On the Cauchy problem for the Hartree approximation in quantum dynamics

Rémi Carles Univ Rennes, CNRS
IRMAR - UMR 6625
F-35000 Rennes, France Remi.Carles@math.cnrs.fr , Clotilde Fermanian Kammerer Univ Paris Est Creteil, CNRS, LAMA, F-94010 Creteil, France
Univ Gustave Eiffel, LAMA, F-77447 Marne-la-Vallée, France clotilde.fermanian@u-pec.fr and Caroline Lasser Technische Universität München, Zentrum Mathematik, Deutschland classer@ma.tum.de

Abstract.

We prove existence and uniqueness results for the time-dependent Hartree approximation arising in quantum dynamics. The Hartree equations of motion form a coupled system of nonlinear Schrödinger equations for the evolution of product state approximations. They are a prominent example for dimension reduction in the context of the the time-dependent Dirac–Frenkel variational principle. We handle the case of Coulomb potentials thanks to Strichartz estimates. Our main result addresses a general setting where the nonlinear coupling cannot be considered as a perturbation. The proof uses a recursive construction that is inspired by the standard approach for the Cauchy problem associated to symmetric quasilinear hyperbolic equations.

RC was supported by Rennes Métropole through its AIS program, and by Centre Henri Lebesgue, program ANR-11-LABX-0020-0. CL was supported by the Centre for Advanced Study in Oslo, Norway, research project Attosecond Quantum Dynamics Beyond the Born-Oppenheimer Approximation .

1. Introduction

We consider the time-dependent Schrödinger equation

(1.1)

i{\partial}_{t}\psi=H\psi,

where the total Hamiltonian is given by

H=H_{x}+H_{y}+w(x,y),\quad H_{x}=-\frac{1}{2}\Delta_{x}+V_{1}(x),\quad H_{y}=-\frac{1}{2}\Delta_{y}+V_{2}(y)

with $x\in{\mathbb{R}}^{d_{1}}$ and $y\in{\mathbb{R}}^{d_{2}}$ , $d_{1},d_{2}\geq 1$ . The potentials $V_{1},V_{2}$ and $w$ are always real-valued, we will make extra regularity and decay assumptions later on . It is common wisdom that for dealing with quantum systems “a solution of the wave equation in many-dimensional space is far too complicated to be practicable” (Dirac 1930) and one aims at approximative methods that effectively reduce the space dimension. Here, we focus on initial data that decouple the space variables,

\psi(0,x,y)=\phi^{x}_{0}(x)\phi^{y}_{0}(y).

Such a form of initial data indeed suggests a dimension reduction approach. Of course, if there is no coupling ( $w(x,y)=0$ ), the full solution is itself a product state $\psi(t,x,y)=\phi_{x}(t,x)\phi_{y}(t,y)$ , with

\begin{cases}i\partial_{t}\phi^{x}=H_{x}\phi^{x},\quad\phi^{x}(0,x)=\phi^{x}_{0}(x),\\ i\partial_{t}\phi^{y}=H_{y}\phi^{y},\quad\phi^{y}(0,y)=\phi^{y}_{0}(y).\end{cases}

When the coupling is present, one seeks for approximate solutions $u(t)\approx\psi(t)$ of product form $u(t,x,y)=\phi^{x}(t,x)\phi^{y}(t,y)$ in order to reduce the initial system (1.1) in ${\mathbb{R}}^{d_{1}+d_{2}}$ to two systems on spaces of smaller dimensions, ${\mathbb{R}}^{d_{1}}$ and ${\mathbb{R}}^{d_{2}}$ . In situations, where the overall configuration space has a natural decomposition of its dimension $d_{1}+\cdots+d_{N}$ , a corresponding product ansatz of $N$ factors is sought. Here we only investigate the case $N=2$ , mentioning that repeated application of the binary construction yields the more general case. Applying the time-dependent Dirac–Frenkel variational principle to the manifold

\mathcal{M}=\left\{u=\varphi^{x}\otimes\varphi^{y}\mid\varphi^{x}\in L^{2}({\mathbb{R}}^{d_{1}}),\ \varphi^{y}\in L^{2}({\mathbb{R}}^{d_{2}})\right\}

yields the so-called time-dependent Hartree approximation,

\psi(t,x,y)\approx\phi^{x}(t,x)\phi^{y}(t,y)\in\mathcal{M},

where the pair $(\phi^{x},\phi^{y})$ solves the nonlinearly coupled system

(1.2)

\begin{cases}i\partial_{t}\phi^{x}=H_{x}\phi^{x}+\langle w\rangle_{y}\phi^{x},\quad\phi^{x}(0,x)=\phi^{x}_{0}(x),\\ i\partial_{t}\phi^{y}=H_{y}\phi^{y}+\langle w\rangle_{x}\phi^{y},\quad\phi^{y}(0,y)=\phi^{y}_{0}(y).\end{cases}

The time-dependent potentials result from the averaging process

	$\displaystyle\langle w\rangle_{y}(t,x):=\int_{{\mathbb{R}}^{d_{2}}}w(x,y)\|\phi^{y}(t,y)\|^{2}dy,$
	$\displaystyle\langle w\rangle_{x}(t,y):=\int_{{\mathbb{R}}^{d_{1}}}w(x,y)\|\phi^{x}(t,x)\|^{2}dx,$

under the assumption, made throughout this paper, that

(1.3)

\|\phi_{0}^{x}\|_{L^{2}({\mathbb{R}}^{d_{1}})}=\|\phi_{0}^{y}\|_{L^{2}({\mathbb{R}}^{d_{2}})}=1.

For any “reasonable” solution (at least with the regularity considered in this paper), the $L^{2}$ -norms of $\phi^{x}(t,\cdot)$ and $\phi^{y}(t,\cdot)$ , respectively, are independent of time, hence

\|\phi^{x}(t)\|_{L^{2}({\mathbb{R}}^{d_{1}})}=\|\phi^{y}(t)\|_{L^{2}({\mathbb{R}}^{d_{2}})}=1,

for all $t$ in the time interval where the solution to (1.2) is well-defined; see §6.3 for a proof.

