counterparts of the right hand side and left hand side of Serfaty’s inequality by means of the classical right hand side and left hand side, respectively. Our main theorem is
Theorem 2.7
Let \((\omega ,u)\in C^{1}([0,T];C^{0,\alpha }({\mathbb {R}}^{2}))\times C^{1}([0,T];C_{\textrm{loc}}^{1,\alpha }({\mathbb {R}}^{2})\cap L^{\infty }({\mathbb {R}}^{2}))\) with \(\alpha \in (0,1)\) be a solution to (1.4), such that
1. \(\omega (t,\cdot )\) is compactly supported with \(\omega \ge 0\) and \(||\omega (t,\cdot )||_{1}=1\) and
2. For all \(1<p\le \infty :\) \(\nabla u\in L^{\infty }([0,T];L^{p}({\mathbb {R}}^{2}))\).
Let \(R_{N,\epsilon ,\hbar }(t)\in {\mathcal {D}}_{s}({\mathfrak {H}}_{N})\) be a solution to equation (1.5) such that \(R_{N,\epsilon ,\hbar }^{in}\) verifies assumption (A). Then \({\mathcal {E}}_{N,\epsilon ,\hbar }(t)\rightarrow 0\) as \(\frac{1}{N}+\epsilon +\hbar \rightarrow 0\), provided \({\mathcal {E}}_{N,\epsilon ,\hbar }(0)\rightarrow 0\) as \(\frac{1}{N}+\epsilon +\hbar \rightarrow 0\). Furthermore, \(\rho _{N:1,\epsilon ,\hbar }\underset{\frac{1}{N}+\epsilon +\hbar \rightarrow 0}{\rightarrow }\omega \) weakly in the sense of measures provided \({\mathcal {E}}_{N,\epsilon ,\hbar }(0)\rightarrow 0\) as \(\frac{1}{N}+\epsilon +\hbar \rightarrow 0\).
Remark 2.8
The assumption \({\mathcal {E}}_{N,\epsilon ,\hbar }(0)\rightarrow 0\) is reasonably typical, at least for \(\epsilon ,\hbar \) satisfying some appropriate relations, as can be seen e.g. through the example constructed in Sect. (5).
Remark 2.9
The assumptions on \((\omega ,u)\) can be realized for suitable initial data-see appendix A for more details.
The functional \({\mathfrak {f}}(X_{N},\mu )\) (and thus \({\mathcal {E}}_{2}(t)\)) is not necessarily a non negative quantity. However, it turns out that it is bounded from below by a term which vanishes asymptotically. Therefore \({\mathcal {E}}(t)\rightarrow 0\) implies that the kinetic term and interaction term vanish separately: \({\mathcal {E}}_{1}(t)\rightarrow 0,{\mathcal {E}}_{2}(t)\rightarrow 0\). We have the following
Proposition 2.10
([22], Proposition 3.3) Let \(\mu \in L^{\infty }({\mathbb {R}}^{2})\) be a probability density. Then
3 Gronwall Estimate
We present here a proof of the asymptotic vanishing of the modulated energy, as stated in the first part of theorem (2.7). In the next section we will explain how to topologize this convergence. The proof below should be regarded as a formal proof. A careful justification of the calculation below follows by the eigenfunction expansion method explained the recent work [9] and is left as an exercise, since in our view the formal calculation makes the essence of the argument more visible. In addition, in some places the dependence on the parameters \(\epsilon ,\hbar \) is implicit. During the last stage of the estimate we will apply Serfaty’s inequality, which is valid provided the function \(\mu \) is a bounded probability density. This is not necessarily true for \(\mu =\omega +\epsilon {\mathfrak {U}}\), and for this reason we will need to consider
Proof of theorem (2.7)
Step 1. Calculation of \(\dot{{\mathcal {E}}_{1}}(t)\).
The second equality is by Eq. (1.5) and the third equality follows by tracing by parts ( i.e. the identity \(\textrm{trace}(A[B,C])=-\textrm{trace}(C[A,B])\)). We start with
In addition
We have
We compute
and
while
Hence
Put \({\textbf{J}}_{kl}=(\partial _{x_{i}^{k}}((x_{i}^{\bot })^{l})-\partial _{x_{i}^{l}}((x_{i}^{\bot })^{k})=(\nabla x^{\bot }-\nabla ^{T}x^{\bot })_{kl}\) (so that \({\textbf{J}}=2\begin{pmatrix}0 &{} 1\\ -1 &{} 0 \end{pmatrix}\)). With this notation we find
So concluding (with Einstein’s summation applied for the indices k, l)
Step 2. Calculation of \(\dot{{\mathcal {E}}_{2}}(t)\). We start by obtaining an evolution equation for the first marginal of \(\rho _{N,\epsilon ,\hbar }\).
