Velocity Averaging for the Wigner Kinetic Equation
in the Semiclassical Regime
March 13, 2025
This paper discusses the possibility of applying the velocity averaging theorems in [F. Golse, P.-L. Lions, B. Perthame, R. Sentis: J. Funct. Anal. 76(1):110–125, 1988] to the Wigner equation governing the quantum evolution of the Wigner transform of quantum density operators. Our first main results address the case of the Wigner function of a special class of density operators associated to mixed states, whose Hilbert-Schmidt norm is of order \(\hbar^{d/2}\), where \(d\) is the space dimension and where \(\hbar\) is the reduced Planck constant. We obtain estimates which are uniform in \(\hbar\) in the semiclassical limit \(\hbar \rightarrow 0\). In space dimension \(d=1\), we prove that the density function belongs to the Sobolev space \(H^s(\mathbb{R})\) for some \(s>0\). In the case of pure states, we first obtain a characterization of the Wigner transform of rank-one quantum density operators, and apply this characterization (1) to analyze a rather general setting in which velocity averaging cannot apply to the Wigner functions of a family of rank-one density operators whose evolution is governed by the von Neumann equation, and (2) to obtain a quick derivation of Madelung’s system of quantum hydrodynamic equations. This derivation provides a physical explanation of one key assumption used in the proof of the negative result (1) described above.
Key Words: Quantum dynamics; Density operator; Wigner transform; Velocity averaging; Bohm potential; Madelung equations
MSC: 81S30; 81Q20; 35B65
The free transport, or advection operator \(\partial_t+c\partial_x\) is the prototype of hyperbolic differential operators, together with the d’Alembert, or wave operator \[\partial^2_t-c^2\partial^2_x=(\partial_t+c\partial_x)(\partial_t-c\partial_x).\] The method of characteristics shows that singularities of the initial data propagate in the solution of the Cauchy problem for the free transport equation, which cannot be more regular than its data.
Kinetic models for gases and plasmas systematically involve the free transport operator \(\partial_t+v\cdot\nabla_x\) on \(\mathbb{R}_t\times\mathbb{R}^d_x\times\mathbb{R}^d_v\), with a velocity \(v\in\mathbb{R}^d\) which is itself a variable. In other words, the (main) unknown function in the kinetic theory of gases is \(f\equiv f(t,x,v)\), the phase space density at time \(t\) of the number of gas molecules at the position \(x\) with velocity \(v\). All kinetic equations for gases or plasmas take the form \[(\partial_t+v\cdot\nabla_x)f(t,x,v)=S[f](t,x,f)\] where \(S[f]\) is a functional of the unknown \(f\), such as the collision integral in the case of the Boltzmann or the Landau equations, or a term of the form \(S[f]=\nabla_v\cdot (fF[f])\), where \(F[f]\) is a mean-field, self-consistent acceleration created by the unknown distribution \(f\) itself, such as gravity in the case of the Vlasov-Poisson equation used in cosmology, or the electromagnetic force in the case of the Vlasov-Maxwell equation used in plasma physics.
If \(S[f]\in L^2_{loc}(\mathbb{R}\times\mathbb{R}^d\times\mathbb{R}^d)\) and if \(\tau+v\cdot\xi\not=0\), then \((t,x)\mapsto f(t,x,v)\) is microlocally in the Sobolev space \(H^1\) in some conical neighborhood of \((\tau,\xi)\) around each \((t,x)\in\mathbb{R}\times\mathbb{R}^d\). On the other hand, if \(f\) itself belongs to \(L^2_{loc}(\mathbb{R}\times\mathbb{R}^d\times\mathbb{R}^d)\), the set of velocities \(v\in\mathbb{R}^d\) such that \(\tau+v\cdot\xi=0\) is an affine plane in \(\mathbb{R}^d\), so that its Lebesgue measure is equal to \(0\). Therefore, one can expect that there exists \(s>0\) such that \[\require{physics} \int_{|v|\le R}f(t,x,v)\dd v\in H^s_{loc}(\mathbb{R}\times\mathbb{R}^d)\] (where \(H^s\) designates the Sobolev space of functions with derivatives of order \(\le s\) in \(L^2\)) for all \(R>0\). In other words, averages in \(v\) of \(f\) are more regular in \(t,x\) than \(f\) itself.
They were first studied independently in [1] and [2] in the \(L^2\) case by studying the Fourier transform of \(\partial_t +v\cdot \nabla\), and more systematically in [3]. The subsequent literature is rich: the \(L^p\) case was studied more completely e.g. in [4]–[6]. Velocity averaging fails in \(L^1\) in general, due to possible concentrations in the \(v\) variable, which offset the benefits of averaging: see for instance Example 1 on pp. 123–124 in [3]. We shall see later that this example is important in the present paper. The concentration in \(v\) is essentially the only obstruction to velocity averaging in \(L^1\). Indeed, under some additional condition of equiintegrability in \(\xi\) on \(\{f\}\), it was proved in [7] that velocity averaging yields strong compactness of averages in \(v\) of families \(\{f\}\) that are bounded in \(L^1_{x,v}\) and such that \(\{v\cdot\nabla_xf\}\) is also bounded in \(L^1_{x,v}\). Very recently, new methods have been used to prove new velocity averaging lemmas: see for instance [8], [9].
The interested reader will find a rather complete survey of the main results in this direction published before 2000 in chapter 1 of [10].
Velocity averaging lemmas have been very useful in the study of kinetic equations because they provide strong compactness in \(L^1_{loc}\), and therefore justify passing to the limit in certain nonlinearities. This is a key step in the construction of global weak solutions for the Boltzmann [11], and the Vlasov-Maxwell equations [12], for instance.
On the other hand, the Wigner transform \(W_{\hbar}\) of quantum density operators, first introduced in [13], is the quantum mechanical analogue of the phase space density in classical kinetic equations. In particular it has the correct macroscopic quantities (or observable densities). For instance, the particle density is the zeroth order moment of the Wigner function in the \(\xi\)-variable, the current density is its first order moment and the energy density its second order moment. However, an important caveat is that the Wigner transform takes negative values in general — coherent states, i.e. Gaussian wave packets being the only pure states whose Wigner transform is positive [14]–[16]. Nevertheless, the Wigner function \(W_{\hbar}\) obeys the Wigner equation, which is the quantum analogue to classical kinetic equations for the phase space density. In the case of point particles accelerated by an external potential \(V\), the Wigner function \(W_{\hbar}\) solves \[\partial_tW_{\hbar} +\xi \cdot \nabla_x W_{\hbar} - \theta[V]W_{\hbar} = 0,\] where \(\theta[V]\) is a pseudodifferential operator applied to \(W_{\hbar}\), whose expression is recalled in the next section. This suggests that velocity averaging results applied to \(W_{\hbar}\) should provide valuable information on observable densities, i.e. on moments in \(\xi\) of the Wigner function \(W_{\hbar}\). For instance, one could think of using velocity averaging in the context of the classical limit of quantum dynamics, e.g. to pass to the limit in the nonlinearities which appear in the system of Madelung equations governing the limiting density function and current density.
In order to pass to the limit \(\hbar \rightarrow 0\) it is necessary to obtain estimates which are uniform in \(\hbar\). An averaging lemma that gives such estimates is called semiclassical averaging lemma, cf. [17], as opposed to estimates that might blow up when \(\hbar \rightarrow 0\) (such averaging lemmas are called quantum averaging lemmas). We prove such a result for the Wigner equation with potential in Theorem 2 for the Wigner function of a special class of mixed states.
The semiclassical limit of Madelung’s equations has been studied in the case of the defocussing, cubic nonlinear Schrödinger equation in space dimension one in [18] by the methods of integrable systems. Velocity averaging does not play any role in [18], and we shall see below that there is a serious objection to using velocity averaging on families of pure quantum states in the semiclassical regime.
The purpose of this paper is to understand how the methods of velocity averaging can be used in the context of quantum dynamics sketched above, where one has to distinguish between quantum and semiclassical velocity averaging on the one hand and pure and mixed states on the other hand. The outline of our discussion of this problem is as follows: section 2 recalls some important features of the Wigner transform, used in the sequel.
Next we study velocity averaging results in an \(L^2\) setting adapted to the equations of quantum dynamics, i.e. the von Neumann equation governing the evolution of quantum density operators associated to quantum states.
Our first main result in this direction is Theorem 2 in section 3, which deals with a special class of mixed states for density operators bounded in Hilbert-Schmidt norm independently, or uniformly in the Planck constant.
Section 4 studies the same problem in the special case of space dimension one with a different approach, leading to a physically more satisfying result, bearing on the “true” density function, instead of moments of the Wigner functions involving a truncation for large values of the momentum variable as in Theorem 2. This result, which is our second main result in this paper, is stated as Theorem 3 in section 4.
Section 6 discusses the case of pure states, left aside in sections 3 and 4, focussed on mixed states only. Proposition 2, our third main result, shows that sequences of pures states with vanishing Bohm pressure in the classical limit, with density functions and current densities converging strongly in \(L^2_{loc}\) and kinetic energy densities converging weakly in \(L^2_{loc}\) have monokinetic Wigner measures. (We recall that Wigner measures are limit points in the sense of distributions of the Wigner functions in the classical limit.) This concentration effect on the momentum variable in the Wigner measure explains why velocity averaging cannot be applied to families of pure states in the semiclassical setting.
The key argument in the proof of Proposition 2 is an identity 31 which follows from a characterization of pure states reported in Lemma 1, which generalizes to the case of all space dimensions (and makes mathematically rigorous) an earlier observation in [19] due to Tatarskiı̆. This characterization of pure states and further consequences thereof are discussed in section 5.
Finally, section 7 explains how 31 coming from our characterization of the Wigner transforms of pure states in Lemma 1 leads to a quick derivation of the system of Madelung fluid dynamical equations for the mass and current densities of pure states governed by the Schrödinger equation with Lipschitz continuous external potential. This clarifies the physical meaning of one condition used in Proposition 2 leading to monokinetic Wigner measures. It also addresses the vacuum issue, i.e. points where the wave function vanishes, whose treatment had been left incomplete in earlier derivations of Madelung’s equations.
Let \(\mathfrak{H}\) be a Hilbert space. The algebra of bounded operators on \(\mathfrak{H}\) is denoted by \(\mathcal{L}(\mathfrak{H)}\). We denote the two-sided ideal of trace-class operators on \(\mathfrak{H}\) by \(\mathcal{L}^1(\mathfrak{H})\). A density matrix \(R\in \mathcal{L}^1(\mathfrak{H})\) is a self-adjoint (\(R=R^*\)), nonnegative (\(\langle R\psi,\psi\rangle \geq 0\) for all \(\psi \in \mathfrak{H}\)) operator with trace \(\tr_{\mathfrak{H}} R = 1\). A rank-one density operator is a pure state — see section 6 below. If \(\mathfrak{H} = L^2(\mathbb{R}^d)\), the operator \(R\) is an integral operator with integral kernel \((X,Y)\mapsto R(X,Y)\) which belongs to \(L^2(\mathbb{R}^d\times \mathbb{R}^d)\). For the sake of simplicity, we shall abusively denote by \(R(X,Y)\) the integral kernel of the density operator \(R\) — so that \(R\) designates both an operator on \(L^2(\mathbb{R}^d)\) and an element of \(L^2(\mathbb{R}^d\times \mathbb{R}^d)\). Since \(R=R^*\) is trace-class, it is compact and by the spectral theorem there is a sequence of real eigenvalues \((\lambda_j)_{j\ge 1}\) and a Hilbert basis (i.e. a complete orthonormal system) \(\{\psi_j\,:\,j\ge 1\} \subset \mathfrak{H}\) such that \[\label{SpecDecR} R(X,Y) = \sum_{j=1}^{\infty} \lambda_j \psi_j(X) \overline{\psi_j(Y)}.\tag{1}\] Since \(R\) is a density operator, \[\label{CondLambda} \lambda_j\ge 0\quad\text{since }R=R^*\ge 0,\quad\text{ and }\sum_{j\ge 1}\lambda_j=\tr_{L^2(\mathbb{R}^d)}R=1.\tag{2}\]
In particular, the integral kernel \(R\) has the property that the map \[\label{IntKerTrCl} z\mapsto(X\mapsto R(X,X+z))\quad\text{ belongs to }C_b(\mathbb{R}^d, L^1(\mathbb{R}^d_X)),\tag{3}\] so that the restriction of the integral kernel to the diagonal, i.e. \(R(X,X)\), is a well-defined element of \(L^1(\mathbb{R}^d_X)\). The density function \(\rho\in L^1(\mathbb{R}^d)\) is \[\label{DefDensityFunc} \rho(X) := R(X,X).\tag{4}\] Let us define \[\require{physics} \widehat{R}(\Xi,H) := \mathcal{F}_{X\rightarrow \Xi}\overline{\mathcal{F}_{Y\rightarrow H}} [R(\cdot,\cdot)](\Xi,H)=\int_{\mathbb{R}^d\times\mathbb{R}^d}e^{-iX\cdot\Xi+iY\cdot H} {R(X,Y)} \dd X \dd Y,\] and set \[\require{physics} \label{DefHatRho} \hat{\rho}(\Xi) := \widehat{R}(\Xi,\Xi) = \int e^{-i(X-Y)\cdot \Xi} {R(X,Y)} \dd X \dd Y.\tag{5}\] (Notice that \(\hat{\rho}\) is not the Fourier transform of \(\rho\).) We recall that the trace of \(R\) is \[\require{physics} \tr_{L^2(\mathbb{R}^d)} R = \int_{\mathbb{R}^d} \rho(X) \dd X\]
In quantum mechanics, the analogue of the Liouville equation satisfied by the phase-space density of particles in classical mechanics is the von Neumann equation \[\label{eq:von32Neumann} \left\{ \begin{align} {}&i\hbar \partial_t R(t) = [\mathcal{H},R(t)], \\ &R(0) = R_{0}, \end{align} \right.\tag{6}\] where \(\mathcal{H}\) is the quantum Hamiltonian \[\mathcal{H}:=-\tfrac{\hbar^2}{2m}\Delta+V,\] and \(V\) is the multiplication operator defined by the formula \((V\psi)(x)=V(x)\psi(x)\), assuming that \(x\mapsto V(x)\) is a real-valued function defined on \(\mathbb{R}^d\) and such that \(\mathcal{H}=\mathcal{H}^*\) on \(L^2(\mathbb{R}^d)\). By Stone’s theorem, \(-i\mathcal{H}/\hbar\) is the generator of a unitary group \(e^{-it\mathcal{H}/\hbar}\) on \(L^2(\mathbb{R}^d)\), and the (generalized) solution of 6 is \[R(t) = e^{-it\mathcal{H}/\hbar}R_{0} e^{it\mathcal{H}/\hbar}, \quad t\in \mathbb{R},\] for each density operator \(R_0\) on \(L^2(\mathbb{R}^d)\).
Define the Weyl variables1 corresponding to \(X,Y\) by \[\label{WeylVar} \left\{ \begin{align} x=\tfrac12(X+Y), \\ y=\tfrac1\hbar(X-Y), \end{align} \right.\tag{7}\] and consider the Weyl-variable density matrix \(\tilde{R}\) via \[\label{eq:weyl32variable32density32matrix} \tilde{R}(t,x,y):= R(t,X,Y)=R(t,x+\tfrac\hbar{2}y,x-\tfrac\hbar{2}y).\tag{8}\] The Wigner transform \(W_{\hbar}\) of \(R\) is defined as \[\require{physics} W_{\hbar}[R](t,x,\xi):=\tfrac1{(2\pi)^d}\mathcal{F}_{y\rightarrow \xi} [\tilde{R}](t,x,\xi)=\tfrac{1}{(2\pi)^d} \int_{\mathbb{R}^d} e^{-i\xi\cdot y} \tilde{R}(t,x,y) \dd y,\] so that \[\label{eq:inverse32relation} \tilde{R}(t,x,y) = (2\pi)^d\mathcal{F}^{-1}_{\xi \rightarrow y}[W_{\hbar}](t,x,y).\tag{9}\] Therefore we can define the moments of the Wigner function in the \(\xi\) variable, such as the particle density \[\require{physics} \label{eq:restriction32density} \rho_{\hbar}(t,x) := R(t,x,x) = \tilde{R}(t,x,y)\Big|_{y=0} = \int_{\mathbb{R}^d}W_{\hbar}(t,x,\xi)\dd \xi,\tag{10}\] the current density \[\require{physics} \label{eq:restriction32current} J_{\hbar}(t,x) := -\tfrac{i}{m}\nabla_y\tilde{R}(t,x,y)\Big|_{y=0} = \tfrac1m\int_{\mathbb{R}^d}\xi W_{\hbar}(t,x,\xi)\dd \xi,\tag{11}\] and the kinetic energy density \[\require{physics} \label{eq:restriction32energy} \mathcal{E}_{\hbar}(t,x) := -\tfrac1{2m}\Delta_y^2\tilde{R}(t,x,y)\Big|_{y=0} = \tfrac1{2m}\int_{\mathbb{R}^d}|\xi|^2 W_{\hbar}(t,x,\xi)\dd \xi.\tag{12}\]
Recall the following definition:
Definition 1. A family of measures \(\rho_{\varepsilon}\) on \(\mathbb{R}^d\), indexed by \(\varepsilon\in(0,1]\), is tight if, \[\require{physics} \sup_{0<\varepsilon\leq 1}\int_{|x|\geq R}\rho_{\varepsilon}(\dd x) \to 0 \quad \text{as } R\to \infty.\] In the present paper, the parameter \(\varepsilon\) is the Planck constant \(\hbar\). Some authors define a family \(\{\psi_{\varepsilon}\,:\,\varepsilon\in(0,1]\}\) as being compact at infinity if the measure with density \(\rho_{\varepsilon}:=|\psi_{\varepsilon}|^2\) (with respect to the Lebesgue measure on \(\mathbb{R}^d\)) is tight.