Even though the time-dependent Hartree approximation is one of the most fundamental approximations in quantum dynamics, mathematical existence and uniqueness proofs are rather scarce. Existence and uniqueness have been studied in the case where the interaction potential is of convolution type, i.e. for $w(x,y)=W(x-y)$ and with one of the subsystems moving by classical mechanics (see [7, 2, 6] for example). A related investigation has targeted the time-dependent self-consistent field system [12] with coupling potentials of Schwartz class. However, our aim here is to discuss the existence and uniqueness of solutions for system (1.2) when the potentials $\langle w\rangle_{x}$ or $\langle w\rangle_{y}$ need not be bounded, and cannot be considered as a perturbation of $V_{2}$ or $V_{1}$ , respectively. This framework requires a different approach. In particular, our result provides the Cauchy theory for the systems discussed in the articles [3, 4] where the accuracy of the Hartree approximation is studied in the broader context of composite quantum dynamics and scale separation.

The steps for our existence and uniqueness proof are strongly inspired by the method which is classical in the study of quasilinear hyperbolic systems, see e.g. [1]. With $n\in{\mathbb{N}}$ , we associate the iterative scheme of recursive equations

(1.4)

\left\{\begin{aligned} i\partial_{t}\phi^{x}_{n+1}=H_{x}\phi^{x}_{n+1}+\langle w_{n}\rangle_{y}\phi^{x}_{n+1},\quad\phi^{x}_{n+1}(0,x)=\phi^{x}_{0}(x),\\ i\partial_{t}\phi^{y}_{n+1}=H_{y}\phi^{y}_{n+1}+\langle w_{n}\rangle_{x}\phi^{y}_{n+1},\quad\phi^{y}_{n+1}(0,y)=\phi^{y}_{0}(y),\end{aligned}\right.

with

(1.5)

\langle w_{n}\rangle_{y}(t,x)=\int_{{\mathbb{R}}^{n}}w(x,y)|\phi^{y}_{n}(t,y)|^{2}dy,\quad\langle w_{n}\rangle_{x}(t,y)=\int_{{\mathbb{R}}^{d}}w(x,y)|\phi^{x}_{n}(t,x)|^{2}dx.

The main steps of the proof of our existence and uniqueness result are then:

(1)

The iterative scheme is well-defined and enjoys bounds in “large” norm, which control second order derivatives and polynomial growth of order two for some finite time horizon.
(2)

The solution of the scheme converges in “small” norm, which is the $L^{2}$ norm.
(3)

It is possible to pass to the limit $n\to+\infty$ in the equation, which leads to the construction of a solution that one then proves to be unique and global in time (provided the initial data is regular enough).

1.1. Outline

In the next Section 2, we recall elementary properties of the time-dependent variational principle and formally derive the Hartree equations (1.2). Then we discuss coupling potentials $w(x,y)$ of Coulombic and of polynomial type in Section 3, where we also present our main result Theorem 3.11, which establishes existence and uniqueness of the solutions to the Hartree system for coupling with polynomial growth. The different steps of the proof of Theorem 3.11 are the subject of Section 4 (analysis of the iterative scheme), Section 5 (convergence in small norms) and Section 6 (passing to the limit). A sufficient condition for the growth of the coupling potential is verified in Section 7. The Appendices A and B summarize some technical arguments.

1.2. Notations

We write $L^{\infty}_{T}$ for $L^{\infty}([0,T])$ . The notations $L^{2}_{x}$ , $L^{2}_{y}$ , $L^{2}_{x,y}$ stand for $L^{2}({\mathbb{R}}^{d_{1}}_{x})$ , $L^{2}({\mathbb{R}}^{d_{2}}_{y})$ , $L^{2}({\mathbb{R}}^{d_{1}+d_{2}}_{x,y})$ , respectively. We denote by ${\langle}\cdot,\cdot{\rangle}_{L^{2}_{x}}$ , ${\langle}\cdot,\cdot{\rangle}_{L^{2}_{y}}$ , ${\langle}\cdot,\cdot{\rangle}_{L^{2}_{x,y}}$ the corresponding inner products. For $f,g\geq 0$ , we write $f\lesssim g$ whenever there exists a “universal” constant (in the sense that it does not depend on time, space, or $n$ , typically) such that $f\leq Cg$ .

2. Variational principle

The time-dependent Hartree approximation results from the Dirac–Frenkel variational principle applied to the manifold

(2.1)

\mathcal{M}=\left\{u=\varphi^{x}\otimes\varphi^{y}\mid\varphi^{x}\in L^{2}_{x},\ \varphi^{y}\in L^{2}_{y}\right\},

see also [15, §II.3.1] for the analogous discussion with Hartree products of $N$ orbitals in $L^{2}({\mathbb{R}}^{3})$ . The reader can also refer to [14]. The principle determines an approximate solution $u(t)\in\mathcal{M}$ for the time-dependent Schrödinger equation

i\partial_{t}\psi=H\psi

with initial data $\psi(0)\in\mathcal{M}$ by requiring that for all times $t$

(2.2)

\left\{\begin{array}[]{l}\partial_{t}u(t)\in\mathcal{T}_{u(t)}\mathcal{M},\\[4.30554pt] \langle v,i\partial_{t}u(t)-Hu(t)\rangle=0\quad\text{for all}\quad v\in\mathcal{T}_{u(t)}\mathcal{M},\end{array}\right.

where $\mathcal{T}_{u(t)}\mathcal{M}$ denotes the tangent space of $\mathcal{M}$ at $u(t)$ . For deriving the Hartree equations, we first have to understand the manifold $\mathcal{M}$ and its tangent space. Note that the representation of a Hartree function is non-unique, since $\varphi^{x}\otimes\varphi^{y}=(a\varphi^{x})\otimes(a^{-1}\varphi^{y})$ for any $a\in{\mathbb{C}}\setminus\{0\}$ . However, we can have unique representations in the tangent space once appropriate gauge conditions are set.

Lemma 2.1 (Tangent space).

For any $u=\varphi^{x}\otimes\varphi^{y}\in\mathcal{M}$ , $u\neq 0$ ,

\mathcal{T}_{u}\mathcal{M}=\left\{v^{x}\otimes\varphi^{y}+\varphi^{x}\otimes v^{y}\mid v^{x}\in L^{2}_{x},\ v^{y}\in L^{2}_{y}\right\}.

Any $v\in\mathcal{T}_{u}\mathcal{M}$ has a unique representation of the form $v=v^{x}\otimes\varphi^{y}+\varphi^{x}\otimes v^{y}$ , if we impose the gauge condition $\langle\varphi^{x},v^{x}\rangle=0$ . The tangent spaces are complex linear subspaces of $L^{2}_{x,y}$ such that $u\in\mathcal{T}_{u}\mathcal{M}$ for all $u\in\mathcal{M}$ .