Claim 3.1
\(\partial _{t}\rho _{N:1}+\nabla .J_{N:1}=0\) in \({\mathcal {D}}'((0,T)\times {\mathbb {R}}^{2})\).
Proof
Let \(a\in C_{b}^{1}({\mathbb {R}}^{2})\).
Thus
\(\square \)
One has
The second term in the right hand side of equation (3.2) is
while the third term in the right hand side of equation (3.2) is
So that concluding
Gathering Eqs. (3.1) and (3.3) we conclude that (applying Einstein’s summation )
Step 3. Rearrangement of terms. By Eq. (2.1) we have
In addition, owing to claim (3.1) and noticing that \(u^{\bot }=\nabla (V\star \omega )\) we have
and
so that
In addition, using that \(\nabla (V\star {\mathfrak {U}})=\nabla p\) we get
so that \(\dot{{\mathcal {E}}}(t)\) is recast as
We now apply conservation of energy.
Claim 3.2
It holds that
.
Proof
We compute the time derivative of the right hand side of equation (2.2).
First
Second
Since the left hand side of equation (2.2) is constant in time we conclude
In view of claim (3.2) \(\dot{{\mathcal {E}}}(t)\) writes
Step 4. Accessing Serfaty’s Inequality. Note that
and
Denote \(\mu :=\omega +\epsilon {\mathfrak {U}}\). We have the identity
By Eq. (3.4) we have
Inserting Eq. (3.5) in the above yields
Step 5. Gronwall inequality and conclusion. We estimate each of the \(J_{i}\) separately:
Estimating \(J_{1}\).
Estimating \(J_{2}\). By lemma (2.6) we can write
where
We observe the following bounds
and
Therefore by theorem (2.4)
By proposition (2.10) we have
To finish the estimate for \(J_{2}\), we shall replace \({\mathcal {E}}_{2}^{*}\) by \({\mathcal {E}}_{2}\), with the expense of adding a term vanishing as \(\epsilon \rightarrow 0\). Indeed, the bound below follows from the remarks following the definition of \({\mathcal {E}}(t)\)
and shows that
Hence
Estimating \(J_{3},J_{4}\) and \(J_{5}\). The terms \(J_{3},J_{4}\) and \(J_{5}\) are of order \(\epsilon \):
Finally, note that
thus
By proposition (2.10) and inequality (3.8) there is a constant \(\gamma \) such that
By inequalities (3.6), (3.10), (3.11), (3.12), (3.13) and Gronwall, \(E(t)\rightarrow 0\) as \(\frac{1}{N}+\epsilon +\hbar \rightarrow 0\). This a fortiori implies \({\mathcal {E}}(t)\rightarrow 0\) as \(\frac{1}{N}+\epsilon +\hbar \rightarrow 0\), which concludes the proof. \(\square \)
Remark 3.3
A modification of the proof presented above would likely allow to obtain the stronger claim that \(\frac{1}{\epsilon }{\mathcal {E}}_{N,\epsilon ,\hbar }\underset{N\rightarrow \infty }{\rightarrow }0\) provided \(\epsilon =\epsilon (N)\) is chosen so that \(\epsilon N^{\beta }\underset{N\rightarrow \infty }{\rightarrow }0\) for some suitable power \(\beta >0\) (which is comparable to the asymptotic relation between \(\epsilon \) and N in [11], theorem 1.5). The main additional ingredient that will be needed to establish such a convergence is the coercivity estimates which will be introduced in the next section. We stress however that for the purpose of proving the convergence \(\rho _{N:1}\underset{\frac{1}{N}+\epsilon +\hbar \rightarrow 0}{\rightarrow }\omega \) no relation between \(\epsilon \) and N is needed. The utility of the convergence to 0 of the quantity \(\frac{1}{\epsilon }{\mathcal {E}}_{N,\epsilon ,\hbar }\) becomes relevant if one seeks to study the limit of the current \((J_{N:1}^{1},J_{N:1}^{2})\), which is expected to converge weakly to the velocity field u. In view of the results of [11] and [18] it seems probable that this convergence can be carried out with no particular difficulties, and we hope to address this matter in a subsequent work in which we study the problem in a more general framework.