The Wigner transform enjoys the following properties, see Theorem III.2 in [22]:
Proposition 1. Let \(R_\hbar\in \mathcal{L}^1(L^2(\mathbb{R}^d))\) be a family of density operators indexed by \(\hbar\in(0,1]\) with density functions \(\rho_{\hbar} \in L^1(\mathbb{R}^d)\). Then
The family \(\{W_{\hbar}[R]\,:\,0<\hbar\le 1\}\) is bounded in \(\mathcal{S}'(\mathbb{R}^d_x\times \mathbb{R}^d_{\xi})\); in particular there exists a subsequence \(\hbar_k\) such that \(W_{\hbar_k}[R]\) converges to a Wigner measure* \(W[R]\) in \(\mathcal{S}'(\mathbb{R}^d_x\times \mathbb{R}^d_{\xi})\) as \(\hbar_k \rightarrow 0\).*
The Wigner measure \(W\) is a bounded, nonnegative Radon measure on \(\mathbb{R}^d_x\times \mathbb{R}^d_{\xi}\).
If the families \(\{\rho_{\hbar}(x)\,:\,0<\hbar\le 1\}\) and \(\{\hbar^{-d}\widehat{\rho}_{\hbar}(\xi/\hbar)\,:\,0<\hbar\le 1\}\), where \(\widehat{\rho}_\hbar\) is defined in 5 , are tight and if \(\rho_{\hbar}\to\rho\) vaguely as \(\hbar\to 0\) (in the sense of Radon measures on \(\mathbb{R}^d\)), then \[\require{physics} \label{equality32wigner32measure32density} \rho = \int_{\mathbb{R}^d}W(\cdot,\xi)\dd \xi,\qquad{(1)}\] and \[\require{physics} \label{limit32wigner32integral} \int_{\mathbb{R}^d\times \mathbb{R}^d}W(x,\xi) \dd x \dd \xi = \lim_{\hbar\rightarrow 0} \int_{\mathbb{R}^d} \rho_{\hbar}(x) \dd x.\qquad{(2)}\]
The Weyl-variable density matrix \(\tilde{R}\), defined in 8 , obeys the von Neumann equation in the form \[\begin{align} {}&\partial_t \tilde{R}_\hbar + \tfrac1m\nabla_x\cdot(-i\nabla_y)\tilde{R}_\hbar -\delta[V] \tilde{R}_\hbar = 0,\tag{13} \\ &\tilde{R}_\hbar(0,x,y) = \tilde{R}_{\hbar,0}(x,y),\tag{14} \end{align}\] where \[\label{eq:def32delta} \delta[V](x,y) := \frac{V(x+\frac{\hbar}{2}y)-V(x-\frac{\hbar}{2}y)}{i\hbar}.\tag{15}\] By taking the Fourier transform in \(y\) of 13 14 one sees that the Wigner transform \(W_{\hbar}:=(2\pi)^{-d}\mathcal{F}_{y\to\xi}\tilde{R}_\hbar\) solves the Wigner equation \[\begin{align} {}&\partial_tW_{\hbar} +\tfrac1m\xi \cdot \nabla_x W_{\hbar} - \theta[V]W_{\hbar} = 0,\tag{16} \\ &W_{\hbar}(0,x,\xi) = W_{\hbar,0}(x,\xi),\tag{17} \end{align}\] where \[W_{\hbar,0}(x,\xi) := (2\pi)^{-d}\mathcal{F}_{y\rightarrow \xi}\tilde{R}_0(x,\xi),\] and where \[\require{physics} \label{eq:def32theta32V} \theta[V]W_{\hbar}(x,\xi) := \tfrac{1}{(2\pi)^d}\int_{\mathbb{R}^d\times\mathbb{R}^d}e^{-i(\xi-\eta) \cdot y}\delta[V](x,y)W_{\hbar}(x,\eta) \dd y \dd\eta.\tag{18}\] Note that this can be rewritten as a convolution in the \(\xi\)-variable. Let \[\require{physics} \label{eq:def K[V]} K[V](x,\xi):= \tfrac{1}{(2\pi)^d}\int_{\mathbb{R}^d}e^{-i\xi \cdot y}\delta[V](x,y)\dd y;\tag{19}\] with this definition \[\label{FlaTheta[V]} \theta[V]W_{\hbar} = K[V]\star_{\xi}W_{\hbar}.\tag{20}\] This structure will be exploited in the proof of the averaging lemma for \(W_{\hbar}\) below.
To begin with, let us recall the basic velocity averaging theorem of interest in connection with the Wigner equation. This is an extension due to DiPerna and Lions [12] of the main result in [3], which is particularly well suited to handle Vlasov-type equations — and was used to prove the global existence of weak solutions to the Cauchy problem for the Vlasov-Maxwell system, for all square-integrable initial data with finite mass and energy.
Theorem 1. Let \(n\ge 0\) and \(f \in L^2(\mathbb{R}\times \mathbb{R}^d_x \times \mathbb{R}^d_{\xi})\) satisfy \[(\partial_t +\xi \cdot \nabla_x)f = g \quad \text{in }\mathcal{D}'(\mathbb{R}\times \mathbb{R}^d_x \times \mathbb{R}^d_{\xi})\] where \(g \in L^2(\mathbb{R} \times \mathbb{R}^d_x, H^{-n}(\mathbb{R}^d_{\xi}))\). Then for each \(\psi\in\mathcal{S}(\mathbb{R}^d)\), \[\require{physics} \int_{\mathbb{R}^d}f(\cdot,\cdot,\xi) \psi(\xi) \dd \xi \in H^{s}(\mathbb{R}\times \mathbb{R}^d),\] where \[s=\frac{1}{2(n+1)}.\]
In order to apply Theorem 1 to the Wigner equation, we first need an \(L^2\) bound on the Wigner transforms of the family of density operators under consideration. Since we are ultimately interested in “semiclassical velocity averaging" – where \(\hbar\to 0\) instead of being a physical constant – this \(L^2\) bound should in particular be uniform in the”small parameter” \(\hbar\). It has been observed in the semiclassical analysis of the Wigner-Poisson equation in [22], [23], that the \(L^2\) norm of the Wigner transform \(W_{\hbar}[R]\) of a density matrix \(R\) is bounded independently of \(\hbar\) if and only if \(R\) belongs a certain class of mixed states, see for instance Theorem IV.5 in [22]. Indeed, by Plancherel’s theorem \[\require{physics} \int_{\mathbb{R}^d}|W_\hbar[R](x,\xi)|^2\dd\xi=\tfrac1{(2\pi)^d}\int_{\mathbb{R}^d}|\tilde{R}(x,y)|^2\dd y,\] so that \[\require{physics} \label{WignerL2Bound} \begin{align} \iint_{\mathbb{R}^d\times\mathbb{R}^d}|W_\hbar[R](x,\xi)|^2\dd x\dd\xi=&\tfrac1{(2\pi)^d}\iint_{\mathbb{R}^d\times\mathbb{R}^d}|\tilde{R}(x,y)|^2\dd x\dd y \\ =&\tfrac1{(2\pi\hbar)^d}\iint_{\mathbb{R}^d\times\mathbb{R}^d}|R(X,Y)|^2\dd X\dd Y, \end{align}\tag{21}\] where the last equality follows from the change of variables 7 . On the other hand, 1 implies that \[\require{physics} \label{TrR2} \tr_{L^2(\mathbb{R}^d)}(R^2)=\iint_{\mathbb{R}^d\times\mathbb{R}^d}|R(X,Y)|^2\dd X\dd Y=\sum_{j\ge 1}\lambda_j^2.\tag{22}\] Applying these formulas to a family \(\{R_\hbar\,:\,0<\hbar\le 1\}\) of density operators on \(L^2(\mathbb{R}^d)\), whose eigenvalues are denoted by \(\lambda_{\hbar,j}\), one finds that \[\sum_{j\ge 1}\lambda_{\hbar,j}^2\le (2\pi\hbar)^d\sup_{0<\hbar\le 1}\|W_\hbar[R_\hbar]\|_{L^2}^2.\] We recall from 2 that \[\lambda_{\hbar,j}\ge 0\quad\text{ and }\quad\sum_{j\ge 1}\lambda_{\hbar,j}=1.\] By the Cauchy-Schwarz inequality \[1=\left(\sum_{j\ge 1}\lambda_{\hbar,j}\right)^2\le\sum_{j\ge 1}\mathbf{1}_{\lambda_{\hbar,j}>0}\sum_{j\ge 1}\lambda_{\hbar,j}^2\le\text{rank}R_\hbar\cdot(2\pi\hbar)^d\sup_{0<\hbar\le 1}\|W_\hbar[R_\hbar]\|_{L^2}^2,\] Therefore, assuming that a family \(\{R_\hbar\,:\,0<\hbar\le 1\}\) of density operators on \(L^2(\mathbb{R}^d)\) satisfies \[\sup_{0<\hbar\le 1}\|W_\hbar[R_\hbar]\|_{L^2}^2=C<\infty\] implies that \[\text{rank}R_\hbar\ge\frac{1}{(2\pi\hbar)^dC},\qquad\text{ so that }\varliminf_{\hbar\to 0}\left(\hbar^d\text{rank}R_\hbar\right)>0.\] In particular, this assumption rules out the possibility that \(\{R_\hbar\,:\,0<\hbar\le 1\}\) is a family of pure states (i.e. rank-one density operators).
The following theorem compiles two results: A quantum averaging lemma, where the constant on the right hand side depends on \(\hbar\) in such a way that it blows up as \(\hbar \rightarrow 0\) and a semiclassical averaging lemma, where the constant on the right hand side is independent of \(\hbar\). It is important to note that in the semiclassical setting the gain in regularity is smaller than in the purely quantum case (\(H^{1/4}\) as opposed to \(H^{1/2}\)).
Theorem 2. Let \(V\equiv V(x)\) be a real-valued continuous function defined on \(\mathbb{R}^d\), and let \(\{R_\hbar\,:\,\hbar\in(0,1]\}\) be a family of time-dependent density operators \(t\mapsto R_\hbar(t)\) defined for all \(t\in[-T,T]\) (with \(T>0\)) which are continuous on \(\mathbb{R}\) for the weak operator topology, are weak solutions of the von Neumann equation \[i\hbar \partial_t R_\hbar(t)= [-\tfrac{\hbar^2}{2m}\Delta + V,R_\hbar(t)],\] and satisfy the bound \[\sup_{|t|\le T}\tr_{L^2(\mathbb{R}^d)}(R_\hbar(t)^2)\le C^2(2\pi\hbar)^d\] for some \(C>0\). For each \(\psi\in\mathcal{S}(\mathbb{R}^d)\), set \[\require{physics} \rho_\psi[R_\hbar](t,x):=\int_{\mathbb{R}^d}W_\hbar[R_\hbar](t,x,\xi)\psi(\xi)\dd\xi.\] (1) If \(V\in L^\infty(\mathbb{R}^d)\), for each \(T>0\), there exists \(C'_T>0\) such that \[\sup_{0<\hbar\le 1}\|\rho_\psi[R_\hbar]\|_{H^{1/2}((-T,T)\times\mathbb{R}^d)}\le C'_T\|V\|_{L^\infty(\mathbb{R}^d)}\hbar^{-1/2}.\] (2) If \(V\) is Lipschitz-continuous on \(\mathbb{R}^d\), for each \(T>0\), \[\sup_{0<\hbar\le 1}\|\rho_\psi[R_\hbar]\|_{H^{1/4}((-T,T)\times\mathbb{R}^d)}<\infty.\]
Proof. First, 21 and 22 imply that \[\label{BoundWhbar} \|W_\hbar[R_\hbar(t)]\|^2_{L^2(\mathbb{R}^d\times\mathbb{R}^d)}\le C\tag{23}\] uniformly in \(\hbar\in(0,1]\) and in \(t\in[-T,T]\). On the other hand \[(\partial_t+\tfrac1m\xi\cdot\nabla_x)W_\hbar[R_\hbar]=\theta[V]W_\hbar[R_\hbar]=K[V]\star_\xi W_\hbar[R_\hbar]\] by 16 and 20 .
By Plancherel’s theorem, \[\begin{align} \|(K[V]\star_\xi W_\hbar[R_\hbar])(t,x,\cdot)\|_{L^2(\mathbb{R}^d)}\le&\sup_{z\in\mathbb{R}^d}|\mathcal{F}_{\xi\to z}K[V](x,z)|\|W_\hbar[R_\hbar](t,x,\cdot)\|_{L^2(\mathbb{R}^d)} \\ \le&\tfrac2\hbar\|V\|_{L^\infty(\mathbb{R}^d)}\|W_\hbar[R_\hbar](t,x,\cdot)\|_{L^2(\mathbb{R}^d)}\le\tfrac{2C}\hbar\|V\|_{L^\infty(\mathbb{R}^d)}, \end{align}\] since \[\mathcal{F}_{\xi\to z}K[V](x,z)=\delta[V](x,-z),\] so that statement (1) follows from Theorem 1 with \(n=0\).
Next define \[\require{physics} \label{DefL[V]} L[V](x,\xi) := \tfrac1{(2\pi)^d}\int_{\mathbb{R}^d}e^{-i\xi \cdot y} \frac{y}{|y|} \frac{V(x+\frac{\hbar}{2}y)-V(x-\frac{\hbar}{2}y)}{\hbar|y|} \dd y,\tag{24}\] and observe that \[\require{physics} \begin{align} \nabla_\xi\cdot L[V](x,\xi)=&\tfrac1{(2\pi)^d}\int_{\mathbb{R}^d}e^{-i\xi \cdot y} \frac{-iy\cdot y}{|y|} \frac{V(x+\frac{\hbar}{2}y)-V(x-\frac{\hbar}{2}y)}{\hbar|y|} \dd y \\ =&\tfrac1{(2\pi)^d}\int_{\mathbb{R}^d}e^{-i\xi \cdot y}\delta[V](x,y)\dd y=K[V](x,\xi)\,. \end{align}\] Therefore \[\label{vNeqDivXi} (\partial_t+\tfrac1m\xi\cdot\nabla_x)W_\hbar[R_\hbar]=\nabla_\xi\cdot L[V]\star_\xi W_\hbar[R_\hbar]=\nabla_\xi\cdot(L[V]\star_\xi W_\hbar[R_\hbar]).\tag{25}\] Reasoning as above, and denoting by \(\mathrm{Lip}(V)\) the Lipschitz constant of \(V\), we find that \[\label{BoundL[V]Wh} \begin{align} \|(L[V]\star_\xi W_\hbar[R_\hbar])(t,x,\cdot)\|_{L^2(\mathbb{R}^d)}\le&\sup_{z\in\mathbb{R}^d}|\mathcal{F}_{\xi\to z}L[V](x,z)|\|W_\hbar[R_\hbar])(t,x,\cdot)\|_{L^2(\mathbb{R}^d)} \\ \le&\mathrm{Lip}(V)\|W_\hbar[R_\hbar])(t,x,\cdot)\|_{L^2(\mathbb{R}^d)}\le C\mathrm{Lip}(V), \end{align}\tag{26}\] since \[\label{BoundL[V]} |\mathcal{F}_{\xi\to z}L[V](x,z)|=\left|i\delta[V](x,y)\tfrac{y}{|y|^2}\right|=\tfrac{|\delta[V]|}{|y|}\le\mathrm{Lip}(V).\tag{27}\] With Theorem 1, both inequalities 23 and 26 , together with 25 , imply statement (2) in the theorem. ◻
Remark 1. Observe that the proof of Theorem 2, and especially 27 , is based on the inequality \[\|f\star g\|_{L^2(\mathbb{R}^d)}\le\|\mathcal{F} f\|_{L^\infty(\mathbb{R}^d)}\|g\|_{L^2(\mathbb{R}^d)}\,,\] a straightforward consequence of Plancherel’s theorem which improves upon Young’s convolution inequality \[\|f\star g\|_{L^2(\mathbb{R}^d)}\le\|f\|_{L^1(\mathbb{R}^d)}\|g\|_{L^2(\mathbb{R}^d)}.\] The latter inequality would lead to more stringent constraints on the potential \(V\), in other words imposing that \(V\) or \(\nabla V\) belong to \(\mathcal{F}L^1(\mathbb{R}^d)\) instead of \(L^\infty(\mathbb{R}^d)\).