The lemma is proved in Appendix A. The following formal arguments show that, in case that the variational solution $u(t)$ is well-defined and sufficiently regular, the $L^{2}$ norm and the energy expectation value are conserved automatically. Indeed, we differentiate with respect to time $t$ and use the variational condition (2.2) for $v=u(t)$ ,

\frac{d}{dt}\|u(t)\|^{2}_{L^{2}_{x,y}}=2\,\mathrm{Re}{\langle}u(t),\partial_{t}u(t){\rangle}_{L^{2}_{x,y}}=2\,\mathrm{Re}{\langle}u(t),\tfrac{1}{i}Hu(t){\rangle}_{L^{2}_{x,y}}=0,

due to self-adjointness of the Hamiltonian. Similarly, using self-adjointness and the variational condition (2.2) for $v=\partial_{t}u(t)$ ,

\frac{d}{dt}\langle u(t),Hu(t)\rangle_{L^{2}_{x,y}}=2\,\mathrm{Re}\langle\partial_{t}u(t),Hu(t)\rangle_{L^{2}_{x,y}}=2\,\mathrm{Re}\langle\partial_{t}u(t),i\partial_{t}u(t)\rangle_{L^{2}_{x,y}}=0.

Let us now formally derive the Hartree system (1.2). We write

u(t)=\varphi^{x}(t)\otimes\varphi^{y}(t),

with $\|\varphi^{x}(t)\|_{L^{2}_{x}}=\|\varphi^{y}(t)\|_{L^{2}_{y}}=1$ . We have

	$\displaystyle i\partial_{t}u$	$\displaystyle=(i\partial_{t}\varphi^{x}(t))\otimes\varphi^{y}(t)+\varphi^{x}(t)\otimes(i\partial_{t}\varphi^{y}(t)),$
	$\displaystyle Hu$	$\displaystyle=H_{x}\varphi^{x}(t))\otimes\varphi^{y}(t)+\varphi^{x}(t)\otimes(H_{y}\varphi^{y}(t))+w(x,y)\varphi^{x}(t)\otimes\varphi^{y}(t).$

Choosing elements $v=v^{x}\otimes\varphi^{y}+\varphi^{x}\otimes v^{y}\in\mathcal{T}_{u(t)}\mathcal{M}$ and evaluating (2.2), we obtain the following necessary and sufficient conditions:

(i)

If $v^{y}$ =0, we obtain that for all $v^{x}\in L^{2}_{x}$ such that $\langle v^{x},\varphi^{x}(t)\rangle_{L^{2}_{x}}=0$ ,

\langle v^{x},(i\partial_{t}-H_{x})\varphi^{x}(t)\rangle_{L^{2}_{x}}=\int_{{\mathbb{R}}^{d_{1}+d_{2}}}w(x,y)v^{x}(x)\varphi^{x}(t,x)|\varphi^{y}(t,y)|^{2}dxdy.

(ii)

If $v^{x}$ =0, we obtain that for all $v^{y}\in L^{2}_{y}$ such that $\langle v^{y},\varphi^{y}(t)\rangle_{L^{2}_{y}}=0$ ,

\langle v^{y},(i\partial_{t}-H_{y})\varphi^{y}(t)\rangle_{L^{2}_{y}}=\int_{{\mathbb{R}}^{d_{1}+d_{2}}}w(x,y)v^{y}(y)\varphi^{y}(t,y)|\varphi^{x}(t,x)|^{2}dxdy.

The choice of $\varphi^{x}(t)$ and $\varphi^{y}(t)$ satisfying the Hartree system (1.2) guarantees (i) and (ii).

3. Main result

We present existence and uniqueness results for the solution of the time-dependent Hartree system (1.2). In §3.1, we discuss how Strichartz estimates may be applied to Coulombic coupling. In §3.2, we give detailed assumptions on polynomial growth conditions. Then, in §3.3 we state our main result Theorem 3.11.

3.1. Coupling potentials of Coulombic form

The case of Coulomb singularities (in combination with classical nuclear dynamics) has already been addressed in [7, 2] by Schauder and Picard fixed point arguments, respectively. We briefly revisit the main result from [7], and show how it may be adapted thanks to Strichartz estimates. We suppose $d_{1}=d_{2}=3$ , and have in mind the case

(3.1)

w(x,y)=\frac{\varepsilon}{|x-y|},\quad\varepsilon\in{\mathbb{R}}.

We consider more generally the case $w(x,y)=W(x-y)$ , for a possibly singular $W$ . We assume that the potentials $V_{1}$ and $V_{2}$ are perturbations of smooth and at most quadratic potentials:

V_{j}={\mathbf{V}}_{j}+v_{j},

where

{\mathbf{V}}_{j}\in\mathcal{Q}=\left\{V\in C^{\infty}({\mathbb{R}}^{3};{\mathbb{R}}),\ {\partial}^{\alpha}V\in L^{\infty}({\mathbb{R}}^{3}),\ \forall|\alpha|\geq 2\right\},

and

v_{j}\in L^{p}({\mathbb{R}}^{3})+L^{\infty}({\mathbb{R}}^{3}),\quad\text{for some }p>3/2.

Typically, we may consider Coulomb potentials, as

\frac{1}{|x|}=\frac{1}{|x|}{\mathbf{1}}_{|x|<1}+\frac{1}{|x|}{\mathbf{1}}_{|x|\geq 1}.

The first term on the right hand side belongs to $L^{p}({\mathbb{R}}^{3})$ for any $1\leq p<3$ , and the second term is obviously bounded. We then make the same assumption on $W$ . Denote

\mathbf{H}_{x}=-\frac{1}{2}\Delta_{x}+\mathbf{V}_{1},\quad\mathbf{H}_{y}=-\frac{1}{2}\Delta_{y}+\mathbf{V}_{2}.

Then $e^{-it\mathbf{H}_{x}}$ and $e^{-it\mathbf{H}_{y}}$ enjoy Strichartz estimates, and (1.2) can be solved at the $L^{2}$ level, by a straightforward adaptation of [10, Corollary 4.6.5]:

Theorem 3.1.

Assume $d_{1}=d_{2}=3$ , $\mathbf{V}_{1},\mathbf{V}_{2}\in\mathcal{Q}$ , $v_{1},v_{2},W\in L^{p}({\mathbb{R}}^{3})+L^{\infty}({\mathbb{R}}^{3})$ for some $p>3/2$ , and $\phi_{0}^{x},\phi_{0}^{y}\in L^{2}({\mathbb{R}}^{3})$ . Then (1.2) has a unique solution $(\phi^{x},\phi^{y})\in C({\mathbb{R}};L^{2}({\mathbb{R}}^{3}))^{2}\cap L^{q}_{\rm loc}({\mathbb{R}};L^{r}({\mathbb{R}}^{3}))^{2}$ , where $1=2/r+1/p$ and $q$ is such that

\frac{2}{q}=3\left(\frac{1}{2}-\frac{1}{r}\right).