4 Weak Convergence
Here we show that the asymptotic vanishing of \({\mathcal {E}}(t)\) implies weak convergence \(\rho _{N:1,\epsilon ,\hbar }\rightarrow \omega \). This fact is essentially already contained in the literature (see e.g. [18], Section 3.4), and is included mainly for the sake of clarity of exposition. In the expense of a bit more technique, one can show that in fact \(\rho _{N:1,\epsilon ,\hbar }\rightarrow \omega \) in some appropriate negative Sobolev space—see [18], Section 3.4 for a guidance on how this is to be done. A key ingredient to establish this weak convergence is the following coercivity estimate for the modulated energy
Proposition 4.1
([22], Proposition 3.6) Let \(0<\alpha \le 1\). There are constants \(C=C(\alpha )>0,\lambda =\lambda (\alpha )>0\) such that for any \(\varphi \in C^{0,\alpha }({\mathbb {R}}^{2})\), any probability density \(\mu \in L^{\infty }({\mathbb {R}}^{2})\) and any \(X_{N}\) with \(\forall i\ne j:x_{i}\ne x_{j}\) it holds that
Let \(X_{N}\) with \(\forall i\ne j:x_{i}\ne x_{j}\). Specifying proposition (4.1) to \(\mu =\omega \) and \(\alpha =1\), we see there are constant \(\lambda ,C>0\) such that for any \(\varphi \in C^{0,1}({\mathbb {R}}^{2})\) we have
In addition we have the following simple identity, observed in [21], Lemma 7.4. The proof is included for the sake of completeness
Lemma 4.2
If \(\rho _{N}\) is a symmetric probability density on \(({\mathbb {R}}^{2})^{N},\) \(\mu \in L^{\infty }({\mathbb {R}}^{2})\cap L^{1}({\mathbb {R}}^{2})\) and \(\varphi \in C_{b}({\mathbb {R}}^{2})\) then
Proof
By symmetry of \(\rho _{N}\) we have the identity
Therefore
\(\square \)
Integrating both sides of inequality (4.1) against \(\rho _{N,\epsilon ,\hbar }(t,X_{N})\) and owing to lemma (4.2) gives
by Cauchy Schwartz. Since \({\mathcal {E}}_{2}(t)+O(\epsilon )={\mathcal {E}}_{2}^{*}(t)\) (see (3.9)) inequality (4.2) is recast as
Since \({\mathcal {E}}_{1}(t)\ge 0,\) theorem (2.7) together with proposition (2.10) evidently imply \({\mathcal {E}}_{2}(t)\rightarrow 0\) as \(\frac{1}{N}+\epsilon +\hbar \rightarrow 0\). Hence, the right hand side of (4.3) goes to 0 as \(\frac{1}{N}+\epsilon +\hbar \rightarrow 0\), which shows that \(\rho _{N:1}\rightarrow \omega \) weakly as \(\frac{1}{N}+\epsilon +\hbar \rightarrow 0\).
5 Typicality of the Assumption \({\mathcal {E}}_{N,\epsilon ,\hbar }(0)\underset{\frac{1}{N}+\epsilon +\hbar }{\rightarrow }0\)
We explain here how to construct initial data \(R_{N,\epsilon ,\hbar }^{in}\) which witnesses the fact that the condition \({\mathcal {E}}(0)\rightarrow 0\) is nonempty. It will be convenient to use Toeplitz operators for the construction of \(R_{N,\epsilon ,\hbar }^{in}\). We borrow some definitions and elementary facts from [6] regarding Toeplitz operators. Let \(z=(q,p)\in {\mathbb {R}}^{d}\times {\mathbb {R}}^{d}\). For each \(\hbar \) we consider the complex valued function on \({\mathbb {R}}^{d}\) defined by
We denote by \(\left| z,\hbar \right\rangle \left\langle z,\hbar \right| :{\mathfrak {H}}\rightarrow {\mathfrak {H}}\) the orthogonal projection on the line \(\left| z,\hbar \right\rangle {\mathbb {C}}\) in \(L^{2}({\mathbb {R}}^{d}\times {\mathbb {R}}^{d})\). For each finite/positive Borel measure \(\nu \) on \({\mathbb {R}}^{d}\times {\mathbb {R}}^{d}\) we define the Toeplitz operator \(\textrm{OP}_{\hbar }^{T}(\nu ):{\mathfrak {H}}\rightarrow {\mathfrak {H}}\) at scale \(\hbar \) with symbol \(\nu \) by the formula
If \(\nu \) is a positive measure
and
In addition, if \(\nu \) is a probability measure then \(\textrm{OP}_{\hbar }^{T}(\nu )\) is a trace class operator. We also recall the definition of the Wigner and Husimi transforms.
Definition 5.1
Let A be an unbounded operator on \(L^{2}({\mathbb {R}}^{d})\) with integral kernel \(k_{A}\in {\mathcal {S}}'({\mathbb {R}}^{d}\times {\mathbb {R}}^{d})\). The Wigner transform of scale \(\hbar \) of A is the distribution on \({\mathbb {R}}^{d}\times {\mathbb {R}}^{d}\) defined by the formula
\(\hbar \)
where \(j_{\hbar }(x,y):=(x+\frac{1}{2}\hbar y,x-\frac{1}{2}\hbar y)\) and \({\mathcal {F}}_{2}\) is the partial Fourier transform with respect to the second variable.The Husimi transform of scale \(\hbar \) is the function on \({\mathbb {R}}^{d}\times {\mathbb {R}}^{d}\) defined by the formula
Remark 5.2
Denoting by \(G_{a}^{d}\) the centered Gaussian density on \({\mathbb {R}}^{d}\) with covariance matrix aI we can equivalently write
If \(\mu \) is a finite/positive Borel probability measure on \({\mathbb {R}}^{d}\times {\mathbb {R}}^{d}\) then
(see formula (51) in [6]).