In this section, we shall study the question of semiclassical velocity averaging in the special case of space dimension \(d=1\), with a slightly different approach. As a matter of fact, we shall be dealing with the integral kernel of the density operator directly, instead of its Wigner transform, so that the term “velocity averaging” is somewhat improper to designate this approach.
Our starting point is 13 , recalled for the reader’s convenience in the form \[\partial_t\tilde{R}(t,x,y)+\tfrac1m\partial_y(-i\partial_x)\tilde{R}(t,x,y)=y\frac{\delta[V](x,y)}{y}\tilde{R}(t,x,y),\] with \(\delta[V](x,y)\) given by 15 . (By comparison with 13 , notice that we have recast the operator \(\nabla_x\cdot(-i\nabla_y)\) as \(\partial_y(-i\partial_x)\), since the discussion in the present section involves the partial Fourier transform in the \(x\)-variable, instead of the Wigner transform of \(R(t)\), which is proportional to the partial Fourier transform of \(\tilde{R}(t,x,y)\) in the \(y\)-variable.) Consider the Laplace transform of \(\tilde{R}\): \[\require{physics} \Lambda\tilde{R}(\omega,x,y):=\int_0^\infty e^{-\omega t}\tilde{R}(t,x,y)\dd t.\] Since \(\tilde{R}\) satisfies 13 , its Laplace transform satisfies, for each \(\omega>0\) \[\tfrac1m\partial_y(-i\partial_x)\Lambda\tilde{R}(\omega,x,y)=y\frac{\delta[V](x,y)}{y}\Lambda\tilde{R}(\omega,x,y)+\tilde{R}^{in}(x,y)-\omega\Lambda\tilde{R}(\omega,x,y)\] One has \[\require{physics} \begin{align} \tfrac1{2\pi}\iint_{\mathbb{R}\times\mathbb{R}}|\tilde{R}(t,x,y)|^2\dd x\dd y=&\tfrac1{2\pi\hbar}\iint_{\mathbb{R}\times\mathbb{R}}|R(t,X,Y)|^2dXdY \\ =&\tfrac1{2\pi\hbar}\mathrm{Tr}_{L^2(\mathbb{R})}\left(R^2(t)\right)=\tfrac1{2\pi\hbar}\mathrm{Tr}_{L^2(\mathbb{R})}\left((R^{in})^2\right)\le C^{in}, \end{align}\] since \(R(t)\) is obtained from \(R^{in}\) by conjugation with the (unitary) Schrödinger group. Hence \[\require{physics} \|\Lambda\tilde{R}(\omega,\cdot,\cdot)\|_{L^2(\mathbb{R}\times\mathbb{R})}\le\int_0^\infty e^{-\omega t}\|\tilde{R}(t,\cdot,\cdot)\|_{L^2(\mathbb{R}\times\mathbb{R})}\dd t\le\frac{2\pi C^{in}}{\omega},\] and therefore \[\tfrac1m\partial_y(-i\partial_x)\Lambda\tilde{R}(\omega,x,y)=u_0(x,y)+yu_1(x,y)\] with \[u_1(x,y):=\tilde{R}^{in}(x,y)-\omega\Lambda\tilde{R}(\omega,x,y),\qquad u_2(x,y):=\frac{\delta[V](x,y)}{y}\Lambda\tilde{R}(\omega,x,y),\] and \[\|u_1\|_{L^2(\mathbb{R}\times\mathbb{R})}\le 4\pi C^{in},\qquad\|u_2\|_{L^2(\mathbb{R}\times\mathbb{R})}\le 2\pi C^{in}\mathrm{Lip}(V).\] We seek to obtain regularity in \(x\) on the Laplace transform of the density function \[\require{physics} \Lambda\rho(\omega,x)=\Lambda\tilde{R}(\omega,x,x)=\int_\mathbb{R}\Lambda W_\hbar[R](\omega,x,\xi)\dd\xi\] Notice the difference with Theorem 2: one seeks information on \(\rho_\psi[R]\) with \(\psi\equiv 1\) instead of \(\psi\in\mathcal{S}(\mathbb{R})\). This situation is a special case of the following result.
Theorem 3. Assume that \(R=R^*\in\mathcal{L}^1(L^2(\mathbb{R}))\) is such that \(\tilde{R}(x,y):=R(x+\tfrac\hbar{2}y,x-\tfrac\hbar{2}y)\) satisfies \[\partial_y(-i\partial_x)\tilde{R}(x,y)=\sum_{k=0}^nu_k(x,y)y^k,\] with \(u_0,\ldots,u_n\in L^2(\mathbb{R}\times\mathbb{R})\). Then the function \(x\mapsto\rho(x):=R(x,x)=\tilde{R}(x,0)\) belongs to the Sobolev space \(H^\frac{1}{2(n+1)}(\mathbb{R})\), and there exists \(C>0\) such that, for all \(R\in\mathcal{L}^1(L^2(\mathbb{R}))\) satisfying the assumptions above \[\|\rho\|_{H^\frac{1}{2(n+1)}(\mathbb{R})}\le C\left(\mathrm{tr}_{L^2(\mathbb{R})}(|R|)+\|\tilde{R}\|_{L^2(\mathbb{R}\times\mathbb{R})}\sum_{k=0}^n\|b_k\|^2_{L^2(\mathbb{R}\times\mathbb{R})}\right),\] where the constant \(C\) is independent of \(\hbar\).
Proof. Applying the Fourier transform \(\mathcal{F}_{x\to\xi}\) to both sides of the equality satisfied by \(\tilde{R}(x,y)\), and setting \(f(\xi,y):=\mathcal{F}_{x\to\xi}[\tilde{R}(\cdot,y)](\xi)\), we find that \[\|f\|_{L^2(\mathbb{R}\times\mathbb{R})}=\sqrt{2\pi}\|\tilde{R}\|_{L^2(\mathbb{R}\times\mathbb{R})},\] by Plancherel’s theorem, while \(b_k(\xi,y):=\mathcal{F}_{x\to\xi}[u_k(\cdot,y)](\xi)\) satisfies \[\|b_k\|_{L^2(\mathbb{R}\times\mathbb{R})}=\sqrt{2\pi}\|u_k\|_{L^2(\mathbb{R}\times\mathbb{R})}.\] Finally \[\xi\partial_yf(\xi,y)=\sum_{k=0}^nb_k(\xi,y)y^k.\] Set \(G(z):=e^{-z^2/2}/\sqrt{2\pi}\), the centered, reduced Gaussian density on \(\mathbb{R}\). For each \({\epsilon}>0\), write the decomposition \[\require{physics} \begin{align} f(\xi,0)=&\int_0^\infty{\epsilon}G({\epsilon}y)\left(f(\xi,y)+f(\xi,-y)+2f(\xi,0)-f(\xi,y)-f(\xi,-y)\right)\dd y \\ =&\int_\mathbb{R}{\epsilon}G({\epsilon}y)f(\xi,y)\dd y\!-\!\int_0^\infty{\epsilon}G({\epsilon}y)\left(\int_0^y(\partial_2f(\xi,z)-\partial_2f(\xi,-z))\dd z\right)\dd y \\ =&\int_\mathbb{R}{\epsilon}G({\epsilon}y)f(\xi,y)\dd y\!-\!\frac{1}{\xi}\int_0^\infty{\epsilon}G({\epsilon}y)\left(\int_0^y\sum_{k=0}^n\beta_k(\xi,z)z^k\dd z\right)\dd y, \end{align}\] where \[\beta_k(\xi,z)=b_k(\xi,z)-(-1)^kb_k(\xi,-z),\qquad k=0,\ldots,n.\] Applying Fubini’s theorem to the summable function \((y,z)\mapsto {\epsilon}G({\epsilon}y)\beta_k(\xi,z)z^k\mathbf{1}_{0<z<y}\) defined a.e. on \(\mathbb{R}\times\mathbb{R}\), leads to \[\require{physics} \begin{align} f(\xi,0)=&\int_\mathbb{R}{\epsilon}G({\epsilon}y)f(\xi,y)\dd y-\frac{1}{\xi}\sum_{k=0}^n\int_0^\infty\left(z^k\int_z^\infty{\epsilon}G({\epsilon}y)\dd y\right)\beta_k(\xi,z)\dd z \\ =&\int_\mathbb{R}{\epsilon}G({\epsilon}y)f(\xi,y)\dd y-\frac{1}{\xi}\sum_{k=0}^n\frac{1}{{\epsilon}^k}\int_0^\infty\left({\epsilon}^kz^k\int_{{\epsilon}z}^\infty G(Y)\dd Y\right)\beta_k(\xi,z)\dd z. \end{align}\] Hence \[\require{physics} \frac{|f(\xi,0)|^2}{n+2}\le\left(\int_\mathbb{R}{\epsilon}G({\epsilon}y)f(\xi,y)\dd y\right)^2 \!\!+\sum_{k=0}^n\frac{1}{{\epsilon}^{2k}|\xi|^2}\left(\int_0^\infty\left({\epsilon}^kz^k\int_{{\epsilon}z}^\infty G(Y)\dd Y\right)\beta_k(\xi,z)\dd z\right)^2\!\!.\] By the Cauchy-Schwarz inequality, \[\require{physics} \left(\int_\mathbb{R}{\epsilon}G({\epsilon}y)f(\xi,y)\dd y\right)^2\le{\epsilon}\int_\mathbb{R} G(Y)^2\dd Y\int_\mathbb{R}|f(\xi,y)|^2\dd y=\frac{\epsilon}{2\sqrt\pi}\int_\mathbb{R}|f(\xi,y)|^2\dd y,\] while \[\require{physics} \begin{align} \left(\int_0^\infty\left({\epsilon}^kz^k\int_{{\epsilon}z}^\infty G(Y)\dd Y\right)\beta_k(\xi,z)\dd z\right)^2 \\ \le\frac{1}{\epsilon}\int_0^\infty Z^{2k}\left(\int_Z^\infty G(Y)\dd Y\right)^2\dd Z\int_0^\infty|\beta_k(\xi,z)|^2\dd z \\ \le\frac{1}{\epsilon}\int_0^\infty Z^{2k}\left(\int_Z^\infty G(Y)\dd Y\right)\dd Z\int_0^\infty|\beta_k(\xi,z)|^2\dd z \\ =\frac{1}{\epsilon}\int_0^\infty\frac{Y^{2k+1}}{2k+1}G(Y)\dd Y\int_0^\infty|\beta_k(\xi,z)|^2\dd z. \end{align}\] Setting \[\require{physics} \gamma_k:=\int_0^\infty\frac{Y^{2k+1}}{2k+1}G(Y)\dd Y=\frac{1}{2k+1}\frac{1}{\sqrt{2\pi}}\int_0^\infty (2U)^ke^{-U}\dd U=\frac{2^kk!}{(2k+1)\sqrt{2\pi}},\] we arrive at the bound \[\frac{|f(\xi,0)|^2}{n+2}\le\frac{\epsilon}{2\sqrt\pi}\|f(\xi,\cdot)\|^2_{L^2(\mathbb{R})}+\sum_{k=0}^n\frac{\gamma_k}{{\epsilon}^{2k+1}|\xi|^2}\|\beta_k(\xi,\cdot)\|^2_{L^2(0,+\infty)}\] which holds for all \(\xi\not=0\), and for all \({\epsilon}>0\).
At this point, we consider separately the cases \(|\xi|\ge 1\) and \(|\xi|<1\). In the first case, we equilibrate the term of order \({\epsilon}\) and the term of highest degree in \(1/{\epsilon}\), and set \[{\epsilon}^{2n+2}=\frac{1}{|\xi|^2}.\] Thus \[\begin{align} \mathbf{1}_{|\xi|\ge 1}\frac{|\xi|^\frac{1}{n+1}}{n+2}|f(\xi,0)|^2\le&\tfrac1{2\sqrt\pi}\|f(\xi,\cdot)\|^2_{L^2(\mathbb{R})} \\ &+\!\sum_{k=0}^{n-1}\frac{\mathbf{1}_{|\xi|\ge 1}\gamma_k}{|\xi|^\frac{2(n-k)}{n+1}}\|\beta_k(\xi,\cdot)\|^2_{L^2(0,+\infty)} \!+\!\gamma_n\|\beta_n(\xi,\cdot)\|^2_{L^2(0,+\infty)} \\ \le&\tfrac1{2\sqrt\pi}\|f(\xi,\cdot)\|^2_{L^2(\mathbb{R})}\!+\!\sum_{k=0}^n\gamma_k\|\beta_k(\xi,\cdot)\|^2_{L^2(0,+\infty)}\,. \end{align}\] In the second case, i.e. \(|\xi|<1\), we equilibrate the term of order \({\epsilon}\) and the term of lowest degree in \(1/{\epsilon}\), and set \[{\epsilon}=\frac{1}{|\xi|},\] so that \[\begin{align} \mathbf{1}_{|\xi|<1}|\xi|\frac{|f(\xi,0)|^2}{n+2}\le&\tfrac1{2\sqrt\pi}\|f(\xi,\cdot)\|^2_{L^2(\mathbb{R})}+\sum_{k=0}^n\gamma_k|\xi|^{2k}\mathbf{1}_{|\xi|<1}\|\beta_k(\xi,\cdot)\|^2_{L^2(0,+\infty)} \\ \le&\tfrac1{2\sqrt\pi}\|f(\xi,\cdot)\|^2_{L^2(\mathbb{R})}+\sum_{k=0}^n\gamma_k\|\beta_k(\xi,\cdot)\|^2_{L^2(0,+\infty)}. \end{align}\] Summarizing, we arrive at the inequality \[\begin{align} \tfrac1{n+2}\min\left(|\xi|,|\xi|^\frac{1}{n+1}\right)|f(\xi,0)|^2\le&\tfrac1{2\sqrt\pi}\|f(\xi,\cdot)\|^2_{L^2(\mathbb{R})}+\sum_{k=0}^n\gamma_k\|\beta_k(\xi,\cdot)\|^2_{L^2(0,+\infty)} \\ \le&\tfrac1{2\sqrt\pi}\|f(\xi,\cdot)\|^2_{L^2(\mathbb{R})}+\sum_{k=0}^n2\gamma_k\|b_k(\xi,\cdot)\|^2_{L^2(\mathbb{R})}. \end{align}\] Now, by Young’s inequality \[|\xi|^\frac{1}{n+1}\le\frac{|\xi|}{n+1}+\frac{n}{n+1},\] so that \[\begin{align} \tfrac1{n+2}|\xi|^\frac{1}{n+1}|f(\xi,0)|^2\le&\tfrac{(|\xi|+n)\mathbf{1}_{|\xi|<1}}{(n+1)(n+2)}|f(\xi,0)|^2+\tfrac{\mathbf{1}_{|\xi|\ge 1}}{(n+2)}|\xi|^\frac{1}{n+1}|f(\xi,0)|^2 \\ \le&\tfrac{\mathbf{1}_{|\xi|<1}}{n+2}|f(\xi,0)|^2+\tfrac1{n+2}\min\left(|\xi|,|\xi|^\frac{1}{n+1}\right)|f(\xi,0)|^2 \\ \le&\tfrac{\mathbf{1}_{|\xi|<1}}{n+2}\|\rho\|^2_{L^1(\mathbb{R})}+\tfrac1{n+2}\min\left(|\xi|,|\xi|^\frac{1}{n+1}\right)|f(\xi,0)|^2, \end{align}\] and hence \[\require{physics} \begin{align} \tfrac1{n+2}\int_\mathbb{R}|\xi|^\frac{1}{n+1}|f(\xi,0)|^2\dd\xi \le&\tfrac{2}{n+2}\|\rho\|^2_{L^1(\mathbb{R})}+\tfrac1{2\sqrt\pi}\|f\|^2_{L^2(\mathbb{R}\times\mathbb{R})}+\sum_{k=0}^n2\gamma_k\|b_k\|^2_{L^2(\mathbb{R}\times\mathbb{R})} \\ \le&\tfrac{2}{n+2}+\sqrt\pi\|\tilde{R}\|_{L^2(\mathbb{R}\times\mathbb{R})}+\sum_{k=0}^n2\gamma_k\|b_k\|^2_{L^2(\mathbb{R}\times\mathbb{R})}. \end{align}\] By the same token \[\require{physics} \begin{align} \int_\mathbb{R}\frac{1+|\xi|^\frac{1}{n+1}}{2(n+2)}|f(\xi,0)|^2\dd\xi\le&\int_\mathbb{R}\frac{\mathbf{1}_{|\xi|<1}+|\xi|^\frac{1}{n+1}\mathbf{1}_{|\xi|\ge 1}}{n+2}|f(\xi,0)|^2\dd\xi \\ \le&\tfrac2{n+2}+\tfrac2{n+2}+\sqrt\pi\|\tilde{R}\|_{L^2(\mathbb{R}\times\mathbb{R})}+\sum_{k=0}^n2\gamma_k\|b_k\|^2_{L^2(\mathbb{R}\times\mathbb{R})}. \end{align}\] This concludes the proof. ◻
Remark 2. Notice that Theorem 3 (and its proof) make use of the fact that \(\rho\in L^1(\mathbb{R})\), implied by the fact that \(R\) is a trace-class operator, to conclude that \(\rho\) belongs to \(H^s(\mathbb{R})\) for some \(s>0\). If one assumes only that \(R\) is a Hilbert-Schmidt operator, the proof only leads to a bound on \[\require{physics} \int_\mathbb{R}\min(|\xi|,|\xi|^{2s})|\mathcal{F}_{x\to\xi}\rho(\xi)|^2\dd\xi\] with \(s=\frac{1}{2(n+1)}\). This control is weaker than a bound on the homogeneous Sobolev norm \(\dot{H}^s(\mathbb{R})\), except in the case \(n=0\), where it is found that \[\|\rho\|_{\dot{H}^{1/2}(\mathbb{R})}\le C\|\tilde{R}\|_{L^2(\mathbb{R}\times\mathbb{R})}^{1/2}\|\partial_x\partial_y\tilde{R}\|_{L^2(\mathbb{R}\times\mathbb{R})}^{1/2}.\] This is the analogue of the basic \(L^2\) case of velocity averaging in space dimension \(1\): for all \(f\equiv f(x,v)\), with \(x,v\in\mathbb{R}\), \[\left\|\int_\mathbb{R} fdv\right\|_{\dot{H}^{1/2}(\mathbb{R})}\le C\|f\|_{L^2(\mathbb{R}\times\mathbb{R})}^{1/2}\|v\partial_xf\|_{L^2(\mathbb{R}\times\mathbb{R})}^{1/2}.\] In that case, no localization in \(v\) is needed: see [3].