The $L^{2}$ -norms of $\phi^{x}$ and $\phi^{y}$ are independent of $t\in{\mathbb{R}}$ , hence in view of (1.3),

\|\phi^{x}(t)\|_{L^{2}({\mathbb{R}}^{3})}=\|\phi^{y}(t)\|_{L^{2}({\mathbb{R}}^{3})}=1,\quad\forall t\in{\mathbb{R}}.

The proof is presented shortly in Appendix B.

Remark 3.2.

The sign of $\varepsilon$ in (3.1) plays no role here. Indeed, the proof relies on local in time Strichartz estimates associated to $\mathbf{H}_{x}$ and $\mathbf{H}_{y}$ , respectively, and the potentials $v_{1},v_{2}$ and $W$ are treated as perturbations, whose sign is irrelevant in order to guarantee the above global existence result. On the other hand, Theorem 3.1 brings no information regarding the quality of the dynamics or the existence of a ground state.

Remark 3.3.

Under extra assumptions on the potentials $v_{1}$ and $v_{2}$ (no extra assumption is needed for $W$ , as it is associated to a convolution), it is possible to consider higher regularity properties. In particular, working at the level of $H^{1}$ -regularity makes it possible to show the conservation of the energy

	$\displaystyle E(t)$	$\displaystyle=\langle H_{x}\phi^{x}(t),\phi^{x}(t)\rangle_{L^{2}_{x}}+\langle H_{y}\phi^{y}(t),\phi^{y}(t)\rangle_{L^{2}_{y}}$
		$\displaystyle\quad+\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}W(x-y)\|\phi^{x}(t,x)\|^{2}\|\phi^{y}(t,y)\|^{2}dxdy,$

provided that $\nabla v_{1}$ and $\nabla v_{2}$ also belong to $L^{p}({\mathbb{R}}^{3})+L^{\infty}({\mathbb{R}}^{3})$ for some $p>3/2$ . We refer to Remark B.5 for more details.

Remark 3.4.

The role of the set $\mathcal{Q}$ is to guarantee that (local in time) Strichartz estimates are available for $\mathbf{H}_{x}$ and $\mathbf{H}_{y}$ . The same would still be true for a larger class of potentials, including for instance Kato potentials ([20]) or potentials decaying like an inverse square ([5]). The choice of this set $\mathcal{Q}$ is made in order to simplify the presentation, and because it is delicate to keep track of all the classes of potentials for which Strichartz estimates have been proved.

3.2. Coupling potentials with polynomial growth

The core of this paper addresses the case where the coupling potential $w$ may grow polynomially. To be more concrete, we recall the example addressed in [4].

Example 3.5.

Assume $d_{1}=d_{2}=1$ and that the potentials are given by

V_{1}(x)=\frac{1}{2}x^{2}\left(\frac{x}{2\ell}-1\right)^{2},\quad\ell>0,\quad V_{2}(y)=\frac{\omega^{2}}{2}y^{2},\quad w(x,y)=\chi(x)y^{2},\;\;\chi\in C_{0}^{\infty}({\mathbb{R}}).

Here, $V_{1}(x)$ corresponds to a double well and $V_{2}(y)$ to a harmonic bath. The coupling $w(x,y)$ could be locally cubic when choosing $\chi(x)=x$ for $x$ in a neighborhood of zero.

We emphasize that in Example 3.5, the average ${\langle}w{\rangle}_{x}$ grows quadratically in $y$ : in terms of growth, ${\langle}w{\rangle}_{x}$ is comparable to $V_{2}$ and cannot be considered as a perturbation as far as the Cauchy problem is concerned. This setting turns out to be very different from the one in [7, 2] (see also Theorem 3.1), and requires a different approach to be developed below.

3.2.1. Restriction to non-negative potentials

In the general case, we assume $d_{1},d_{2}\geq 1$ . First, the potentials $V_{1}$ and $V_{2}$ are smooth, real-valued, $V_{1}\in C^{\infty}({\mathbb{R}}^{d_{1}};{\mathbb{R}})$ , $V_{2}\in C^{\infty}({\mathbb{R}}^{d_{2}};{\mathbb{R}})$ , and bounded from below:

\forall x\in{\mathbb{R}}^{d_{1}},\forall y\in{\mathbb{R}}^{d_{2}},\;\;V_{1}(x)\geq-C_{1}\;\;\mbox{and}\;\;V_{2}(y)\geq-C_{2},

for some constants $C_{1},C_{2}>0$ . The operators $H_{x}$ and $H_{y}$ then are self-adjoint operators. Up to changing $\phi^{x}(t,x)$ to $\phi^{x}(t,x)e^{itC_{1}}$ (which amounts to replacing $V_{1}$ by $V_{1}+C_{1}$ in (1.2)), and $\phi^{y}(t,y)$ to $\phi^{y}(t,y)e^{itC_{2}}$ , we may actually assume

(H1)

V_{1}(x)\geq 1,\ \forall x\in{\mathbb{R}}^{d_{1}},\quad\text{and}\quad V_{2}(y)\geq 1,\ \forall y\in{\mathbb{R}}^{d_{2}},

as we are only interested in existence results for the Cauchy problem (1.2). Thus $H_{x}$ and $H_{y}$ are sums of a nonnegative operator (Laplacian in $x$ and $y$ , respectively) and of a nonnegative potential. We use them to measure the regularity of the solutions of the system (1.2).

3.2.2. Functional setting

For $k\in{\mathbb{N}}$ , we define the Hilbert spaces

\displaystyle\mathcal{H}_{x}^{k}

\displaystyle=\left\{\phi\in L^{2}({\mathbb{R}}^{d_{1}}),\;\;H^{k/2}_{x}\phi\in L^{2}({\mathbb{R}}^{d_{1}})\right\}\;\;\mbox{and}\;\;\mathcal{H}_{y}^{k}=\left\{\phi\in L^{2}({\mathbb{R}}^{d_{2}}),\;\;H^{k/2}_{y}\phi\in L^{2}({\mathbb{R}}^{d_{2}})\right\},

which are the natural analogues of Sobolev spaces $H^{k}$ in the presence of (nonnegative) potentials (in view of (H1)), equipped with the norms given by

\|\phi\|_{\mathcal{H}_{x}^{k}}^{2}=\|\phi\|_{L^{2}_{x}}^{2}+\|H^{k/2}_{x}\phi\|_{L^{2}_{x}}^{2},\quad\|\phi\|_{\mathcal{H}_{y}^{k}}^{2}=\|\phi\|_{L^{2}_{y}}^{2}+\|H^{k/2}_{y}\phi\|_{L^{2}_{y}}^{2}.