We gather a few elementary formulas in the following
Theorem 5.3
([6], formulas (46) and (48)) 1. \(\textrm{OP}_{\hbar }^{T}(1)=I_{{\mathfrak {H}}}\).
2. If f is a quadratic form on \({\mathbb {R}}^{d}\) then
3. If \(\nu \) is a finite/positive Borel measure on \({\mathbb {R}}^{d}\times {\mathbb {R}}^{d}\) then
We will need the following
Lemma 5.4
Let \(\nu \in {\mathcal {P}}({\mathbb {R}}^{2}\times {\mathbb {R}}^{2})\) have finite second moments. Then
Proof
Calculation of the cross terms. Put
and denote by \(k_{A}(x,y)\) the kernel of A. It is readily checked that
which implies
hence
where the last equality is due to the formula
for any polynomial g of degree \(\le m\). By 3. of theorem (5.3), it follows that
Calculation of the dominant terms. We utilize theorem (5.3) with \(f(p)=|p|^{2}\).
Owing to formula (5.3) we find
so that
Next, we utilize theorem (5.3) with \(f(q)=|q|^{2}\).
and once again by formula (5.3)
so that
Gathering (5.4), (5.5) and (5.6) produces the asserted identity. \(\square \)
Remark 5.5
Considerations similar to the ones demonstrated in the proof of lemma 5.4 show that the regularity condition in assumption (A) is indeed satisfied.
We can now choose \(\hbar =\hbar (\epsilon )\) such that \(\frac{\hbar }{\epsilon }\rightarrow 0\). Under this assumption we can exemplify the assumption \({\mathcal {E}}(0)\rightarrow 0\). We proceed through the following steps
Step 1. The kinetic part. Pick \(\theta \in L^{1}({\mathbb {R}}^{2})\cap L^{\infty }({\mathbb {R}}^{2})\) and take
where \(\omega \) is a \(C_{0}^{\infty }({\mathbb {R}}^{2})\) probability density, and let
It is evident from (5.1)–(5.2) that this is indeed a density matrix. Moreover by lemma (5.4) we have
Clearly, by assumption
We have
In addition
and
The above inequalities evidently entail
As for the quadratic term, by formulas (52) and (53) in [6] (or equivalently by a similar calculation to the one demonstrated in lemma (5.4))
This takes care of the kinetic part.
Step 2. The interaction part. Note that if \(\rho _{N}\) is given as a pure tensor product \(\rho _{N}=\rho ^{\otimes N}\) then
Let \(\rho _{\epsilon ,\hbar }(x)\) denote the integral kernel of \(\textrm{OP}_{\hbar }^{T}((2\pi \hbar )^{2}\nu _{\epsilon })\) evaluated at the diagonal (x, x). By a straightforward calculation
Substep 2.1. \(J\underset{\hbar +\epsilon \rightarrow 0}{\rightarrow }0\). We claim that \(J\underset{\hbar +\epsilon \rightarrow 0}{\rightarrow }0\). Assuming that \(\omega \) has compact support we have
Clearly the terms in the second line are of order \(\epsilon \). The term in the first line is handled as follows. Decompose V into its negative and positive parts \(V=V^{-}+V^{+}\). We have that
Since \(V^{+}\in L^{p}({\mathbb {R}}^{2})\) for all \(1\le p<\infty \), the same is true for the convolution \(V^{+}\star \omega \), and so since \(G_{\frac{\hbar }{2}}^{2}\) is an approximation to the identity we get
In addition, observing that \(|V^{-}|(x)\le |x|^{2}\), one finds
Now pick \(\omega (x)=\frac{1}{\Lambda }\chi G_{\frac{1}{2}}^{2}(x)\) where \(\chi \in C_{0}^{\infty }({\mathbb {R}}^{2})\) with \(0\le \chi \le 1\) and \(\Lambda :=||\chi G_{\frac{1}{2}}^{2}||_{1}\). We split the last integral as
Owing to formula (5.3), the first integral is mastered as follows
By Plancherel the right hand side of equation (5.7) is
where the last inequality is thanks to the elementary inequality \(1-e^{-r}\le r\) for \(r\in [0,\infty )\). Taking \(0<\beta <\frac{1}{3}\) yields
Finally, the formula for convolution of Gaussians gives
so that
as a tail of a converging integral.