In view of the remark preceding the statement of Theorem 2, the problem of velocity averaging for the Wigner transform of pure states cannot be addressed by the method used in section 3 in the classical limit, and certainly not by using Theorem 1. The case of pure states in the semiclassical regime must therefore be discussed separately.
As a preparation, the present section discusses a few properties of Wigner functions of pure states. These properties have several important implications, especially in semiclassical velocity averaging.
We first recall the definition of a pure quantum state.
Definition 2. A density operator \(R\) on \(\mathfrak H\) defines a pure state if it is a projection, in other words if it is an idempotent element of the algebra \(\mathcal{L}(\mathfrak H)\), i.e. if \(R^2=R\).
Since a density operator is self-adjoint, \(R\) defines a pure state if and only if it is an orthogonal projection, i.e. \(R^2=R=R^*\). Since the rank of an orthogonal projection in \(\mathfrak H\) is its trace, and since a the trace of a density operator is \(1\), a pure state corresponds to a rank-one orthogonal projection in \(\mathfrak H\). Pick \(\psi\in\mathfrak H\setminus\{0\}\) so that the range of the rank-one orthogonal projection \(R\) is \(\mathbb{C}\psi\); without loss of generality, one can assume that \(\|\psi\|_\mathfrak H=1\). Thus the operator \(R\) can be written in Dirac’s bra-ket notation as \[\label{Rk1bra-ket} R = \ketbra{\psi}{\psi},\tag{28}\] where we recall that \(|\psi\rangle\) designates the vector \(\psi\in\mathfrak H\), while \(\langle\psi|\) designates the continuous linear functional \(\phi\mapsto(\psi|\phi)_\mathfrak H\) on \(\mathfrak H\). When \(\mathfrak H=L^2(\mathbb{R}^d)\), this is equivalent to the fact that the pure state density operator \(R\) has integral kernel \[\label{Rk1Kernel} R(X,Y) = \psi(X)\overline{\psi(Y)}.\tag{29}\]
There are several well-known characterizations of pure states in quantum mechanics, besides the condition \(R^2=R\) in the definition above. For instance, a density operator represents a pure state if and only if its von Neumann entropy \(-\mathrm{tr}(R\ln R)=0\). However, for the purpose of our discussion, we are chiefly interested in characterizations of pure states involving the Wigner function of the associated density operator. One could try to use the identity \(W_\hbar[R]=W_\hbar[R^2]=W_\hbar[R]\circ_\hbar W_\hbar[R]\) where \(\circ_\hbar\) designates the Moyal product. A quick glance at the formula for the Moyal product (see for instance formula (18.5.6) in chapter XVIII of [24]) suggests that this approach could prove somewhat unpractical.
We shall appeal to the following criterion. As a matter of fact, we shall not use exactly this criterion, but a consequence thereof, stated below as Corollary 1.
Lemma 1. Let \(R=R^*\in\mathcal{L}^1(L^2(\mathbb{R}^d))\) with Wigner function \(W_\hbar[R](x,\xi)\). Assume that \((x,y)\mapsto\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)\) (the partial Fourier transform of \(W_\hbar[R]\) in the \(\xi\)-variable) is of class \(C^1\) on \(\mathbb{R}^d\times\mathbb{R}^d\) and that \(\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)\not=0\) for all \(x,y\in\mathbb{R}^d\). Then \(R\) is a rank-one density operator if and only if, for all \(j,k=1,\ldots,d\), \[\label{IdentTatarskii} \left\{ \begin{align} \tfrac4{\hbar^2}\partial_{y_j}\left(\frac{\partial_{y_k}\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}{\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}\right) &\!=\!\partial_{x_j}\left(\frac{\partial_{x_k}\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}{\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}\right), \\ \partial_{y_j}\left(\frac{\partial_{x_k}\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}{\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}\right) &\!=\!\partial_{x_j}\left(\frac{\partial_{y_k}\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}{\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}\right), \end{align} \right. \quad\text{ in }\mathcal{D}'(\mathbb{R}^d\times\mathbb{R}^d).\tag{30}\]
Notice that (a formal variant of) Lemma 1 in the special case of space dimension \(d=1\) was already stated in [19].
Proof. Assume first that \(R=\ketbra{\psi}{\psi}\), whose integral kernel is \(R(X,Y)=\psi(X)\overline{\psi(Y)}\). Then \[\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)=\psi(x-\tfrac\hbar{2}y)\overline{\psi(x+\tfrac{\hbar}2y)},\] and our assumptions imply that \(\psi\in C^1(\mathbb{R}^d,\mathbb{C}\setminus\{0\})\). Then, for \(\ell=1,\ldots,d\), \[\left\{ \begin{align} \frac{\partial_{x_\ell}\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}{\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)} &=\frac{\partial_\ell\psi(x-\tfrac\hbar{2}y)}{\psi(x-\tfrac\hbar{2}y)}+\frac{\partial_\ell\overline{\psi(x+\tfrac\hbar{2}y)}}{\overline{\psi(x+\tfrac\hbar{2}y)}}, \\ \frac{\partial_{y_\ell}\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}{\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)} &=-\tfrac\hbar{2}\frac{\partial_\ell\psi(x-\tfrac\hbar{2}y)}{\psi(x-\tfrac\hbar{2}y)}+\tfrac\hbar{2}\frac{\partial_\ell\overline{\psi(x+\tfrac\hbar{2}y)}}{\overline{\psi(x+\tfrac\hbar{2}y)}}. \end{align} \right.\] Set \(\omega_+=\mathbb{C}\setminus[0,+\infty)\) and \(\omega_-=\mathbb{C}\setminus(-\infty,0]\). Let \(\log_-\) be the principal determination of the logarithm on \(\omega_-\), and set \(\log_+z=\log_-(-z)+i\pi\). One has \(\omega_+\cap\omega_-=H_+\cup H_-\) where \(H_\pm\) is the open upper (resp. lower) half-plane in \(\mathbb{C}\). One easily checks that \[\begin{align} \log_+z-\log_-z&=0,&&\qquad z\in H_+, \\ \log_+z-\log_-z&=2i\pi,&&\qquad z\in H_-. \end{align}\] Set \(\Omega_\pm=\psi^{-1}(\omega_\pm)\), which is an open subset of \(\mathbb{R}^d\) since \(\psi\) is continuous. One has obviously \(\mathbb{R}^d=\Omega_+\cup\Omega_-\) since \(\psi\) takes its values in \(\mathbb{C}\setminus\{0\}\). (Indeed \(\Omega_+\cup\Omega_-=\psi^{-1}(\omega_+\cup\omega_-)=\psi^{-1}(\mathbb{C}\setminus\{0\})=\mathbb{R}^d\).) Besides \[\Omega_+\cap\Omega_-=\psi^{-1}(H_+)\cup\psi^{-1}(H_-)\quad\text{ with }\psi^{-1}(H_\pm)\text{ open and disjoint in }\mathbb{R}^d.\] Since \(\log_+\psi-\log_-\psi\) is a constant multiple of \(2i\pi\) on \(\psi^{-1}(H_\pm)\), one has \[\partial_\ell\log_+\psi=\frac{\partial_\ell\psi}{\psi}=\partial_\ell\log_-\psi\quad\text{ on }\Omega_+\cap\Omega_-.\] Hence \[\left\{ \begin{align} \frac{\partial_{x_\ell}\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}{\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)} &=\partial_\ell(\log_\pm\psi)(x-\tfrac\hbar{2}y)+\partial_\ell(\log_\pm\overline{\psi})(x+\tfrac\hbar{2}y), \\ \frac{\partial_{y_\ell}\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}{\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)} &=-\tfrac\hbar{2}\partial_\ell(\log_\pm\psi)(x-\tfrac\hbar{2}y)+\tfrac\hbar{2}\partial_\ell(\log_\pm\overline{\psi})(x+\tfrac\hbar{2}y), \end{align} \right.\] where the determination \(\log_+\) or \(\log_-\) of the logarithm is chosen according to whether \(x-\tfrac\hbar{2}y\) and \(x+\tfrac\hbar{2}y\) belong to \(\Omega_+\) or \(\Omega_-\), so that \[\left\{ \begin{align} \partial_{x_j}\left(\frac{\partial_{x_k}\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}{\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}\right) &=\partial_j\partial_k(\log_\pm\psi)(x-\tfrac\hbar{2}y)+\partial_j\partial_k(\log_\pm\overline{\psi})(x+\tfrac\hbar{2}y), \\ \partial_{x_j}\left(\frac{\partial_{y_k}\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}{\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}\right) &=-\tfrac\hbar{2}\partial_j\partial_k(\log_\pm\psi)(x-\tfrac\hbar{2}y)+\tfrac\hbar{2}\partial_j\partial_k(\log_\pm\overline{\psi})(x+\tfrac\hbar{2}y), \\ \partial_{y_j}\left(\frac{\partial_{y_k}\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}{\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}\right) &=\tfrac{\hbar^2}{4}\partial_j\partial_k(\log_\pm\psi)(x-\tfrac\hbar{2}y)+\tfrac{\hbar^2}{4}\partial_j\partial_k(\log_\pm\overline{\psi})(x+\tfrac\hbar{2}y). \end{align} \right.\] Comparing the first and last line in the system above gives the first \(d^2\) identities in the lemma. The second group of identities in the lemma follows from the symmetry in \(j,k\) of the partial derivatives \(\partial_j\partial_k(\log_\pm\psi)\) and \(\partial_j\partial_k(\log_\pm\overline{\psi})\) in the sense of distributions on \(\Omega_\pm\). Hence the conditions in the lemma are necessary for the density operator \(R\) to be of rank equal to one.
Conversely, assume that \(\tilde{R}:=\mathcal{F}_{\xi\to y}W_\hbar[R]\in C^1(\mathbb{R}^d\times\mathbb{R}^d,\mathbb{C}\setminus\{0\})\) satisfies the identities in the lemma, and set \[R(X,Y)=\tilde{R}\left(\tfrac{X+Y}2,\tfrac{Y-X}\hbar\right),\qquad X,Y\in\mathbb{R}^d,\] which is an integral kernel for the density operator \(R\). Set \(U_\pm=R^{-1}(\omega_\pm)\), which is open in \(\mathbb{R}^d\times\mathbb{R}^d\) since \(R\) is continuous, and satisfy \(\mathbb{R}^d\times\mathbb{R}^d=U_+\cup U_-\), since \(R\) takes its values in \(\mathbb{C}\setminus\{0\}=\omega_+\cup\omega_-\). (Indeed \(U_+\cup U_-=R^{-1}(\omega_+\cup\omega_-) =R^{-1}(\mathbb{C}\setminus\{0\})=\mathbb{R}^d\times\mathbb{R}^d\).) As above, \[U_+\cap U_-= R^{-1}(H_+)\cup R^{-1}(H_-)\quad\text{ with }R^{-1}(H_\pm)\text{ open and disjoint in }\mathbb{R}^d\times\mathbb{R}^d,\] and \[\nabla_{X,Y}\log_+R(X,Y)=\frac{\nabla_{X,Y}R(X,Y)}{R(X,Y)}=\nabla_{X,Y}\log_-R(X,Y)\,,\qquad(X,Y)\in U_+\cap U_-,\] since \(\log_+R-\log_-R\) is a constant multiple of \(2i\pi\) on \(R^{-1}(H_\pm)\). The identities in the lemma are recast as \[\left\{ \begin{align} \tfrac4{\hbar^2}\partial_{y_j}\partial_{y_k}\log_\pm\tilde{R}(x,y)&=\partial_{x_j}\partial_{x_k}\log_\pm\tilde{R}(x,y) \\ \partial_{y_j}\partial_{x_k}\log_\pm\tilde{R}(x,y)&=\partial_{x_j}\partial_{y_k}\log_\pm\tilde{R}(x,y) \end{align} \right. \qquad\text{ in }\mathcal{D}'(V_\pm)\qquad\text{ for }1\le j,k\le d,\] where \(V_\pm\) is the inverse image of \(U_\pm\) by the linear transformation \((x,y)\mapsto(x-\tfrac\hbar{2}y,x+\tfrac\hbar{2}y)\). Since this linear transformation is one-to-one and onto, its inverse maps \(U_+\cup U_-=\mathbb{R}^d\times\mathbb{R}^d\) onto \(V_+\cup V_-=\mathbb{R}^d\times\mathbb{R}^d\). By the chain rule \[\left\{ \begin{align} \partial_{x_\ell}\log_\pm\tilde{R}(x,y)&=\partial_{x_\ell}\log_\pm R(x-\tfrac\hbar{2}y,x+\tfrac\hbar{2}y)=(\partial_{X_\ell}+\partial_{Y_\ell})\log_\pm R(x-\tfrac\hbar{2}y,x+\tfrac\hbar{2}y), \\ \partial_{y_\ell}\log_\pm\tilde{R}(x,y)&=\partial_{y_\ell}\log_\pm R(x-\tfrac\hbar{2}y,x+\tfrac\hbar{2}y)=\tfrac\hbar{2}(\partial_{Y_\ell}-\partial_{X_\ell})\log_\pm R(x-\tfrac\hbar{2}y,x+\tfrac\hbar{2}y), \end{align} \right.\] so that, for \(1\le j,k\le d\) \[\left\{ \begin{align} (\partial_{Y_j}-\partial_{X_j})(\partial_{Y_k}-\partial_{X_k})\log_\pm R(X,Y)&=(\partial_{X_j}+\partial_{Y_j})(\partial_{X_k}+\partial_{Y_k})\log_\pm R(X,Y), \\ \tfrac\hbar{2}(\partial_{Y_j}-\partial_{X_j})(\partial_{Y_k}+\partial_{X_k})\log_\pm R(X,Y)&=\tfrac\hbar{2}(\partial_{X_j}+\partial_{Y_j})(\partial_{Y_k}-\partial_{X_k})\log_\pm R(X,Y), \end{align} \right.\] which is recast as \[\left\{ \begin{align} (\partial_{Y_j}\partial_{X_k}+\partial_{X_j}\partial_{Y_k})\log_\pm R(X,Y)=0, \\ (\partial_{Y_j}\partial_{X_k}-\partial_{X_j}\partial_{Y_k})\log_\pm R(X,Y)=0, \end{align} \right.\] or, equivalently \[\partial_{X_j}\partial_{Y_k}\ln_\pm R(X,Y)=0\quad\text{ in }\mathcal{D}'(U_\pm)\quad\text{ for all }1\le j,k\le d.\] This implies that \[\partial_{X_j}\left(\frac{\partial_{Y_k}R(X,Y)}{R(X,Y)}\right)=0\quad\text{ in }\mathcal{D}'(\mathbb{R}^d\times\mathbb{R}^d)\quad\text{ for all }1\le j,k\le d.\] Since \(\mathbb{R}^d\times\mathbb{R}^d\) is connected, this implies the existence of \(b_1,\ldots,b_d\in C(\mathbb{R}^d)\) such that \[\frac{\partial_{Y_k}R(X,Y)}{R(X,Y)}=b_k(Y)\quad\text{ for all }(X,Y)\in\mathbb{R}^d\times\mathbb{R}^d\,.\] Besides \[\partial_{Y_j}b_k(Y)=\partial_{Y_k}b_j(Y)\quad\text{ in }\mathcal{D}'(\mathbb{R}^d)\quad\text{ for all }1\le j,k\le d.\] Indeed \[b_k(Y)=\partial_{Y_k}\log_\pm R(X,Y)\quad\text{ for all }(X,Y)\in U_\pm\quad\text{ and all }k=1,\ldots,d,\] so that \[\partial_{Y_j}b_k(Y)=\partial_{Y_k}b_j(Y)\quad\text{ in }\mathcal{D}'(U_+\cup U_-)\quad\text{ for all }1\le j,k\le d,\] and \(U_+\cup U_-=\mathbb{R}^d\times\mathbb{R}^d\). Hence there exists \(B\in C^1(\mathbb{R}^d)\) such that \[\nabla B(Y)=(b_1(Y),\ldots,b_d(Y))\,,\quad Y\in\mathbb{R}^d.\] Therefore \[\log_\pm R(X,Y)-B(Y)=\mathfrak A_\alpha^\pm(X)\quad\text{ for all }(X,Y)\in U^\pm_\alpha,\] where \(U^\pm_\alpha\) is the family of connected components of \(U_\pm\).