For $\alpha,\beta\in{\mathbb{N}}$ , $\Phi=(\phi^{x},\phi^{y})\in\mathcal{H}_{x}^{\alpha}\times\mathcal{H}_{y}^{\beta}$ , we set

\|\Phi\|_{\alpha,\beta}^{2}=\|\phi^{x}\|_{\mathcal{H}_{x}^{\alpha}}^{2}+\|\phi^{y}\|_{\mathcal{H}_{y}^{\beta}}^{2}=\|\phi^{x}\|_{L^{2}_{x}}^{2}+\|H_{x}^{\alpha/2}\phi^{x}\|_{L^{2}_{x}}^{2}+\|\phi^{y}\|_{L^{2}_{y}}^{2}+\|H_{y}^{\beta/2}\phi^{y}\|_{L^{2}_{y}}^{2}.

All along the paper, we use that in view of (H1), $0\leq H_{x}^{\alpha}\leq H_{x}^{\alpha+1}$ and $0\leq H_{y}^{\beta}\leq H_{y}^{\beta+1}$ .

As (1.2) is reversible, from now on we consider positive time only. We shall work with the time-dependent functional spaces

X^{\alpha,\beta}_{T}=\left\{\Phi(t)=(\phi^{x}(t),\phi^{y}(t)),\;\phi^{x}\in L^{\infty}\left([0,T],\mathcal{H}_{x}^{\alpha}\right),\;\phi^{y}\in L^{\infty}\left([0,T],\mathcal{H}_{y}^{\beta}\right)\right\}.

If $\Phi=(\phi^{x},\phi^{y})\in X^{\alpha,\beta}_{T}$ , we set

\|\Phi\|_{X_{T}^{\alpha,\beta}}=\sup_{t\in[0,T]}\|\Phi(t)\|_{\alpha,\beta}.

We choose to consider integer exponents $\alpha$ and $\beta$ for the sake of simplicity. We emphasize however that our approach requires $\alpha,\beta\geq 2$ ; see Section 4 for a more precise discussion on this aspect. We note that Theorem 3.11 allows $\alpha=\beta=2$ .

3.2.3. Main assumptions

We assume that the coupling potential $w\in C^{\infty}({\mathbb{R}}^{d_{1}+d_{2}};{\mathbb{R}})$ satisfies

(H2) There exist $c_{0},C>0$ with $c_{0}<1$ such that for all $(x,y)\in{\mathbb{R}}^{d_{1}}\times{\mathbb{R}}^{d_{2}}$ ,

|w(x,y)|\leq c_{0}(V_{1}(x)+V_{2}(y)+C).

We emphasize that no condition is required concerning the above constant $C$ : for instance if $V_{1}$ and $V_{2}$ are bounded, then we may always pick $c_{0}<1$ so that (H2) is satisfied. For unbounded potentials, the requirement $c_{0}<1$ can be understood as some smallness property, in the sense that $w(x,y)$ is a perturbation of $V_{1}(x)+V_{2}(y)$ . This actually corresponds to the physical framework where the system (1.2) is introduced in order to approximate the exact solution $\psi$ of (1.1) through the formula $\psi\approx\phi^{x}\otimes\phi^{y}$ ; see [3] for a derivation of error estimates. We also note that the assumption $c_{0}<1$ implies that $w$ is $(H_{x}+H_{y})$ -bounded with relative bound $c_{0}<1$ , hence by Kato–Rellich Theorem (see e.g. [18, Theorem X.12]), $H$ is self-adjoint.

Remark 3.6.

If $w$ is at most quadratic, in the sense that $w\in C^{\infty}({\mathbb{R}}^{d_{1}+d_{2}};{\mathbb{R}})$

{\partial}_{x,y}^{\gamma}w\in L^{\infty}({\mathbb{R}}^{d_{1}+d_{2}}),\quad\forall\gamma\in{\mathbb{N}}^{d_{1}+d_{2}},\ |\gamma|\geq 2,

then the assumption (H2) is not needed to guarantee that $H$ is self-adjoint (see the Faris–Lavine Theorem, [18, Theorem X.38]). Such a framework corresponds to the assumptions made for the error analysis in [3, 4]. For such potentials $w$ , the assumption $c_{0}<1$ in (H2) is needed only in order to ensure that the Hartree solutions are global in time.

We also assume some conditions on the regularity of commutators of the coupling potential with the operators $H_{x}$ and $H_{y}$ . For integers $\alpha,\beta\geq 1$ , we consider the condition:

(H3)_α,β. There exist $c_{1},c_{2}>0$ such that for all $k\in\{1,\cdots,\alpha\}$ , $\ell\in\{1,\cdots,\beta\}$ , for all $f_{j}=f_{j}(x)$ , $g_{j}=g_{j}(y)$ in the Schwartz class ( $j\in\{1,2\}$ ),

	$\displaystyle\left\|{\langle}H_{x}^{k-1}[w(\cdot,y),H_{x}]f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right\|$	$\displaystyle+\left\|{\langle}[w(\cdot,y),H_{x}]H_{x}^{k-1}f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right\|$
		$\displaystyle\leq c_{1}\left(1+V_{2}(y)\right)\\|f_{1}\\|_{\mathcal{H}_{x}^{k}}\\|f_{2}\\|_{\mathcal{H}_{x}^{k}},\quad\text{for a.a. }y\in{\mathbb{R}}^{d_{2}},$
	$\displaystyle\left\|{\langle}H_{y}^{\ell-1}[w(x,\cdot),H_{y}]g_{1},g_{2}{\rangle}_{L^{2}_{y}}\right\|$	$\displaystyle+\left\|{\langle}[w(x,\cdot),H_{y}]H_{y}^{\ell-1}g_{1},g_{2}{\rangle}_{L^{2}_{y}}\right\|$
		$\displaystyle\leq c_{2}\left(1+V_{1}(x)\right)\\|g_{1}\\|_{\mathcal{H}_{y}^{\ell}}\\|g_{2}\\|_{\mathcal{H}_{y}^{\ell}},\quad\text{for a.a. }x\in{\mathbb{R}}^{d_{1}}.$

Assumption (H3)_α,β is made in order to generalize the framework of Example 3.5. The subsequent proofs do not use the special form of the Hamiltonians $H_{x}$ , $H_{y}$ , so that our result extends as soon as they are self-adjoint operators and assumptions (H1), (H2), (H3)_α,β are satisfied. This applies in particular for magnetic Schrödinger operators.