Substep 2.2. We utilize the formula \(G_{\frac{\hbar }{2}}^{2}\star G_{\frac{\hbar }{2}}^{2}=G_{\hbar }^{2}\) to find
The first term is recognized as a Cauchy difference (by substep 2.1) and therefore vanishes as \(\hbar \rightarrow 0\), while the second and third terms are of order \(\frac{1}{N}\) and \(\epsilon \) respectively, because \(\left| \underset{{\mathbb {R}}^{2}}{\int }G_{\frac{\hbar }{2}}^{2}\star (V\star \omega )\omega (x)dx\right| \) is uniformly bounded in \(\hbar \) (again by substep 2.1).
Remark 5.6
We make no claim of optimality of the assumption \(\frac{\hbar }{\epsilon }\rightarrow 0\), although it is not obvious how to modify the example constructed above in a way which avoids a relation between \(\epsilon ,\hbar \). Of course, the utility of this assumption is reflected merely for the purpose of the construction of initial data.
6 Essential Self-adjointness of the Hamiltonian: Conservation of Energy
We elaborate on the self-adjointness of the quantum Hamiltonian and give a proof of lemma (2.2). Recall the notation \(A(x):=\frac{1}{2\epsilon }x^{\bot }\) and the operators defined by the formulas
on \(C_{0}^{\infty }({\mathbb {R}}^{2N})\). For each fixed \(1\le p\le N\) and \(k=1,2\) denote by \(\Pi ^{p,k}\) the closure of the operator \(-i\hbar \partial _{x_{p}^{k}}+A^{k}(x_{p})\) in \(C_{0}^{\infty }({\mathbb {R}}^{2N})\) (it is closable as a symmetric operator). With slight abuse of notation, denote by \(K_{N}\) the operator
which is self-adjoint according to Theorem X.25 in [17]. EquivalentlyFootnote 1 we may view \(K_{N}\) (and \({\mathscr {K}}_{N}\)) as unbounded self-adjoint operators through the use of the terminology of quadratic forms, which we now briefly review following Section 2.3 in [26]. Given a non-negative quadratic form \(q:{\mathfrak {Q}}\rightarrow {\mathbb {C}}\) (\({\mathfrak {Q}}\) is a dense subspace of the underlying Hilbert space \({\mathfrak {H}}\)) corresponding to a sesquilinear form \(s:{\mathfrak {Q}}\times {\mathfrak {Q}}\rightarrow {\mathbb {C}}\) define a scalar product \(\left\langle \cdot ,\cdot \right\rangle _{q}:{\mathfrak {Q}}\times {\mathfrak {Q}}\rightarrow {\mathbb {C}}\) by letting
The norm inherited from this scalar product is denoted \(||\cdot ||_{q}\) (i.e. \(||\varphi ||_{q}^{2}:=q(\varphi )+||\varphi ||^{2}\)). If q is closable, the completion of \({\mathfrak {Q}}\) with respect to \(||\cdot ||_{q}\) is a subspace of \({\mathfrak {H}}\) denoted \({\mathfrak {H}}_{q}\).
Theorem 6.1
Let q be a non-negative closed quadratic form on \({\mathfrak {Q}}\) corresponding to a sesquilinear form s. There is a unique self-adjoint operator T with form domain \({\mathfrak {Q}}(T)\) such that \({\mathfrak {Q}}={\mathfrak {Q}}(T)\) and \(q=q_{T}\). In addition
Consider the sesquilinear form
on \(C_{0}^{\infty }({\mathbb {R}}^{2N})\times C_{0}^{\infty }({\mathbb {R}}^{2N})\) whose corresponding quadratic form is Q. The completion of \(C_{0}^{\infty }({\mathbb {R}}^{2N})\) with respect to \(||\cdot ||_{Q}\) is denoted \({\mathfrak {Q}}\) and the extension of Q to \({\mathfrak {Q}}\) is denoted q. With slight abuse of notation, denote by \(K_{N}\) the self-adjoint operator arising via the above theorem with \({\mathfrak {Q}}\) and q as defined in the previous line. Similarly (with slight abuse of notation) \({\mathscr {K}}_{N}\) is the self-adjoint operator arising from the form
with \({\mathfrak {Q}}\) being the completion of \(C_{0}^{\infty }({\mathbb {R}}^{2N})\) with respect to \(||\cdot ||_{{\mathscr {Q}}}\), where \({\mathscr {Q}}\) is the quadratic form corresponding to \({\mathscr {S}}\). Since both \(K_{N}\) and\({\mathscr {K}}_{N}\) (as operators on \(C_{0}^{\infty }({\mathbb {R}}^{2N})\)) are known to be essentially self-adjoint (Theorem (1.1) in [24]), the above self-adjoint extensions are in fact unique. The Kato-Rellich perturbation theory for self-adjoint operators is the most fundamental tool used for the purpose of viewing the quantum Hamiltonian, which is a perturbation of \({\mathscr {K}}_{N}\), as an unbounded self-adjoint operator.