Exchanging the variables \(X\) and \(Y\), and arguing as above shows that there exists \(A\in C^1(\mathbb{R}^d)\) so that \[\log_\pm R(X,Y)-A(X)=\mathfrak B_\alpha^\pm(Y)\quad\text{ for all }(X,Y)\in U^\pm_\alpha.\] In other words \[\mathfrak A_\alpha^\pm(X)+B(Y)=A(X)+\mathfrak B_\alpha(Y)\quad\text{ for all }(X,Y)\in U^\pm_\alpha,\] so that \[A(X)-\mathfrak A_\alpha^\pm(X)=B(Y)-\mathfrak B_\alpha^\pm(Y)\quad\text{ for all }(X,Y)\in U^\pm_\alpha.\] Therefore, there exists a family of constants \(C_\alpha^\pm\) indexed by the family of connected components \(U_\alpha^\pm\) such that \[A(X)-\mathfrak A_\alpha^\pm(X)=B(Y)-\mathfrak B_\alpha^\pm(Y)=C_\alpha^\pm\quad\text{ for all }(X,Y)\in U^\pm_\alpha,\] so that \[\log_\pm R(X,Y)=A(X)+B(Y)-C_\alpha^\pm\quad\text{ for all }(X,Y)\in U^\pm_\alpha.\] Hence \[R(X,Y)=\exp(-C_\alpha^\pm)e^{A(X)}e^{B(Y)}\quad\text{ for all }(X,Y)\in U^\pm_\alpha.\] Thus \[R(X,Y)e^{-A(X)}e^{-B(Y)}=\exp(-C_\alpha^\pm)\quad\text{ for all }(X,Y)\in U^\pm_\alpha,\] and the right-hand side of this equality is locally constant on \(\mathbb{R}^d\times\mathbb{R}^d\) which is connected, while the left-hand side is continuous, since \(A\) and \(B\) are continuous on \(\mathbb{R}^d\). Hence both sides of this equality are equal to a constant on \(\mathbb{R}^d\times\mathbb{R}^d\): \[R(X,Y)=re^{A(X)}e^{B(Y)}\quad\text{ for all }(X,Y)\in\mathbb{R}^d\times\mathbb{R}^d,\] where \(r\not=0\) is a constant on \(\mathbb{R}^d\times\mathbb{R}^d\). In particular \[|r|\|e^A\|_{L^2(\mathbb{R}^d)}\|e^B\|_{L^2(\mathbb{R}^d)}=\|R\|_{L^2(\mathbb{R}^d\times\mathbb{R}^d)}\le\|R\|_{\mathcal{L}^1(L^2(\mathbb{R}^d))},\] so that \(e^A\in L^2(\mathbb{R}^d)\) and \(\mathrm{Ran}(R)=\mathbb{C} e^A\), which implies that \(R\) is a rank-one operator. ◻
Remark 3. In the case of space dimension \(d=1\), the identities in Lemma 1 boil down to the single equality \[\tfrac4{\hbar^2}\partial_y\left(\frac{\partial_y\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}{\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}\right) =\partial_x\left(\frac{\partial_x\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}{\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)}\right),\] or, equivalently, to the d’Alembert equation (wave equation in space dimension \(1\)) \[\tfrac4{\hbar^2}\partial_y^2\log_\pm\tilde{R}(x,y)=\partial_x^2\log_\pm\tilde{R}(x,y)\qquad\text{ in }\mathcal{D}'(V_\pm),\] with speed of propagation \(2/\hbar\). We recall that \(V_\pm\) is the inverse image of \(U_\pm\) by the linear transformation \((x,y)\mapsto(x+\tfrac\hbar{2}y,x-\tfrac\hbar{2}y)\), while \(U_\pm=R^{-1}(\omega_\pm)\), with \(\omega_+:=\mathbb{C}\setminus[0,+\infty)\) and \(\omega_-:=\mathbb{C}\setminus(-\infty,0]\).
We shall later need the following implication of Lemma 1: if \(R=R^*\in\mathcal{L}^1(L^2(\mathbb{R}^d))\) is a rank-one density operator satisfying the assumptions of Lemma 1, \[\tilde{R}(x,-y)=\mathcal{F}_{\xi\to y}W_\hbar[R](x,y)\] satisfies, for all \(j,k=1,\ldots,d\), the identity \[\label{IdentPure} \begin{align} \tilde{R}(x,y)\partial_{y_j}\partial_{y_k}\tilde{R}(x,y)-&(\partial_{y_j}\tilde{R}(x,y))(\partial_{y_k}\tilde{R}(x,y)) \\ =&\tilde{R}(x,y)^2\partial_{y_j}\left(\frac{\partial_{y_k}\tilde{R}(x,y)}{\tilde{R}(x,y)}\right)=\tfrac14\hbar^2\tilde{R}(x,y)^2\partial_{x_j}\left(\frac{\partial_{x_k}\tilde{R}(x,y)}{\tilde{R}(x,y)}\right) \\ =&\tfrac14\hbar^2\left(\tilde{R}(x,y)\partial_{x_j}\partial_{x_k}\tilde{R}(x,y)-(\partial_{x_j}\tilde{R}(x,y))(\partial_{x_k}\tilde{R}(x,y))\right)\,. \end{align}\tag{31}\]
However, since both sides of the second equality above, which follows from Lemma 1, are multiplied by \(\tilde{R}(x,y)\), one can hope to avoid the assumption that \(\tilde{R}(x,y)\not=0\) required to apply Lemma 1.
That this is indeed the case follows from the observation below, stated in an abstract setting.
Lemma 2. Let \(D,D'\) be commuting derivations on a commutative algebra \(\mathcal{A}\). Set \[T_{D,D'}(f):=DD'(f^2)-4(Df)(D'f)\,.\] For all \(f,g\in\mathcal{A}\), one has \[T_{D,D'}(fg)=T_{D,D'}(f)g^2+f^2T_{D,D'}(g)\,.\]
Proof. By the Leibniz formula, one has indeed \[\begin{align} T_{D,D'}(fg)=&D(g^2D'(f^2)+f^2D'(g^2))-4(gDf+fDg)(gD'f+fD'g) \\ =&g^2DD'(f^2)+f^2DD'(g^2)+D(g^2)D'(f^2)+D(f^2)D'(g^2) \\ &-4(gDf+fDg)(gD'f+fD'g) \\ =&g^2DD'(f^2)+f^2DD'(g^2)+4gf(Dg)(D'f)+4fg(Df)(D'g) \\ &-4g^2(Df)(D'f)-4f^2(Dg)(D'g)-4fg(Dg)(D'f)-4gf(Df)D'g) \\ =&g^2DD'(f^2)+f^2DD'(g^2)-4g^2(Df)(D'f)-4f^2(Dg)(D'g) \\ =&T_{D,D'}(f)g^2+f^2T_{D,D'}(g)\;. \end{align}\] ◻
With this, we shall establish the equality between the leftmost and rightmost sides of 31 without using the nonvanishing condition \(\mathcal{F}_{\xi\to y}W_\hbar[R]\not=0\) everywhere on \(\mathbb{R}^d\times\mathbb{R}^d\) assumed in Lemma 1. This equality is at the core of our discussion of semiclassical velocity averaging for pure states in the next section.
Corollary 1. Consider the pure state density operator \(R:=|\psi\rangle\langle\psi|\), where \(\psi\in H^2(\mathbb{R}^d)\). Then \[\tilde{R}(x,y):=\mathcal{F}_{\xi\to y}W_\hbar[R](x,-y)\] satisfies, for all \(j,k=1,\ldots,d\), the identity \[\label{IdentBisPure} \begin{align} \tilde{R}(x,y)\partial_{y_j}\partial_{y_k}\tilde{R}(x,y)-&(\partial_{y_j}\tilde{R}(x,y))(\partial_{y_k}\tilde{R}(x,y)) \\ &=\tfrac14\hbar^2\left(\tilde{R}(x,y)\partial_{x_j}\partial_{x_k}\tilde{R}(x,y)-(\partial_{x_j}\tilde{R}(x,y))(\partial_{x_k}\tilde{R}(x,y))\right)\,. \end{align}\tag{32}\]
Proof. Let us first treat the case where \(\psi\) is smooth. With the notations of Lemma 2, set \[T_{x_j,x_k}:=T_{\partial_{x_j},\partial_{x_k}}\,,\qquad T_{y_j,y_k}:=T_{\partial_{y_j},\partial_{y_k}}\,,\qquad j,k=1,\ldots,d\,.\] Obviously \[\partial_{y_\ell}\psi(x\pm\tfrac12\hbar y)=\pm\tfrac12\hbar\partial_{x_\ell}\psi(x\pm\tfrac12\hbar y)\,,\] so that \[T_{y_j,y_k}(\psi(x\pm\tfrac12\hbar y))=\tfrac14\hbar^2T_{x_j,x_k}(\psi(x\pm\tfrac12\hbar y))\,,\qquad j,k=1,\ldots,d\,.\] Applying Lemma 2 with \(f=\psi(x\pm\tfrac12\hbar y)\) and \(g=\overline{\psi(x-\tfrac12\hbar y)}\) while \(D=\partial_{y_j}\) or \(\partial_{x_j}\) and \(D'=\partial_{y_k}\) or \(\partial_{x_k}\) shows that \[\tilde{R}(x,y):=\psi(x+\tfrac12\hbar y)\overline{\psi(x-\tfrac12\hbar y)}\] satisfies \[T_{y_j,y_k}(\tilde{R}(x,y))=\tfrac14\hbar^2T_{x_j,x_k}(\tilde{R}(x,y))\,,\qquad j,k=1,\ldots,d\,.\] The conclusion follows from observing that \[\begin{align} T_{y_j,y_k}(\tilde{R}(x,y))=&\partial_{y_j}\partial_{y_k}(\tilde{R}(x,y)^2)-4(\partial_{y_j}\tilde{R}(x,y))(\partial_{y_k}\tilde{R}(x,y)) \\ =&2\partial_{y_j}(\tilde{R}(x,y)\partial_{y_k}\tilde{R}(x,y))-4(\partial_{y_j}\tilde{R}(x,y))(\partial_{y_k}\tilde{R}(x,y)) \\ =&2\tilde{R}(x,y)\partial_{y_j}\partial_{y_k}\tilde{R}(x,y)-2(\partial_{y_j}\tilde{R}(x,y))(\partial_{y_k}\tilde{R}(x,y))\,, \end{align}\] and similarly \[T_{x_j,x_k}(\tilde{R}(x,y))=2\tilde{R}(x,y)\partial_{x_j}\partial_{x_k}\tilde{R}(x,y)-2(\partial_{x_j}\tilde{R}(x,y))(\partial_{x_k}\tilde{R}(x,y))\,.\]
If \(\psi\in H^2(\mathbb{R}^d)\), then \((X,Y)\mapsto\psi(X)\overline{\psi(Y)}\) belongs to \(H^2(\mathbb{R}^d\times\mathbb{R}^d)\). This is most easily seen in Fourier variables, since \[(1+|\xi|+|\eta|)^2|\hat{\psi}(\xi)||\hat{\psi}(\eta)|\le(1+|\xi|)^2|\hat{\psi}(\xi)|(1+|\eta|)^2|\hat{\psi}(\eta)|\in L^2(\mathbb{R}^d_\xi\times\mathbb{R}^d_\eta)\,.\] Hence \(\tilde{R}\in H^2(\mathbb{R}^d\times\mathbb{R}^d)\) (as the composition of the function \((X,Y)\mapsto\psi(X)\overline{\psi(Y)}\) which belongs to \(H^2(\mathbb{R}^d\times\mathbb{R}^d)\) with the linear diffeomorphism \((x,y)\mapsto(x+\tfrac\hbar{2}y,x-\tfrac\hbar{2}y)\)). Therefore \(\tilde{R}\), \(\partial_{y_\ell}\tilde{R}\) and \(\partial_{y_j}\partial_{y_k}\tilde{R}\) belong to \(L^2(\mathbb{R}^d\times\mathbb{R}^d)\), and the same is true of \(\partial_{x_\ell}\tilde{R}\) and \(\partial_{x_j}\partial_{x_k}\tilde{R}\). Hence all the products involved in the identity 32 belong to \(L^1(\mathbb{R}^d\times\mathbb{R}^d)\), and 32 is an equality a.e. between functions in \(L^1(\mathbb{R}^d\times\mathbb{R}^d)\).
Since we have seen that 32 holds true if \(\psi\) is smooth, the case of \(\psi\in H^2(\mathbb{R}^d)\) follows by density of the Schwartz space \(\mathcal{S}(\mathbb{R}^d)\) of smooth functions with rapidly decaying derivatives of all orders in \(H^2(\mathbb{R}^d)\). ◻
Remark 4. Notice that 32 is a consequence of the assumption that \(R\) is a pure state density operator, which does not require the condition \(\psi\not=0\) used in Lemma 1. As we explained in 31 , this identity is a straightforward consequence of Lemma 1 if one knows that \(\psi\not=0\). We do not claim that 32 can be used to obtain a characterization of pure states (i.e. the converse of that implication) without assuming \(\psi\not=0\) as in Lemma 1. Although this converse implication is not used in this paper, we think that Lemma 1 is a result of independent interest as it (a) makes Tatarskiı̆’s formal argument in [19] fully rigorous and (b) extends it to space dimensions higher than one.