Remark 3.7.

It is not necessary to assume that the potential $w$ is smooth, $w\in C^{\infty}({\mathbb{R}}^{d_{1}+d_{2}};{\mathbb{R}})$ . We only need enough regularity in order to write assumption ${\bf(H3)}_{\alpha,\beta}$ for the $\alpha$ and $\beta$ that we consider (recalling that $H_{x}^{k-1}$ and $H_{y}^{\ell-1}$ are self-adjoint). For example, if $w\in C^{2}({\mathbb{R}}^{d_{1}+d_{2}};{\mathbb{R}})$ , then Theorem 3.11 holds with $\alpha=\beta=2$ . We make this regularity assumption for simplicity, as the most important properties are those discussed in this subsection.

Remark 3.8.

Whenever $w(x,y)$ is a Coulomb potential as in Section 3.1, the assumptions (H3)_α,β are not satisfied. One then needs to take advantage of the convolution feature of the coupling and of the properties of $H_{x}$ and $H_{y}$ such as the Strichartz estimates in Proposition B.2.

We next present sufficient conditions on the potentials guaranteeing that assumptions (H2) and (H3)_2,2 hold.

Lemma 3.9 (Sufficient conditions).

Let $V_{1}\in C^{\infty}({\mathbb{R}}^{d_{1}};{\mathbb{R}})$ and $V_{2}\in C^{\infty}({\mathbb{R}}^{d_{2}};{\mathbb{R}})$ such that $V_{1},V_{2}\geq 0$ . The above assumptions (H2) and (H3)_2,2 are satisfied provided that the following estimates hold:

•

There exists $C>0$ such that

(3.2)

|\nabla V_{1}(x)|\leq C\left(1+V_{1}(x)\right),\forall x\in{\mathbb{R}}^{d_{1}};\quad|\nabla V_{2}(y)|\leq C\left(1+V_{2}(y)\right),\forall y\in{\mathbb{R}}^{d_{2}}.

•

There exist $0<c_{0}<1$ and $c>0$ independent of $x\in{\mathbb{R}}^{d_{1}}$ and $y\in{\mathbb{R}}^{d_{2}}$ such that

(3.3)

\begin{cases}&|w(x,y)|\leq c_{0}(V_{1}(x)+V_{2}(y)+c),\\ &|\nabla_{x}w(x,y)|\leq c(\sqrt{V_{1}(x)}+V_{2}(y)+1),\\ &|\nabla_{y}w(x,y)|\leq c(V_{1}(x)+\sqrt{V_{2}(y)}+1),\\ &|\Delta_{x}w(x,y)|+|\Delta_{y}w(x,y)|\leq c(V_{1}(x)+V_{2}(y)+1).\end{cases}

The proof of this lemma is given in Section 7.

Remark 3.10.

We note that Example 3.5 meets the requirements stated in Lemma 3.9, provided that $\|\chi\|_{L^{\infty}({\mathbb{R}})}<\omega^{2}/2$ . Thus it satisfies the assumptions of Theorem 3.11 below for $\alpha=\beta=2$ .

3.3. Main result and comments

Before stating our main result we informally summarize the previous assumptions on the potentials for the case of polynomial coupling:

	$\displaystyle{\bf(H1)}:\text{boundedness from below of the potentials $V_{1}(x)$ and $V_{2}(y)$;}$
	$\displaystyle{\bf(H2)}:\text{control of $w(x,y)$ in terms of $V_{1}(x)+V_{2}(y)$;}$
	$\displaystyle{\bf(H3)}_{\alpha,\beta}:\text{control of commutators involving $w(x,y)$, in terms of $H_{x}$ and $H_{y}$.}$

We have the following result on existence and uniqueness as well norm and energy conservation of the time-dependent Hartree approximation.

Theorem 3.11.

Let $d_{1},d_{2}\geq 1$ , $\alpha,\beta\geq 2$ and $\phi_{0}^{x}\in\mathcal{H}_{x}^{\alpha}$ , $\phi_{0}^{y}\in\mathcal{H}_{y}^{\beta}$ . Suppose that (H1), (H2) and (H3)_α,β are satisfied.

•

(1.2) possesses a unique, global solution in $\Phi\in C({\mathbb{R}}_{+};L^{2}\times L^{2})\cap\bigcap_{T>0}X^{\alpha,\beta}_{T}$ .

•

Conservations: the $L^{2}$ -norms of $\phi^{x}$ and $\phi^{y}$ are independent of $t\geq 0$ , hence in view of (1.3),

\|\phi^{x}(t)\|_{L^{2}({\mathbb{R}}^{d_{1}})}=\|\phi^{y}(t)\|_{L^{2}({\mathbb{R}}^{d_{2}})}=1,\quad\forall t\geq 0.

In addition, the following total energy is also independent of $t\geq 0$ :

	$\displaystyle E(t)$	$\displaystyle:=\langle H_{x}\phi^{x}(t),\phi^{x}(t)\rangle_{L^{2}_{x}}+\langle H_{y}\phi^{y}(t),\phi^{y}(t)\rangle_{L^{2}_{y}}$
		$\displaystyle\quad+\iint_{{\mathbb{R}}^{d_{1}}\times{\mathbb{R}}^{d_{2}}}w(x,y)\|\phi^{x}(t,x)\|^{2}\|\phi^{y}(t,y)\|^{2}dxdy.$

We will see that the assumption $c_{0}<1$ in (H2) arises in two steps of the proof of Theorem 3.11. First, to make sure that the approximating scheme (1.4) introduced below is well-defined, we invoke Kato–Rellich Theorem, to show essentially that ${\langle}w{\rangle}_{y}$ (or more precisely, ${\langle}w_{n}{\rangle}_{y}$ ) is $H_{x}$ -bounded with relative bound smaller than one, and that the same holds when the roles of $x$ and $y$ are swapped. Second, the assumption $c_{0}<1$ guarantees that the conserved energy $E$ , defined in Theorem 3.11, is a coercive functional, so the conservation of $E$ provides uniform in times a priori estimates, which in turn allow to show that the local in time solutions are actually global in time solutions.