Theorem 6.2
Let \(T,D(T)\subset {\mathfrak {H}}\) be a (essentially) self-adjoint operator and \(S,D(S)\subset {\mathfrak {H}}\) a symmetric operator such that \(D(T)\subset D(S)\). Suppose there exist \(0<a<1,b>0\) such that for each \(\varphi \in D(T)\)
Then \(T+S,D(T+S)=D(T)\) is (essentially) self-adjoint. In the case where T is essentially self-adjoint we have \(D({\overline{T}})\subset D({\overline{S}})\) and \(\overline{T+S}={\overline{T}}+{\overline{S}}\).
If T, S satisfy the condition (6.2) for some \(a>0\) and \(b>0\), we say that S is T-bounded. The infimum over all \(a>0\) for which there exist a \(b>0\) such that (6.2) holds is called the relative bound of S. The theorem stated below, which is one of the central conclusions from Kato-Rellich theory, shows that the self-adjointness of the N-body Laplacian is invariant under a perturbation by a symmetric N-body potential satisfying a certain integrability condition.
Theorem 6.3
Let \(V\in L^{\infty }({\mathbb {R}}^{2})+L^{2}({\mathbb {R}}^{2})\) be a real valued function. Let \({\mathscr {V}}_{N}\) be the multiplication operator defined by the formula (6.1) with domain \(D({\mathscr {V}}_{N})=H^{2}({\mathbb {R}}^{2N})\). Let
viewed as a self-adjoint operator with domain \(H^{2}({\mathbb {R}}^{2N})\). Then \({\mathscr {V}}_{N}\) is \(-\Delta _{N,\hbar }\)-bounded with relative bound 0. Consequently, \(-\Delta _{N,\hbar }+{\mathscr {V}}_{N}\) is self-adjoint on \(H^{2}({\mathbb {R}}^{2N})\).
Although \(\log (x)\) fails to satisfy the assumption stated in (6.3), the negative part of \(\log (x)\) evidently does verify this assumption, and this observation will be important in the argument that we employ. The work of Avron-Herbst-Simon [1] extends theorem 6.3 for the case where a magnetic field is included. In particular they prove the following result
Theorem 6.4
(Theorem 2.4 in [1])]. Let \({\mathscr {V}}_{N}\) be a multiplication operator on \(L^{2}({\mathbb {R}}^{2N})\). Suppose that \({\mathscr {V}}_{N}\) is \(-\Delta _{N}\)-bounded with relative bound \(\alpha \). Then \({\mathscr {V}}_{N}\) is \(K_{N}\)-bounded with relative bound \(\le \alpha \).
In contrast to [1], the Hamiltonian of interest contains an attractive potential of the form 
. It is the aim of the forthcoming lemmata to show that the essential self-adjointness of the Hamiltonian in this case is obtained as a corollary of the result reported in theorem (6.4). As we will see, the reason for this is that a bound on \({\mathscr {V}}_{N}\) with respect to \(K_{N}\) is sharper than a bound with respect to \({\mathscr {K}}_{N}\). The first step is
Lemma 6.5
For each \(\varphi \in C_{0}^{\infty }({\mathbb {R}}^{2})\) one has the following estimates
and
Proof
With the abbreviation \({\mathscr {K}}={\mathscr {K}}_{1}\), \(K=K_{1}\) and \(\Pi =(\Pi ^{1,1},\Pi ^{1,2})\), one has
which is the first inequality. The second inequality is implied from (6.3) as follows
\(\square \)
The next lemma shows that the norm associated with \({\mathscr {K}}_{N}\) controls the norm associated with \(K_{N}\) up to a constant
Lemma 6.6
There is a constant \(C=C(\hbar ,\epsilon ,N)\) such that for all \(\varphi _{N}\in C_{0}^{\infty }({\mathbb {R}}^{2N})\) it holds that
Proof
Let us first consider the case \(N=1\) (with the abbreviation \({\mathscr {K}}={\mathscr {K}}_{1}\), \(K=K_{1}\) and \(\Pi ^{1,k}=\Pi ^{k}\)). For each \(\varphi \in C_{0}^{\infty }({\mathbb {R}}^{2})\) we expand
The second term is estimated from below as follows (applying Einstein’s summation for k, l)
where we used the simple observation that \(\Pi ^{l}\) and \(\overline{\Pi ^{k}}\) commute. In addition, for each l, k the following relation is easily observed
where we recall
Therefore
Moreover
To conclude, Eqs. (6.5), (6.6) and inequality (6.7) entail
hence
Integration by parts shows that
so that in light of lemma (6.5)
where
Therefore identity (6.4) implies
that is
which is the same as
where
The case of arbitrary \(N\ge 1\) is deduced from the case \(N=1\). Indeed, putting
for each \(\varphi _{N}\in C_{0}^{\infty }({\mathbb {R}}^{2N})\) we have
because for any \(p\ne q\) we have that
So in view of inequality (6.8) we have that the right hand side of (6.9) is
hence
where
\(\square \)
By theorem (6.2), in order to assert that \({\mathscr {H}}_{N},D({\mathscr {H}}_{N})=C_{0}^{\infty }({\mathbb {R}}^{2N})\) is essentialy self-adjoint it is sufficient to show that \({\mathscr {V}}_{N}\), viewed as a multiplication operator with domain \(D({\mathscr {V}}_{N})=C_{0}^{\infty }({\mathbb {R}}^{2N})\), is relatively bounded (with relative bound \(<1\)) with respect to \({\mathscr {K}}_{N}\). This is achieved in the following
Lemma 6.7
For each \(0<a<1\) there is some \(b=b(N,\epsilon ,\hbar )>0\) such that for all \(\varphi _{N}\in C_{0}^{\infty }({\mathbb {R}}^{2N})\)
In particular \({\mathscr {H}}_{N}={\mathscr {K}}_{N}+{\mathscr {V}}_{N}\) is essentially self-adjoint on \(C_{0}^{\infty }({\mathbb {R}}^{2N})\).