With Corollary 1, we shall now discuss the possibility of obtaining a semiclassical velocity averaging result similar to Theorem 2 in the case of a family of pure quantum states.
Proposition 2. Let \(\psi_\hbar\in H^2(\mathbb{R}^d)\) be a family of wave functions such that \(\|\psi_\hbar\|_{L^2(\mathbb{R}^d)}=1\), and let \(w_\hbar:=W_\hbar[\ketbra{\psi_\hbar}]\). Assume that a subsequence \(\psi_{\hbar_n}\) satisfies \[w_{\hbar_n}\to w\text{ in }\mathcal{S}'(\mathbb{R}^d\times\mathbb{R}^d),\quad\text{ and }\quad\hbar_n\|\nabla\rho_{\hbar_n}\|_{L^2(B(0,R))}\to 0\] for all \(R>0\) as \(\hbar_n\to 0\). Assume that \[\require{physics} \label{Def43LimE} \mathcal{E}_{\hbar_n}=\tfrac1{2m}\int_{\mathbb{R}^d}|\xi|^2w_{\hbar_n}\dd\xi\to\mathcal{E}:=\tfrac1{2m}\int_{\mathbb{R}^d}|\xi|^2w\dd\xi\quad\text{ weakly in }L^2(\mathbb{R}^d)\qquad{(3)}\] as \(\hbar_n\to 0\), while \[\require{physics} \label{Def43LimRhoJ} \rho_{\hbar_n}:=\int_{\mathbb{R}^d}w_{\hbar_n}\dd\xi\to\rho:=\int_{\mathbb{R}^d}w\dd\xi\quad\text{ and }J_{\hbar_n}:=\tfrac1m\int_{\mathbb{R}^d}\xi w_{\hbar_n}\dd\xi\to J:=\tfrac1m\int_{\mathbb{R}^d}\xi w\dd\xi\qquad{(4)}\] in \(L^2_{loc}(\mathbb{R}^d)\) as \(\hbar_n\to 0\) for each \(R>0\). Set \[u(x):=\frac{\mathbf{1}_{\rho(x)>0}}{\rho(x)}J(x)\in\mathbb{R}^d.\] Then \(w\) is a monokinetic, positive Borel measure on the phase space \(\mathbb{R}^d\times\mathbb{R}^d\), i.e. \[w=\rho(x)\delta(\xi-u(x)).\]
That \(w_{\hbar_n}\) converges to a monokinetic Wigner measure \(w=\rho(x)\delta(\xi-u(x))\) is a very strong indication that the strong convergence of the first moments \(\rho_{\hbar_n}\) and \(J_{\hbar_n}\) in \(L^2_{loc}(\mathbb{R}^d)\) cannot be deduced from a velocity averaging lemma. Indeed, velocity averaging is based on the fact that, at the kinetic level of description, the orthogonal projections of the variable \(\xi\) on each line through the origin are regularly distributed: see condition (2.1) in [3]. This is incompatible with a situation where the \(\xi\) variable concentrates on a single value at each position \(x\), which is precisely the case of a monokinetic distribution function.
Thus, the conclusions expected from a velocity averaging lemma in the present setting, namely the strong convergence of the moments of order \(0\) and \(1\), i.e. of \(\rho_{\hbar_n}\) and \(J_{\hbar_n}\), lead to a conclusion which precludes using velocity averaging!
Put in other words, this observation does obviously not prevent \(\rho_\hbar\) and \(J_\hbar\) to be strongly relatively compact families of \(L^2_{loc}\), but in that case, the strong compactness of these families cannot be expected to follow from velocity averaging.
Remark 5. The impossibility to deduce the strong compactness in \(L^2_{loc}\) of the density functions and momentum densities of a family of pure states on \(L^2\) from a velocity averaging lemma is based on the observation that this strong compactness implies that Wigner measures of such states must be monokinetic. Conditions under which the Wigner measures of a family of pure states are monokinetic have been studied in detail in section 4.3 of [25]: see Theorem 4.5 and Corollary 4.7 there. However, this reference assumes that \(\psi_{\epsilon}\) is of the form \[\psi_{\epsilon}(x)=\sqrt{\rho_{\epsilon}(x)}e^{iS_{\epsilon}(x)/{\epsilon}}\] and that \(\nabla S_{\epsilon}\) converges uniformly on some open neighborhood of the support of \(\rho\), the limit of \(\rho_{\epsilon}\) in \(L^1\) to a \(C^1\) function, or on some open neighborhood of the union of the supports of all the \(\rho_{\epsilon}\). These assumptions are much stronger than what can be deduced from velocity averaging. (Indeed, velocity averaging implies at best a gain of a \(1/2\) derivative in \(L^2_{loc}\), and never \(C^1\) regularity.) And the difference between the assumptions needed in section 4.3 of [25] and our Proposition 2 are not only minor technicalities. Indeed, it is absolutely essential for our purpose in the present paper that the assumptions implying that the Wigner measure of a sequence is monokinetic coincide exactly with the compactness conclusions of velocity averaging in order to arrive at a contradiction. This is why we cannot appeal to the results from section 4.3 in [25] to reach this contradiction.
Proof. Call \(\tilde{R}_\hbar(x,y):=\psi(x+\tfrac{\hbar}2y)\overline{\psi(x-\tfrac{\hbar}2y)}\). Observe that \[\label{FlaRhoJE} \rho_{\hbar_n}(x)=\tilde{R}_{\hbar_n}(x,0)\,,\quad J_{\hbar_n}(x)=\tfrac1m(-i\nabla_y\tilde{R}_{\hbar_n}(x,0))\,,\quad\mathcal{E}_{\hbar_n}(x)=\tfrac1{2m}(-\Delta_y\tilde{R}_{\hbar_n}(x,0)).\tag{33}\] Next we use the identity in Corollary 1 for \(j=k\) and sum over \(j=1,\ldots,d\), to find that \[\rho_{\hbar_n}(x)\Delta_y\tilde{R}_{\hbar_n}(x,0)=|\nabla_y\tilde{R}_{\hbar_n}(x,0)|^2+\tfrac{\hbar_n^2}4\rho_{\hbar_n}(x)^2\Delta_x\ln\rho_{\hbar_n}(x),\] or, equivalently \[2m\rho_{\hbar_n}(x)\mathcal{E}_{\hbar_n}(x)-m^2|J_{\hbar_n}(x)|^2=-\tfrac{\hbar_n^2}8\Delta_x\left(\rho_{\hbar_n}(x)^2\right)+\tfrac{\hbar_n^2}2|\nabla_x\rho_{\hbar_n}(x)|^2.\] Since \(\rho_{\hbar_n}\to\rho\) and \(J_{\hbar_n}\to J\) in \(L^2(B(0,R))\) while \(\mathcal{E}_{\hbar_n}\to\mathcal{E}\) weakly in \(L^2(\mathbb{R}^d)\), one has \[\rho_{\hbar_n}(x)\mathcal{E}_{\hbar_n}\to\rho\mathcal{E}\text{ in }\mathcal{D}'(\mathbb{R}^d)\quad\text{ while }|J_{\hbar_n}|^2\to|J|^2\text{ in }L^1(B(0,R)),\] and \[\hbar_n^2\rho_{\hbar_n}^2\to 0\text{ in }L^1(B(0,R)),\quad\text{ so that }\hbar_n^2\Delta_x\left(\rho_{\hbar_n}^2\right)\to 0\text{ in }\mathcal{D}'(\mathbb{R}^d).\] Since \(\hbar_n\|\nabla\rho_{\hbar_n}\|_{L^2(B(0,R))}\to 0\), passing to the limit in both sides of the equality above as \(\hbar_n\to 0\) implies that \[\require{physics} 2m\rho\mathcal{E}-m^2|J|^2=\int_{\mathbb{R}^d}w\dd\xi\int_{\mathbb{R}^d}|\xi|^2w\dd\xi-\left|\int_{\mathbb{R}^d}\xi w\dd\xi\right|^2=0\qquad\text{ a.e. on }\mathbb{R}^d.\] Multiplying both sides of this identity by \(\mathbf{1}_{\rho(x)>0}/\rho(x)^2\) shows that \[\require{physics} \frac{\mathbf{1}_{\rho>0}}{\rho}\int_{\mathbb{R}^d}\left|\xi-\frac{\mathbf{1}_{\rho>0}}{\rho}\int_{\mathbb{R}^d}\zeta w\dd\zeta\right|^2w\dd\xi =\frac{\mathbf{1}_{\rho>0}}{\rho}\int_{\mathbb{R}^d}|\xi|^2w\dd\xi-\left|\frac{\mathbf{1}_{\rho>0}}{\rho}\int_{\mathbb{R}^d}\xi w\dd\xi\right|^2=0.\] so that, since \(w\) is a positive (Borel) measure on \(\mathbb{R}^d\times\mathbb{R}^d\), \[\require{physics} w=\rho(x)\delta(\xi-u(x)),\quad\text{ with }u:=\frac{\mathbf{1}_{\rho>0}}{\rho}\int_{\mathbb{R}^d}\zeta w\dd\zeta.\] ◻
Remark 6. The semiclassical regime is an essential feature in the negative result reported in Proposition 2. In situations other than the semiclassical regimes, velocity averaging can be used even in the case of pure states as in Theorem 2. Returning to the discussion preceding Theorem 2, and assuming that \(\hbar\ge\hbar_0>0\), the inequality \[\sup_{\hbar_0\le\hbar\le 1}\|W_\hbar[R_\hbar]\|_{L^2}^2=C<\infty\] is obviously compatible with the constraint \[1\le\text{rank}R_\hbar\cdot(2\pi\hbar)^d\sup_{\hbar_0\le\hbar\le 1}\|W_\hbar[R_\hbar]\|_{L^2}^2\] in the case \(\text{rank}R_\hbar=1\), as can be seen by taking \[C>\frac{1}{(2\pi\hbar_0)^d}\,.\] In fact, the idea of applying velocity averaging to the Wigner equation away from the semiclassical regime and in the case of pure states goes back to the work of Lions and Perthame [26]. This reference explains the connection between dispersion for the free Schrödinger equation and velocity averaging for the Wigner equation — see also [27] for an extension of these results to the case of the Schrödinger equation with an external potential. However, this last reference is not based on velocity averaging itself, but on dispersion effects on the Wigner equation formulated in terms of moment estimates (another issue discussed in [26]).
Remark 7. Let us recall Theorem III.1 (5) in [22]: given any Borel probability measure \(\mu\) on \(\mathbb{R}^d\times\mathbb{R}^d\), there exists a family \(\psi_\hbar\) of \(L^2(\mathbb{R}^d)\) such that \(\|\psi_\hbar\|_{L^2(\mathbb{R}^d)}=1\) and \(\psi_\hbar\to 0\) weakly in \(L^2(\mathbb{R}^d)\) while \(W_\hbar[\ketbra{\psi_\hbar}]\to\mu\) in \(\mathcal{S}'(\mathbb{R}^d\times\mathbb{R}^d)\) as \(\hbar\to 0\). Since \(\mu\) is not necessarily monokinetic, this means that the additional assumptions in Proposition 2 are in general not satisfied by the family \(\psi_\hbar\), and that these additional assumptions are essential for Proposition 2 to hold.
Remark 8. The condition on \(\nabla\rho_\hbar\) in Proposition 2 is obviously an essential feature of this result. We shall return to its physical meaning at the end of this paper. On the other hand, assume that the kinetic energy corresponding to the pure state defined by the wave function \(\psi_\hbar\in L^2(\mathbb{R}^d)\) such that \(\|\psi_\hbar\|_{L^2(\mathbb{R}^d)}=1\) is uniformly bounded in \(\hbar\): \[\require{physics} (\psi_\hbar|-\tfrac{\hbar^2}{2m}\Delta\psi_\hbar)_{L^2(\mathbb{R}^d)}=\tfrac{\hbar^2}{2m}\|\nabla\psi_\hbar\|^2_{L^2(\mathbb{R}^d)} =\tfrac1{(2\pi)^d}\int_{\mathbb{R}^d}\tfrac{\hbar^2}{2m}|\xi|^2|\mathcal{F}_{x\to\xi}\psi_\hbar(\xi)|^2\dd \xi\le C.\] Then \[\sup_{\hbar>0}\int_{|\xi|>R/\hbar}|\mathcal{F}_{x\to\xi}\psi_\hbar(\xi)|^2d\xi\le(2\pi)^d\frac{2mC}{R^2}\to 0\quad\text{ as }R\to+\infty.\] See (1.28) and (1.26) in [28] for a more general sufficient condition leading to that conclusion than the boundedness of the kinetic energy. In other words the family \(\psi_\hbar\) is \(\hbar\)-oscillatory in the terminology of Definition 1.6 in [28]. Since \(\widehat{\rho}_{\hbar}(\xi)=|\mathcal{F}_{x\to\xi}\psi_\hbar(\xi)|^2\) (see 5 for the definition of \(\widehat{\rho}_\hbar\)), one has \[\int_{|\zeta|>R}\frac{1}{\hbar^d}\widehat{\rho}_\hbar\left(\frac{\zeta}{\hbar}\right)d\zeta=\int_{|\xi|>R/\hbar}|\mathcal{F}_{x\to\xi}\psi_\hbar(\xi)|^2d\xi.\] Thus, the tightness of \(\hbar^{-d}\widehat{\rho}_{\hbar}(\xi/\hbar)\) is equivalent to \(\psi_\hbar\) being \(\hbar\)-oscillatory.
Here are some examples of families \(\psi_\hbar\) satisfying the condition \(\hbar\|\nabla\rho_\hbar\|_{L^2(B(0,R))}\to 0\) as \(\hbar\to 0^+\).
Example 1. (WKB states) Set \[\psi_\hbar(x):=a_\hbar(x)e^{iS_\hbar(x)/\hbar},\] with \(a_\hbar\) and \(S_\hbar\) of class \(C^1\) and real-valued on \(\mathbb{R}^d\). Then \(\rho_\hbar(x)=a_\hbar(x)^2\) and \[\require{physics} \hbar^2\|\nabla\rho_\hbar(x)\|^2_{L^2(B(0,R))}=4\hbar^2\int_{B(0,R)}a_\hbar(x)^2|\nabla a_\hbar(x)|^2\dd x.\] This is to be compared with \[\require{physics} \begin{align} \hbar^2\|\nabla\psi_\hbar\|^2_{L^2(B(0,R))}=&\int_{B(0,R)}|\hbar\nabla a_\hbar(x)+ia_\hbar(x)\nabla S_\hbar(x)|^2\dd x \\ =&\int_{B(0,R)}(\hbar^2|\nabla a_\hbar(x)|+a_\hbar(x)^2|\nabla S_\hbar(x)|^2)\dd x. \end{align}\]
Example 2. (Coherent states) Set \[\psi_{q,p}(x):=(\pi\hbar)^{-d/4}e^{-|x-q|^2/2\hbar}e^{ip\,\cdot(x-q/2)/\hbar}\] for \(q,p\in\mathbb{R}^d\). Then \[\rho_\hbar(x)=(\pi\hbar)^{-d/2}e^{-|x-q|^2/\hbar}\] so that \[\hbar\nabla\rho_\hbar(x)=2(x-q)\rho_\hbar(x)\] and \[\require{physics} \begin{align} \hbar^2\|\nabla\rho_\hbar\|^2_{L^2(\mathbb{R}^d)}=&4\int_{\mathbb{R}^d}|z|^2(\pi\hbar)^de^{-2|z|^2/\hbar}\dd z \\ =&\hbar^{1-\frac{d}{2}}\int_{\mathbb{R}^d}\tfrac{4}{\pi^d}|y|^2e^{-2|y|^2}\dd y=\tfrac{d}{(4\pi)^{d/2}}\hbar^{1-\frac{d}{2}}\to 0 \end{align}\] if and only if \(d=1\).