The property $\Phi\in C({\mathbb{R}}_{+};L^{2}\times L^{2})\cap\bigcap_{T>0}X^{\alpha,\beta}_{T}$ means that $t\mapsto\|\Phi(t)\|_{\alpha,\beta}$ is locally bounded on ${\mathbb{R}}_{+}$ . The map $t\mapsto\|\Phi(t)\|_{1,1}$ is bounded on ${\mathbb{R}}_{+}$ in view of the conservation of the coercive energy $E$ , but higher order norms may not be bounded as $t$ goes to infinity (recall that Theorem 3.11 requires $\alpha,\beta\geq 2$ ).

4. Analysis of the iterative scheme: existence and uniform bounds

This section is devoted to the analysis of the system (1.4). For $n\in{\mathbb{N}}$ , we denote by $\Phi_{n}=(\phi_{n}^{x},\phi_{n}^{y})$ , the solution to the scheme (1.4) and we prove local in time uniform estimates. At this stage, we only need that $\phi_{0}^{x}\in\mathcal{H}_{x}^{\alpha}$ , $\phi_{0}^{y}\in\mathcal{H}_{y}^{\beta}$ for integers $\alpha,\beta\geq 1$ .

Lemma 4.1.

Let $\alpha,\beta\geq 1$ . Assume that (H1), (H2) and (H3)_α,β are satisfied. Assume $\phi_{0}^{x}\in\mathcal{H}_{x}^{\alpha}$ , and $\phi_{0}^{y}\in\mathcal{H}_{y}^{\beta}$ . Then, the sequence $(\Phi_{n})_{n\in{\mathbb{N}}}$ solution to (1.4) is well-defined and there exists $T>0$ such that for all $n\in{\mathbb{N}}$ , the solution $\Phi_{n}\in X_{T}^{\alpha,\beta}$ of the scheme (1.4) satisfies

(4.1)

\|\Phi_{n}\|_{X_{T}^{\alpha,\beta}}\leq 2\|\Phi_{0}\|_{\alpha,\beta}.

The proof of this lemma relies on the fact that the control of $\Phi_{n+1}$ involves terms which are linear in $\Phi_{n+1}$ and quadratic in $\Phi_{n}$ , and require the $X_{T}^{1,1}$ -norm of $\Phi_{n}$ . For this reason, the Lemma holds as soon as $\alpha,\beta\geq 1$ . However in Theorem 3.11 , we require at least an $X^{2,2}_{T}$ regularity. The reason will appear in Section 5, as we do need uniform (in $n$ ) estimates in $X^{2,2}_{T}$ to show that the sequence $(\Phi_{n})_{n}$ converges in $X^{0,0}_{T}=L^{\infty}_{T}L^{2}$ .

In Section 4.1, we address the construction of the family $(\Phi_{n})_{n\in{\mathbb{N}}}$ , which relies on a commutation lemma that we prove in Section 4.3. Section 4.2 is devoted to the proof of the uniform bound stated in Lemma 4.1.

4.1. Well-posedness of the scheme

Before entering into the proof of Lemma 4.1, let us discuss why the scheme is indeed well-defined: as $\Phi_{n+1}$ solves a decoupled system of linear Schrödinger equations, it suffices to study the properties of the time-dependent potentials ${\langle}w_{n}{\rangle}_{y}$ and ${\langle}w_{n}{\rangle}_{x}$ . We fix $T>0$ arbitrary and take $\phi_{0}^{x}\in\mathcal{H}_{x}^{\alpha}$ , and $\phi_{0}^{y}\in\mathcal{H}_{y}^{\beta}$ , with $\alpha,\beta\geq 1$ . For $n=0$ , $\Phi_{0}$ is obviously well-defined with $\Phi_{0}\in X_{T}^{\alpha,\beta}$ , and

(4.2)

\|\phi_{n}^{x}(t)\|_{L^{2}_{x}}=\|\phi_{n}^{y}(t)\|_{L^{2}_{y}}=1,\quad\forall t\in{\mathbb{R}},

holds for $n=0$ . We argue by induction. If $\Phi_{n}\in X_{T}^{1,1}$ satisfies (4.2), then in view of (H2), $\langle w_{n}\rangle_{y}(t,x)$ and $\langle w_{n}\rangle_{x}(t,y)$ are well-defined. In addition, for $t\in[0,T]$ , (H2) yields

(4.3)			$\displaystyle\|\langle w_{n}\rangle_{y}(t,x)\|\leq c_{0}V_{1}(x)\\|\phi^{y}_{n}(t)\\|_{L^{2}}^{2}+C\\|\phi_{n}^{y}(t)\\|^{2}_{\mathcal{H}_{y}^{1}},\quad\text{ a.e. }x,$
(4.3)			$\displaystyle\|\langle w_{n}\rangle_{x}(t,y)\|\leq c_{0}V_{2}(y)\\|\phi^{x}_{n}(t)\\|^{2}_{L^{2}}+C\\|\phi_{n}^{x}(t)\\|^{2}_{\mathcal{H}_{x}^{1}},\quad\text{a.e. }y,$

for some constant $C$ whose value is irrelevant here, unlike the fact that we assume $c_{0}<1$ . Indeed, together with (4.2), this implies that $\langle w_{n}\rangle_{y}$ is $H_{x}$ -bounded with relative bound at most $c_{0}$ . By Kato–Rellich Theorem (see e.g. [18, Theorem X.12]), $\Phi_{n+1}\in X_{T}^{0,0}$ is well-defined (see e.g. [19, Section VIII.4]), and (4.2) holds at level $n+1$ . Next, we prove that $\Phi_{n+1}\in X_{T}^{1,1}$ . Applying the operator $H_{x}$ to the first equation in (1.4), we find

(4.4)

(i{\partial}_{t}-H_{x})(H_{x}\phi_{n+1}^{x})=\langle w_{n}\rangle_{y}(t)(H_{x}\phi_{n+1}^{x})+[H_{x},\langle w_{n}\rangle_{y}(t)]\phi^{x}_{n+1}.