Proof
Set \(V^{+}:={\textbf{1}}_{|x|\ge 1}\log (|x|)\) and \(V^{-}:={\textbf{1}}_{|x|\le 1}\log (|x|)\).
Step 1. Relative bound for \(\frac{1}{N\epsilon }\underset{p<q}{\sum }V_{pq}^{+}\). For each \(\varphi _{N}\in C_{0}^{\infty }({\mathbb {R}}^{2N})\) one has
Therefore, given \(a>0\) pick \(\eta =\frac{a\epsilon ^{2}}{16(N-1)}\) and \(b_{1}=\frac{4(N-1)}{\epsilon ^{2}\eta }\) in order to find
Step 2. Relative bound for \(\frac{1}{N\epsilon }\underset{p<q}{\sum }V_{pq}^{-}\). The combination of theorems (6.3) and (6.4) with lemma (6.6) implies that for each \(0<a<1\) there is some \(b_{2}=b_{2}(\epsilon ,\hbar ,N)\) such that for all \(\varphi _{N}\in C_{0}^{\infty }({\mathbb {R}}^{2N})\) it holds that
Therefore the combination inequalities (6.12) and (6.13) entails
with \(b=2(b_{1}+b_{2}+1)\). \(\square \)
The following elementary observation will be used to prove lemma (2.2)
Fact 6.8
(Footnote 3 in [7]) If \(S=S^{*}\ge 0\) is unbounded with domain \(D(S)\subset {\mathfrak {H}}\) and \(T=T^{*}\ge 0\) is trace class with eigenvectors \((e_{m})_{m\ge 1}\subset D(S)\) and eigenvalues \((\lambda _{m})_{m\ge 1}\) respectively, then \(TST\ge 0\) and
Proof of lemma (2.2)
Step 1. \(\mathcal {{\mathscr {H}}}_{N}R_{N}(t)^{\frac{1}{2}}\) is Hilbert–Schmidt. Let \(\{e_{m}\}_{m\ge 1}\) be a complete system of eigenfunctions of \(R_{N}^{in}\) with \(R_{N}^{in}e_{m}=\lambda _{m}e_{m}\). By fact (6.8)
Owing to lemma (6.7) we get
which implies that the right hand side of (6.14) is
by assumption.
Step 2. \(\textrm{trace}_{{\mathfrak {H}}_{N}}(\sqrt{R_{N}(t)}\mathcal {{\mathscr {H}}}_{N}\sqrt{R_{N}(t)})\) is constant. As a result of step 1 and Cauchy-Schwarz we see that for all t the operator \(\sqrt{R_{N}(t)}\mathcal {{\mathscr {H}}}_{N}\sqrt{R_{N}(t)}\) is trace class. Furthermore
Step 3. Energy Conservation. Denote by \(S_{N}(t,X_{N},Y_{N})\) the integral kernel of \(\sqrt{R_{N}(t)}\). Then we have
The claim follows from Eq. (6.15). \(\square \)
Notes
This equivalence is shown in [27]
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Acknowledgements
This paper is part of the author’s work towards a PhD. The author would like to express his deepest gratitude towards his supervisor François Golse for many fruitful discussions and for carefully reading previous versions of this manuscript and providing insightful comments and improvements. The author would also like to thank Daniel Han-Kwan and Mikaela Iacobelli for a helpful correspondence, as well as to Matthew Rosenzweig for suggesting a few valuable references and offering comments which improved clarity of exposition. The comments of an anonymous referee have improved the quality of this work, and are gratefully acknowledged. The author declares no conflict of interest. This work was partially supported by École Polytechnique and by the research grant “Stability analysis for nonlinear partial differential equations across multiscale applications”. The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Appendix A
Appendix A
Following closely [16] and [2], we aim to explain here how a classical solution to equation (1.4) \((\omega ,u)\in C^{1}([0,T];C^{0,\alpha }({\mathbb {R}}^{2}))\times C^{1}([0,T];C_{\textrm{loc}}^{1,\alpha }({\mathbb {R}}^{2})\cap L^{\infty }({\mathbb {R}}^{2}))\) with \(\alpha \in (0,1)\), where \(\omega (t,\cdot )\) is a compactly supported probability density on \({\mathbb {R}}^{2}\) and \(\nabla u\in L^{\infty }([0,T];L^{\infty }\cap L^{2}({\mathbb {R}}^{2}))\), can be realized for some suitable initial data. First we recall that given a bounded and summable vorticity \(\omega \), the \(L^{\infty }\) norm of the velocity field attached to \(\omega \) is controlled by the \(L^{\infty }\cap L^{1}\) norm of \(\omega \), as summarized in the following simple lemma.