Example 3. Here is another class of examples, very close to the case of coherent states: \[\psi_\alpha(x):=\hbar^{-d\alpha/2}a(x/\hbar^\alpha)e^{ip\,\cdot x/\hbar},\] assuming that \[\require{physics} \|\psi\|^2_{L^2(\mathbb{R}^d)}=\hbar^{-d\alpha}\int_{\mathbb{R}^d}a(x/\hbar^\alpha)^2dx=\int_{\mathbb{R}^d}a(y)^2\dd y=1.\] Then \[\rho_\hbar(x)=\hbar^{-d\alpha}a(x/\hbar^\alpha)^2\,,\quad\nabla\rho_\hbar(x)=2\hbar^{-(d+1)\alpha}a(x/\hbar^\alpha)\nabla a(x/\hbar^\alpha)\] and \[\require{physics} \begin{align} \hbar^2\|\nabla\rho_\hbar\|^2_{L^2(\mathbb{R}^d)}=&4\hbar^2\hbar^{-2(d+1)\alpha}\int_{\mathbb{R}^d}a(x/\hbar^\alpha)^2|\nabla a(x/\hbar^\alpha)|^2\dd x \\ =&4\hbar^2\hbar^{-2(d+1)\alpha}\hbar^{d\alpha}\int_{\mathbb{R}^d}a(y)^2|\nabla a(y)|^2\dd y \\ =&\hbar^{2-(d+2)\alpha}\|\nabla(a^2)\|^2_{L^2(\mathbb{R}^d)}\to 0 \end{align}\] if and only if \(\alpha<\frac{2}{d+2}\) provided that \(a^2\in H^1(\mathbb{R}^d)\).
Here is another application of Corollary 1 leading to a a quick and very natural derivation of Madelung’s quantum hydrodynamic equations. These equations were obtained for the first time in [29] (see especially equations (4’) and (3”) in that reference). In Madelung’s own words, his work [29] corresponds mostly to an analogy between Schrödinger’s approach to quantum dynamics with hydrodynamics (see [29] on p. 322).
The presentation of Madelung’s hydrodynamic equations below is closer in spirit to [30], with however significant differences — see the remark following Theorem 4 below.
Although Madelung’s system of hydrodynamic equations is essentially independent of the question of velocity averaging for the Wigner equation at first sight, we shall see that the condition used in Proposition 2 \[\hbar_n\|\nabla\rho_{\hbar_n}\|_{L^2(B(0,R))}\to 0,\] already discussed in Remark 8, but whose physical meaning remains somewhat mysterious, has a very natural interpretation in terms of the Bohm potential that appears in Madelung’s system.
Let \(\psi\equiv\psi(t,x)\in\mathbb{C}\) be a solution of the Schrödinger equation \[\label{SchroV} i\hbar\partial_t\psi(t,x)=-\tfrac{\hbar^2}{2m}\Delta_x\psi(t,x)+V(x)\psi(t,x).\tag{34}\] The von Neumann equation 13 for \(R(t)=\ketbra{\psi(t,\cdot)}\) is recast as \[\partial_t\tilde{R}(t,x,y)+\tfrac1m\sum_{j=1}^d \partial_{x_j}(-i\partial_{y_j})\tilde{R}(t,x,y)=\frac{V(x+\frac{\hbar}{2}y)-V(x-\frac{\hbar}{2}y)}{i\hbar}\tilde{R}(t,x,y), \label{eq:von32Neumann32Madelung}\tag{35}\] where we recall that \[\label{R61FW} \tilde{R}(t,x,y)=\psi(t,x+\tfrac\hbar{2}y)\overline{\psi(t,x-\tfrac\hbar{2}y)}=\mathcal{F}_{\xi\to y}W_\hbar[R](t,x,-y).\tag{36}\]
Let us first discuss the case where \(\psi\) is smooth and \(\psi(t,x)\not=0\) for all \(t\) and all \(x\). Evaluating both sides of 35 at \(y=0\), we find that \[\label{vNy610} \partial_t\tilde{R}(t,x,0)+\tfrac1m\sum_{j=1}^d\partial_{x_j}(-i\partial_{y_j}\tilde{R}(t,x,0))=0.\tag{37}\] Define the density function \(\rho(t,x)\) as in 4 , and the velocity field \(u(t,x)\in\mathbb{R}^d\) by \[\label{RhoU} \rho(t,x):=\tilde{R}(t,x,0)=|\psi(t,x)|^2>0,\quad\text{ and }\quad u(t,x):=\frac{-i\nabla_y\tilde{R}(t,x,0)}{m\tilde{R}(t,x,0)}\,.\tag{38}\] With these definitions, 37 takes the form of the continuity equation \[\partial_t\rho(t,x)+\nabla_x\cdot(\rho u)(t,x)=0. \label{eq:continuity}\tag{39}\] For the Euler (momentum) equation, we apply \(-\tfrac{i}m\partial_{y_k}\) to both sides of 35 , and arrive at \[\partial_t(-\tfrac{i}m\partial_{y_k}\tilde{R}(t,x,y))\!-\!\tfrac1{m^2}\sum_{j=1}^d\partial_{x_j}(\partial_{y_j}\partial_{y_k} \tilde{R}(t,x,y))\!=\!-\tfrac1m\partial_{y_k}\left(\!\tfrac{V(x+\frac{\hbar}{2}y)-V(x-\frac{\hbar}{2}y)}{\hbar}\tilde{R}(t,x,y)\!\right)\,. \label{eq:euler1}\tag{40}\] Specializing this equality to \(y=0\), and using the quantities defined in 38 shows that \[\label{eq:euler2} \partial_t(\rho u_k)(t,x)-\tfrac1{m^2}\sum_{j=1}^d\partial_{x_j}(\partial_{y_j}\partial_{y_k}\tilde{R}(t,x,0)) =-\tfrac1m\partial_kV(x)\rho(t,x)\,.\tag{41}\] The computations leading to 39 and 41 is classical and have been recalled here for the reader’s convenience. It remains to find a closure relation expressing \(\partial_{y_j}\partial_{y_k}\tilde{R}(t,x,0)\) in terms of \(\rho\) and \(u\). This step is quite straightforward in Madelung’s original approach [29], but much less so in the Gasser-Markowich approach.
This task is greatly simplified by the identity 31 following from Lemma 1, recast as \[\label{PrePreClo} \begin{align} -\tilde{R}(t,x,y)\partial_{y_j}\partial_{y_k}\tilde{R}(t,x,y)-(-i\partial_{y_j}\tilde{R}(t,x,y))(-i\partial_{y_k}\tilde{R}(t,x,y)) \\ =-\tfrac14\hbar^2\left(\tilde{R}(t,x,y)\partial_{x_j}\partial_{x_k}\tilde{R}(t,x,y)-(\partial_{x_j}\tilde{R}(t,x,y))(\partial_{x_k}\tilde{R}(t,x,y))\right)&\,, \end{align}\tag{42}\] which is evaluated to \(y=0\), to find \[\label{PreClo} \begin{align} -\rho(t,x)\partial_{y_j}\partial_{y_k}\tilde{R}(t,x,0)-m^2\rho(t,x)^2u_j(t,x)u_k(t,x) \\ =-\tfrac14\hbar^2\left(\rho(t,x)\partial_{x_j}\partial_{x_k}\rho(t,x)-(\partial_{x_j}\rho(t,x))(\partial_{x_k}\rho(t,x))\right) \\ =-\tfrac14\hbar^2\rho(t,x)\left(\partial_{x_j}\partial_{x_k}\rho(t,x)-4\left(\partial_{x_j}\sqrt{\rho(t,x)}\right)\left(\partial_{x_k}\sqrt{\rho(t,x)}\right)\right)&\,. \end{align}\tag{43}\] Dividing the leftmost and rightmost sides of 43 by \(\rho>0\) leads to the closure relation \[\label{Closure} \begin{align} -\partial_{y_j}\partial_{y_k}\tilde{R}(t,x,0)=m^2\rho(t,x)u_j(t,x)u_k(t,x) \\ -\tfrac14\hbar^2\left(\partial_{x_j}\partial_{x_k}\rho(t,x)-4\left(\partial_{x_j}\sqrt{\rho(t,x)}\right)\left(\partial_{x_k}\sqrt{\rho(t,x)}\right)\right)&\,. \end{align}\tag{44}\] We next insert this expression of \(-\partial_{y_j}\partial_{y_k}\tilde{R}(t,x,0)\) in 41 to find the momentum equation \[\label{eq:Madelung32Euler} \begin{align} \partial_t(\rho u)(t,x)+&\nabla_x\cdot(\rho u^{\otimes 2})(t,x)=-\tfrac1m\rho\nabla V(x) \\ +&\tfrac{\hbar^2}{4m^2}\nabla_x\cdot\left(\nabla_x^2\rho(t,x)-4\left(\nabla_x\sqrt{\rho(t,x)}\right)^{\otimes 2}\right)\,. \end{align}\tag{45}\] Summarizing, we have explained how Lemma 1 and its consequence the identity 31 lead to the closure relation needed in the derivation of Madelung’s quantum hydrodynamical system 39 and 45 . Unfortunately, the argument above uses the smoothness of \(\psi\) and the fact that \(\psi\not=0\) everywhere, which does not improve on [29].
However, we shall see that the same approach as above, using Corollary 1 instead of Lemma 1, allows us weaken the assumptions on regularity and vacuum (\(\psi=0\)).
Theorem 4. Let \(\psi\in C([0,\tau);H^2(\mathbb{R}^d))\) be a solution of 34 with \(V\in W^{1,\infty}(\mathbb{R}^d)\). Assume that \(\psi\) satisfies, for all \(t>0\), the condition \[\label{psinot610} \psi(t,x)\not=0,\qquad\text{ for a.e. }x\in\mathbb{R}^d.\tag{46}\] Then the density function \(\rho:=|\psi|^2\) as in 4 , and the velocity field defined by \[\label{DefVelField} u(t,x):=\mathbf{1}_{\rho(t,x)>0}\frac{-i\nabla_y\tilde{R}(t,x,0)}{m\tilde{R}(t,x,0)}\tag{47}\] satisfy Madelung’s system of quantum hydrodynamics on \([0,\tau)\times\mathbb{R}^d\): \[\label{MadelungSyst} \left\{ \begin{align} {}&\partial_t\rho+\nabla_x\cdot(\rho u)=0, \\ &\partial_t(\rho u)+\nabla_x\cdot(\rho u^{\otimes 2})=\tfrac{\hbar^2}{4m^2}\nabla_x\!\cdot\big(\nabla^2_x\rho-4(\nabla_x\sqrt{\rho})^{\otimes 2}\big)-\tfrac1m\rho\nabla_xV. \end{align} \right.\tag{48}\]
Remark 9. Our assumption that \(\psi\in C([0,\tau);H^2(\mathbb{R}^d))\) is quite natural for solutions of the Cauchy problem for 34 . Indeed, if \(\psi(0,\cdot)\) is in the domain of \(\mathcal{H}:=-\tfrac{\hbar^2}{2m}\Delta+V\), that is a self-adjoint operator on \(L^2(\mathbb{R}^d)\), the solution \(\psi(t,\cdot)=\exp(-it\mathcal{H}/\hbar)\psi(0,\cdot)\) of the Cauchy problem for 34 is continuous and bounded in \(t\) with values in the domain of \(\mathcal{H}\) by Stone’s theorem. Besides, under the assumption that \(V\in L^\infty(\mathbb{R}^d)\), the domain of \(\mathcal{H}\) is \(H^2(\mathbb{R}^d)\). In that case, our assumption that \(\psi\in C([0,\tau);H^2(\mathbb{R}^d))\) is satisfied provided that the initial data for 34 belongs to \(H^2(\mathbb{R}^d)\).
Proof. First we recall that, if \(\phi\in L^2(\mathbb{R}^d)\), the function \(y\mapsto\phi(x+\tfrac\hbar{2}y)\overline{\phi(x\!-\!\tfrac\hbar{2}y)}\) belongs2 to \(C_b(\mathbb{R}^d_y;L^1(\mathbb{R}^d_x))\). Applying this to the functions \(\partial_\ell\psi(t,\cdot)\) and \(\partial_j\partial_k\psi(t,\cdot)\in L^2(\mathbb{R}^d)\) justifies setting \(y=0\) in 35 and in 40 to arrive at 37 and 41 , once these equalities are written in the sense of distributions in \((t,x)\), meaning that both sides of each equality are integrated against smooth, compactly supported test functions in \((t,x)\).
Then, the equality 43 is justified in the same manner from 42 , which follows from Corollary 1, only assuming Sobolev \(H^2\) regularity in the position variable.
It remains to justify 44 . Since we have assumed that \(\psi\in C_b([0,\tau);H^2(\mathbb{R}^d_x))\), the functions \(\partial_{y_j}\partial_{y_k}\tilde{R}\big|_{y=0}\) and \(\partial_{x_j}\partial_{x_k}\rho\) both belong to \(C_b([0,\tau);L^1(\mathbb{R}^d))\). Next we use the Sobolev embedding theorem to see that \[\psi\text{ and }\partial_{x_\ell}\psi\in C_b(\mathbb{R}_t;H^{1/4}(\mathbb{R}^d))\subset C_b([0,\tau);L^p(\mathbb{R}^d))\,,\qquad p:=\tfrac{4d}{2d-1}\,,\] so that \[\partial_{x_j}\partial_{x_k}\rho=\overline{\psi}\partial_{x_j}\partial_{x_k}\psi+\psi\partial_{x_j}\partial_{x_k}\overline{\psi} +\partial_{x_j}\overline{\psi}\partial_{x_k}\psi+\partial_{x_j}\psi\partial_{x_k}\overline{\psi} \in C_b([0,\tau);L^q(\mathbb{R}^d))\] with \(q:=\tfrac{4d}{4d-1}>1\).
At this point, we apply Theorem 1 (ii) of [31] and conclude that \[\nabla\sqrt{\rho}\in C_b([0,\tau);W^{1,2q}(\mathbb{R}^d)\,,\quad\text{ so that }(\partial_{x_j}\sqrt{\rho})(\partial_{x_k}\sqrt{\rho})\in C_b([0,\tau);L^q(\mathbb{R}^d))\,.\] Therefore 43 can be put in the form \[m^2\rho^2u_ju_k=\rho\left(\tfrac14\hbar^2\partial_{x_j}\partial_{x_k}\rho-\hbar^2\left(\partial_{x_j}\sqrt{\rho}\right)\left(\partial_{x_k}\sqrt{\rho}\right)-\partial_{y_j}\partial_{y_k}\tilde{R}\Big|_{y=0}\right)\,,\] and we have seen that the term between parenthesis on the right-hand side above satisfies \[\tfrac14\hbar^2\partial_{x_j}\partial_{x_k}\rho-\hbar^2\left(\partial_{x_j}\sqrt{\rho}\right)\left(\partial_{x_k}\sqrt{\rho}\right)-\partial_{y_j}\partial_{y_k}\tilde{R}\Big|_{y=0}\in L^1_{loc}([0,\tau)\times\mathbb{R}^d_x)\,.\] Since \(\rho>0\) a.e. in \(\mathbb{R}_t\times\mathbb{R}^d_x\), we conclude that \[m^2\rho u_ju_k=\tfrac14\hbar^2\partial_{x_j}\partial_{x_k}\rho-\hbar^2\left(\partial_{x_j}\sqrt{\rho}\right)\left(\partial_{x_k}\sqrt{\rho}\right)-\partial_{y_j}\partial_{y_k}\tilde{R}\Big|_{y=0}\text{ a.e. in }\mathbb{R}_t\times\mathbb{R}^d_x\,.\] In particular \(\rho u_ju_k\in L^1_{loc}([0,\tau)\times\mathbb{R}^d)\), and 44 holds as an equality between elements of \(L^1_{loc}([0,\tau)\times\mathbb{R}^d)\subset\mathcal{D}'((0,\tau)\times\mathbb{R}^d)\).
The expression of \(-\partial_{y_j}\partial_{y_k}\tilde{R}(t,x,0)\) in 44 is substituted in the left-hand side of 41 to arrive at 45 , which concludes the proof. ◻
Remark 10. Notice that our proof of Theorem 4 does not require knowing in advance that \(\rho u_ju_k\in L^1_{loc}(\mathbb{R}_t\times\mathbb{R}^d)\). Indeed, since we already know that the quantity \(\partial_{y_j}\partial_{y_k}\tilde{R}(t,x,0)\) belongs to \(L^1_{loc}(\mathbb{R}_t\times\mathbb{R}^d)\), it is enough to compute it on the complement of a set of Lebesgue measure \(0\), specifically on the complement of \(\rho^{-1}(\{0\})\).
However, one has the following information on \(\rho u_ju_k\). Let us recall the following well-known formula for the current \[J_\ell=-\tfrac{i\hbar}{2m}(\overline{\psi}\partial_{x_\ell}\psi-\psi\partial_{x_\ell}\overline{\psi})\,,\] so that \[\begin{align} |J|^2=&\tfrac{\hbar^2}{4m^2}\sum_{\ell=1}^d\left|\overline{\psi}\partial_{x_\ell}\psi-\psi\partial_{x_\ell}\overline{\psi}\right|^2 \\ =&\tfrac{\hbar^2}{4m^2}\sum_{\ell=1}^d\left(2|\psi|^2|\partial_{x_\ell}\psi|^2-\psi^2\left(\partial_{x_\ell}\overline{\psi}\right)^2-\overline{\psi}^2\left(\partial_{x_\ell}\psi\right)^2\right)\,. \end{align}\] At this point, we use the Sobolev embedding \(H^1(\mathbb{R}^d)\subset L^p(\mathbb{R}^d)\) for all \(p\in[2,\tfrac{2d}{d-2})\) if \(d\ge 2\) while \(H^1(\mathbb{R})\subset L^\infty(\mathbb{R})\), to find that \[\frac{|J|^2}{\rho}=\tfrac{\hbar^2}{4m^2}\sum_{\ell=0}^d\left(2|\partial_{x_\ell}\psi|^2-\frac{\psi{\overline{\psi}}}{\left}(\partial_{x_\ell}\overline{\psi}\right)^2-\frac{\overline{\psi}}{\psi}\left(\partial_{x_\ell}\psi\right)^2\right) \in L^\infty(\mathbb{R}_t;L^p(\mathbb{R}^d_x))\] with \(2\le p<\tfrac{2d}{d-2}\) if \(d\ge 2\) and \(p=\infty\) if \(d=1\), since \(\psi\in C_b(\mathbb{R}_t;H^2(\mathbb{R}^d))\) and \(\psi\not=0\) a.e. on \(\mathbb{R}_t\times\mathbb{R}^d\). Indeed, \(\psi/\overline{\psi}\) and \(\overline{\psi}/\psi\) are well-defined measurable functions on the complement of \(\psi^{-1}(\{0\})\), assumed to be a set of Lebesgue measure \(0\), so that \(\psi/\overline{\psi}\) and \(\overline{\psi}/\psi\in L^\infty(\mathbb{R}_t\times\mathbb{R}^d_x)\) — see next footnote. By the Cauchy-Schwarz inequality, this implies in turn that \(\rho u_ju_k\in L^\infty(\mathbb{R}_t;L^p(\mathbb{R}^d_x))\) for the same values of \(p\) as above.
Remark 11. Madelung’s original derivation of 48 is based on a representation of the wave function in the form \(\psi(t,x)=\alpha(t,x)e^{im\phi(t,x)/\hbar}\) with real-valued \(\alpha\) and \(\phi\), and his momentum equation (equation (3’) in [29]) takes the form of a Hamilton-Jacobi equation with a potential that is a linear combination of \(V\) and \(\Delta_x\alpha/\alpha\). Madelung’s derivation implicitly assumes that \(\psi\not=0\) everywhere. It seems that Gasser and Markowich [30] were the first to attempt deriving 48 as the system of moments in the momentum variable of the Wigner function of \(\psi\). This is a very natural idea if one thinks of the Wigner equation as a bona fide kinetic equation. In particular, this approach leads to a momentum equation in 48 in conservation form, which is more natural from the hydrodynamic viewpoint, and may alleviate requirements on the positivity of the density function \(|\psi|^2\).
However the derivation of the Madelung system in [30] is incorrect and/or incomplete for the following reasons.
Firstly, Lemma 2.1 in [30] does not make any assumption on the set \(\psi^{-1}(\{0\})\), where \(\psi\) is the solution of 34 whose density function and momentum density are expected to satisfy the Madelung system. Yet the first step in the proof of Lemma 2.1 in [30] is a claim that \((\nabla_x\rho)^{\otimes 2}/\rho\in L^\infty(\mathbb{R}_t;L^1(\mathbb{R}^d))\), based on the fact that the ratio \(\overline{\psi}/\psi\) is a well-defined element of \(L^\infty(\mathbb{R}_t\times\mathbb{R}^d)\), a statement known to be true if and only if3 the set \(\psi^{-1}(\{0\})\) is of Lebesgue measure \(0\).
Secondly, even if this missing assumption is added to the statement of Lemma 2.1 in [30], the remaining part of its proof is left to the reader as an “easy exercise” — in the words of the authors of [30]. Deriving the local conservation law of momentum 41 from 34 is classical — even though the second term on the right-hand side is more often written in the equivalent form \[-\partial_{y_j}\partial_{y_k}\tilde{R}\big|_{y=0}=\int_{\mathbb{R}^d}\xi_j\xi_kW_\hbar[R]d\xi\,.\] Starting from 41 , the closure relation 44 is an almost straightforward consequence of our identity 32 , already used in the proof of our Proposition 2.
Another merit of this approach is that it leads to a structure of the Madelung pressure tensor (see 50 below) involving the expression \[\nabla_x^2\rho-4(\nabla_x\sqrt{\rho})^{\otimes 2}\quad\text{ instead of }\quad\rho\nabla_x^2\ln\rho\,,\] which is seen to be a \(L^1_{loc}\) function as a consequence of the Lions-Villani observation [31] on the Sobolev regularity of square roots of nonnegative functions. This property seems less obvious on the expression \(\rho\nabla_x^2\ln\rho\) more commonly found in the literature as in [29].
Finally, allowing \(\psi(t,\cdot)\) to vanish on a set of measure \(0\) is not only a minor technical improvement over the case where \(\psi(t,x)\not=0\) for all \(x\in\mathbb{R}^d\). There are situations of physical interest involving nonlinear variants of 34 and isolated points \(x_j(t)\) where \(\psi(t,x_j(t))=0\), such as vortex dynamics in the Ginzburg-Landau equation, where Madelung’s quantum hydrodynamics appear naturally: see for instance [32].
If moreover \(\rho\in C^2([0,\tau)\times\mathbb{R}^d_x)\) is such that \(\rho>0\) everywhere on \([0,\tau)\times\mathbb{R}^d_x\), there is another equivalent form of the Euler equation 45 , \[\label{eq:Madelung32Euler32QuantPress} \partial_t(\rho u)+\nabla_x\cdot(\rho (u\otimes u+\tfrac1m\Pi))=-\tfrac1m\rho\nabla_xV,\tag{49}\] with pressure tensor \[\label{DefQuantPress} \Pi:=-\tfrac{\hbar^2}{4m}\nabla^2_x\ln\rho.\tag{50}\] Indeed, if \(\rho\in C^2([0,\tau)\times\mathbb{R}^d_x)\) satisfies \(\rho>0\) on \([0,\tau)\times\mathbb{R}^d_x\), one has \[\rho\nabla^2_x\ln\rho=\rho\nabla_x\left(\frac{\nabla_x\rho}{\rho}\right)=\nabla_x^2\rho-\frac{(\nabla_x\rho)^{\otimes 2}}{\rho}=\nabla_x^2\rho-4(\nabla_x\sqrt{\rho})^{\otimes 2}.\]
Still another equivalent form of Madelung’s system requires checking the formula for the Bohm quantum potential. Assuming that \(\rho\in C^3([0,\tau)\times\mathbb{R}^d_x)\) is such that \(\rho>0\) everywhere on \([0,\tau)\times\mathbb{R}^d_x\), observe that \[\begin{align} \nabla_x\cdot(\rho\nabla^2_x\ln\rho)=&\nabla^2_x\ln\rho\cdot\nabla_x\rho+\rho\nabla_x\cdot\nabla^2_x\ln\rho \\ =&\rho(\nabla^2_x\ln\rho\cdot\nabla_x\ln\rho+\nabla_x\Delta_x\ln\rho) \\ =&\rho\nabla_x(\tfrac12|\nabla_x\ln\rho|^2+\Delta_x\ln\rho) \\ =&2\rho\nabla_x\left(\left|\frac{\nabla_x\sqrt{\rho}}{\sqrt{\rho}}\right|^2+\nabla_x\cdot\left(\frac{\nabla_x\sqrt\rho}{\sqrt\rho}\right)\right)=2\rho\nabla_x\left(\frac{\Delta_x\sqrt\rho}{\sqrt\rho}\right)\,. \end{align}\] Introducing the Bohm quantum potential \[\label{BohmQuantPot} P:=-\tfrac{\hbar^2}{2m}\frac{\Delta_x\sqrt\rho}{\sqrt\rho}\,,\tag{51}\] the Euler equation in Madelung’s system is recast as \[\label{eq:Madelung32Euler32BohmPot} \partial_tu+u\cdot\nabla_xu=-\tfrac1m\nabla_x(P+V)\,.\tag{52}\]
Finally, let us discuss the condition \(\hbar^2\|\nabla\rho_\hbar\|^2_{L^2(\mathbb{R}^d)}\to 0\) in Proposition 2. Let us compute \[\rho^2_\hbar\mathrm{Tr}(\nabla^2\ln\rho_\hbar)=\rho^2_\hbar\Delta\ln\rho_\hbar=2\rho^2_\hbar\nabla\cdot\left(\frac{\nabla\sqrt{\rho_\hbar}}{\sqrt{\rho_\hbar}}\right) =2\rho^2_\hbar\left(\frac{\Delta\sqrt{\rho_\hbar}}{\sqrt{\rho_\hbar}}-\frac{|\nabla\sqrt{\rho_\hbar}|^2}{\rho_\hbar}\right),\] and \[\rho^2_\hbar\frac{\Delta\sqrt{\rho_\hbar}}{\sqrt{\rho_\hbar}}=\rho^{3/2}_\hbar\Delta\sqrt{\rho_\hbar}=\Delta(\rho^2_\hbar)-3\rho_\hbar|\nabla\sqrt{\rho_\hbar}|^2=\Delta(\rho^2_\hbar)-\tfrac34|\nabla\rho_\hbar|^2.\] These identites can be recast as \[\rho^2_\hbar\mathrm{Tr}\Pi_\hbar=\rho^2_\hbar P_\hbar+\tfrac{\hbar^2}{8m}|\nabla\rho_\hbar|^2 \quad\text{ and }\quad\rho^2_\hbar P_\hbar=-\tfrac{\hbar^2}{2m}\Delta(\rho^2_\hbar)+\tfrac38\tfrac{\hbar^2}m|\nabla\rho_\hbar|^2,\] or, equivalently \[\hbar^2|\nabla\rho_\hbar|^2=\tfrac43\hbar^2\Delta(\rho^2_\hbar)+\tfrac83m\rho^2_\hbar P_\hbar=\hbar^2\Delta(\rho^2_\hbar)+2m\rho^2_\hbar\mathrm{Tr}\Pi_\hbar.\]
Returning to Proposition 2, observe that the sequence \(\hbar^2_n\rho^2_{\hbar_n}\to 0\) since \(\rho_{\hbar_n}\) is bounded in \(L^2(B(0,R))\). Hence \[\hbar^2_n\Delta(\rho^2_{\hbar_n})\to 0\quad\text{ in }\mathcal{D}'(\mathbb{R}^d)\qquad\text{ as }\hbar_n\to 0.\] Therefore \[\hbar_n^2\|\nabla\rho_{\hbar_n}\|^2_{L^2(B(0,R)}\to 0\iff\rho^2_{\hbar_n}P_{\hbar_n}\to 0\text{ in }\mathcal{D}'(\mathbb{R}^d)\iff\rho^2_{\hbar_n}\mathrm{Tr}\Pi_{\hbar_n}\to 0\text{ in }\mathcal{D}'(\mathbb{R}^d).\] In other words, the condition \(\hbar_n^2\|\nabla\rho_{\hbar_n}\|^2_{L^2(B(0,R)}\to 0\) is equivalent to the fact that the quantum pressure \(\Pi_{\hbar_n}\), or the Bohm potential \(P_{\hbar_n}\), multiplied by \(\rho_{\hbar_n}^2\) converges to \(0\) in the sense of distributions on \(\mathbb{R}^d\).
Remark 12. In view of the formulas 10 , 11 and 12 , one can think of the Madelung system as equations for the moments in the \(\xi\)-variable of \(W_\hbar[\ketbra{\psi_\hbar}]\) in the classical (vanishing \(\hbar\)) limit. By comparison with the way in which fluid dynamical equations are derived from the kinetic theory of gases, one should think of Wigner measures \(w\), i.e. limit of converging subsequences \(W_{\hbar_n}[\ketbra{\psi_{\hbar_n}}]\) in \(\mathcal{S}'(\mathbb{R}^d\times\mathbb{R}^d)\), as the analogue of the local Maxwellian distribution function in gas dynamics, i.e. \[\frac{\rho(t,x)}{(2\pi\theta(t,x))^{3/2}}\exp\left(-\frac{|v-u(t,x)|^2}{2\theta(t,x)}\right).\] For ideal gases, such as are described by the Boltzmann equation, the equation of state gives the pressure \(p=\rho\theta\) in terms of the density \(\rho\) and temperature \(\theta\) (choosing units such that the Boltzmann constant is \(1\)). Therefore, if \(p(t,x)=0\) and \(\rho(t,x)>0\), one has \(\theta(t,x)=0\), in which case the Maxwellian must be replaced with \[\rho(t,x)\delta(v-u(t,x)).\] since \(e^{-|v|^2/2\theta}/(2\pi\theta)^{3/2}\to\delta_0(v)\) in \(\mathcal{S}'(\mathbb{R}^3)\) as \(\theta\to 0^+\).
The situation described in Proposition 2 does not involve thermodynamical quantities such as a temperature. However the quantum pressure, or the Bohm potential, plays a role similar to that of the classical pressure in gas dynamics: the vanishing of the quantum pressure implies that the limiting Wigner measure is monokinetic. However, this result is obtained in a completely different manner, since one does not have any explicit formula representing the distribution of \(\xi\) in \(w\) in terms of the density function \(\rho\), the current density \(J\) of the velocity field \(u\) in 47 and the quantum pressure \(\Pi\) in 50 .
Acknowledgements. We wish to thank Shi Jin for introducing us to the problem discussed in this paper, and Norbert Mauser for suggesting several references closely related to our approach (especially the work of Gasser and Markowich on Madelung’s quantum hydrodynamics). We are also grateful to Thomas Alazard and Herbert Koch for valuable insight on the vacuum problem in Madelung’s equations.
This terminology is not standard, and is introduced in [20] by analogy with Weyl’s quantization: see formula (1.1.9) of [21].↩︎
Indeed, for all \(u\in L^2(\mathbb{R}^d)\), one has \[\int_{\mathbb{R}^d}|u(x+z)-u(x)|^2dx\to 0\quad\text{ as }|z|\to 0\,.\]↩︎
Obviously, if \(\psi\) is Borel measurable on \(\mathbb{R}^d\) and \(\psi\not=0\) a.e., then \(\overline{\psi}/\psi\) is Borel measurable on the complement of the set \(\mathcal{N}:=\psi^{-1}(\{0\})\) which is of Lebesgue measure \(0\), so that \(\overline{\psi}/\psi\) is measurable on \(\mathbb{R}^d\), and \(|\overline{\psi}/\psi|=1\) a.e. on \(\mathbb{R}^d\) so that \(\overline{\psi}/\psi\in L^\infty(\mathbb{R}^d)\). If \(\mathcal{N}\) is of Lebesgue measure \(|\mathcal{N}|>0\), one has \(\psi=|\psi|u\) with \[u=\frac{\psi+\mathbf{1}_\mathcal{N}e^{i\Theta}}{|\psi|+\mathbf{1}_\mathcal{N}}\,,\qquad|u|=1\,,\quad\text{ so that one can set } \frac{\overline{\psi}}{\psi}:=\frac{\overline{u}}{u}=\frac{\overline{\psi}+\mathbf{1}_\mathcal{N}e^{i\Theta}}{\psi+\mathbf{1}_\mathcal{N}e^{i\Theta}}=\frac{\overline{\psi}}{\psi}\mathbf{1}_{\mathcal{N}^c}+e^{-2i\Theta}\mathbf{1}_\mathcal{N}\,,\] where \(\Theta\) is an arbitrary real-valued function. Here again \(|\overline{\psi}/\psi|=1\), but \(\overline{\psi}/\psi\) is not a well-defined element of \(L^\infty(\mathbb{R}^d)\), since this definition of \(\overline{\psi}/\psi\) explicitly involves the function \(\Theta\) which is arbitrary on the set \(\mathcal{N}\) of positive Lebesgue measure. One could even choose a non measurable \(\Theta\), in which case \(\overline{\psi}/\psi\notin L^\infty(\mathbb{R}^d)\).↩︎