Lemma 7.1
Let \(\omega \in L^{\infty }({\mathbb {R}}^{2})\cap L^{1}({\mathbb {R}}^{2})\) and let u be the velocity attached to it via the Biot-Savart law, i.e.
Then
Proof
Let \(0\le \eta \le 1,\eta \in C_{0}^{\infty }({\mathbb {R}}^{2})\) such that \(\eta \equiv 1\) on \(B_{1}(0)\) and \(\eta \equiv 0\) outside \(B_{2}(0)\). Then
\(\square \)
Next, below is stated a uniqueness and existence theorem for equation (1.4)
Theorem 7.2
([16], Theorem 8.2) Let the initial vorticity \(\omega _{0}\in L_{0}^{\infty }({\mathbb {R}}^{2})\). Then there exists a unique weak solution (in the sense of Definition 8.1 in [16]) to Eq. (1.4) with \(\omega \in L^{\infty }([0,T];L_{0}^{\infty }({\mathbb {R}}^{2})).\)
Denoting by \(X^{t}(\alpha )\) the particle trajectory map of the velocity (see 1.3 in [16]), we have
Proposition 7.3
([16], Proposition 8.3) Suppose that \(\omega _{0}\in L^{1}({\mathbb {R}}^{2})\cap L^{\infty }({\mathbb {R}}^{2})\) such that \(\omega _{0}\in C^{m,\alpha }(\overline{\Omega _{0}})\) where \(\Omega _{0}\) is open and bounded. Then \(\omega (t,\cdot )\in C_{\textrm{loc}}^{m,\alpha }(\Omega _{t}),u(t,\cdot )\in C_{\textrm{loc}}^{m+1,\alpha }(\Omega _{t})\) where
Therefore given a compactly supported probability density \(\omega _{0}\in C_{\textrm{loc}}^{0,\alpha }({\mathbb {R}}^{2})\) the unique solution provided by theorem (7.2) will be in fact a classical solution \((\omega ,u)\in C^{1}([0,T],C_{\textrm{loc}}^{0,\alpha }({\mathbb {R}}^{2}))\times C^{1}([0,T],C_{\textrm{loc}}^{1,\alpha }({\mathbb {R}}^{2}))\) with compactly supported vorticity. Indeed, since \(\omega (t,x)=\omega _{0}(X^{-t}(x))\), we see that the support of \(\omega (t,x)\) is included in \(\Omega _{t}\) and is locally Hölder regular on \(\Omega _{t}\) and therefore this is true on the entire space \({\mathbb {R}}^{2}\). The \(C_{\textrm{loc}}^{1,\alpha }({\mathbb {R}}^{2})\) regularity for u follows from elliptic regularity and that \(X^{t}\) is a homeomorphism. Of course, that \(\omega (t,\cdot )\) is a probability density for all times \(t\in [0,T]\) is because it satisfies a transport equation (and thus the initial sign and \(L^{1}\) norm are both preserved). Since the Gronwall estimate of Sect. 3 utilizes the fact that \(\nabla u(t,\cdot )\) is square summable, we also recall the Calderon-Zygmund inequality
Lemma 7.4
([16], Lemma 8.3) Suppose \(\omega \in L^{\infty }([0,T];L_{0}^{\infty }({\mathbb {R}}^{2})\) is a weak solution to Eq. (1.4). Then for all \(1<p<\infty \)
When Hölder regularity on \(\omega \) is available, the limit case \(p=\infty \) can also be covered via the following
Lemma 7.5
([2], Proposition 7.7) Assume \(\omega \in L^{a}({\mathbb {R}}^{2})\cap C^{0,\alpha }({\mathbb {R}}^{2})\) for some \(1\le a<\infty \) and \(\alpha \in (0,1)\). Let u be the velocity attached to it. There is a constant C such that
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Porat, I.B. Derivation of Euler’s Equations of Perfect Fluids from von Neumann’s Equation with Magnetic Field. J Stat Phys 190, 121 (2023). https://doi.org/10.1007/s10955-023-03131-5
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DOI: https://doi.org/10.1007/s10955-023-03131-5