Since $H_{x}$ is self-adjoint, we deduce

	$\displaystyle\\|H^{1/2}_{x}\phi^{x}_{n+1}(t)\\|_{L^{2}_{x}}^{2}$	$\displaystyle={\rm Re}{\langle}H_{x}\phi^{x}_{n+1}(t),\phi^{x}_{n+1}(t){\rangle}_{L^{2}_{x}}$
		$\displaystyle=\\|H^{1/2}_{x}\phi^{x}_{0}\\|_{L^{2}_{x}}^{2}+{\rm Re}\left(\int_{0}^{t}\frac{d}{ds}{\langle}H_{x}\phi^{x}_{n+1}(s),\phi^{x}_{n+1}(s){\rangle}_{L^{2}_{x}}ds\right)$
		$\displaystyle=\\|H^{1/2}_{x}\phi^{x}_{0}\\|_{L^{2}_{x}}^{2}-{\rm Re}\left(\int_{0}^{t}{\langle}i[H_{x},\langle w_{n}\rangle_{y}]\phi^{x}_{n+1}(s),\phi^{x}_{n+1}(s){\rangle}_{L^{2}_{x}}ds\right).$

Minkowski inequality yields, in view of (H3)_1,1,

(4.5)

\left|{\langle}[H_{x},\langle w_{n}\rangle_{y}(t)]f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right|\lesssim\|\phi_{n}^{y}\|_{L^{\infty}_{T}\mathcal{H}^{1}_{y}}^{2}\|f_{1}\|_{\mathcal{H}_{x}^{1}}\|f_{2}\|_{\mathcal{H}_{x}^{1}}.

We infer the existence of a universal constant $C>0$ such that

\displaystyle\|H^{1/2}_{x}\phi^{x}_{n+1}(t)\|_{L^{2}_{x}}^{2}

\displaystyle\leq\|H^{1/2}_{x}\phi^{x}_{0}\|_{L^{2}_{x}}^{2}+C\|\Phi_{n}\|_{X^{1,1}_{T}}^{2}\int_{0}^{t}\|\phi^{x}_{n+1}(s)\|^{2}_{\mathcal{H}_{x}^{1}}ds.

We deduce

(4.6)

\sup_{t\in[0,T]}\|\phi^{x}_{n+1}(t)\|^{2}_{\mathcal{H}_{x}^{1}}\lesssim\|\phi^{x}_{0}\|^{2}_{\mathcal{H}_{x}^{1}}+C\|\Phi_{n}\|_{X^{1,1}_{T}}^{2}\int_{0}^{t}\|\phi^{x}_{n+1}(s)\|^{2}_{\mathcal{H}_{x}^{1}}ds,

We have a similar estimate for $\|H^{1/2}_{y}\phi^{y}_{n+1}(t)\|_{L^{2}_{x}}^{2}$ :

(4.7)

\sup_{t\in[0,T]}\|\phi^{y}_{n+1}(t)\|^{2}_{\mathcal{H}_{y}^{1}}\lesssim\|\phi^{y}_{0}\|^{2}_{\mathcal{H}_{y}^{1}}+C\|\Phi_{n}\|_{X^{1,1}_{T}}^{2}\int_{0}^{t}\|\phi^{y}_{n+1}(s)\|^{2}_{\mathcal{H}_{y}^{1}}ds,

and so Gronwall lemma and the inductive assumption yields $\Phi_{n+1}\in X_{T}^{1,1}$ and completes the construction of the sequence $(\Phi_{n})_{n\in{\mathbb{N}}}$ .

4.2. Uniform bounds

We conclude the proof of Lemma 4.1 in analyzing the regularity of the solutions.

Proof of Lemma 4.1.

In view of the definition of the scheme and of the conservations

\frac{d}{dt}\|\phi_{n}^{x}\|_{L^{2}}^{2}=\frac{d}{dt}\|\phi_{n}^{y}\|_{L^{2}}^{2}=0,

we need now consider $H_{x}^{\alpha}\phi^{x}_{n}$ and $H_{y}^{\beta}\phi^{y}_{n}$ for $\alpha,\beta\geq 1$ . Let

R=2\|\Phi_{0}\|_{\alpha,\beta},

and introduce

B_{R,T}=\{\Phi\in X_{T}^{\alpha,\beta},\ \|\Phi\|_{X_{T}^{\alpha,\beta}}\leq R\}.

We distinguish two cases for the ease of presentation.

First case: $\alpha=\beta=1$ . In that case, if $\Phi_{n}\in B_{R,T}$ , then estimates (4.6) and (4.7) imply

\|\Phi_{n+1}(t)\|_{X_{T}^{1,1}}^{2}\leq\|\Phi_{0}\|_{1,1}^{2}+CTR^{2}\|\Phi_{n+1}(t)\|_{X_{T}^{1,1}}^{2}.

We infer that choosing $T>0$ sufficiently small in terms of $R$ , but independently of $n$ , $\Phi_{n}\in B_{R,T}$ implies $\Phi_{n+1}\in B_{R,T}$ .

Higher regularity: The control of higher order regularity is obtained by a similar recursive argument which uses an iterated commutator estimate. Let $\alpha,\beta\geq 1$ . We have

(4.8)

\left\{\begin{aligned} i\partial_{t}H_{x}^{k}\phi^{x}_{n+1}=(H_{x}+\langle w_{n}\rangle_{y})H_{x}^{k}\phi^{x}_{n+1}+[H^{k}_{x},\langle w_{n}\rangle_{y}]\phi^{x}_{n+1},\quad H_{x}^{k}\phi^{x}_{n+1\mid t=0}=H_{x}^{k}\phi^{x}_{0},\\ i\partial_{t}H_{y}^{\ell}\phi^{y}_{n+1}=(H_{y}+\langle w_{n}\rangle_{x})H_{y}^{\beta}\phi^{y}_{n+1}+[H_{y}^{\ell},\langle w_{n}\rangle_{x}]\phi^{y}_{n+1},\quad H_{y}^{\ell}\phi^{y}_{n+1\mid t=0}=H_{y}^{\ell}\phi^{y}_{0}.\end{aligned}\right.

The next lemma allows to control the commutators.

Lemma 4.2.

Let $\Phi_{n}\in X^{1,1}_{T}$ for some $T>0$ , and $k,\ell\geq 1$ be integers. Suppose that (H3)_k,ℓ is satisfied. For all $t\in[0,T]$ ,

	$\displaystyle\left\|{\langle}[H_{x}^{k},\langle w_{n}\rangle_{y}(t)]f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right\|$	$\displaystyle\lesssim\\|\phi_{n}^{y}\\|^{2}_{L^{\infty}_{T}\mathcal{H}^{1}_{y}}\\|f_{1}\\|_{\mathcal{H}^{k}_{x}}\\|f_{2}\\|_{\mathcal{H}^{k}_{x}},\qquad\forall f_{1},f_{2}\in\mathcal{H}_{x}^{k}$
	$\displaystyle\left\|{\langle}[H_{y}^{\ell},\langle w_{n}\rangle_{x}(t)]g_{1},g_{2}{\rangle}_{L^{2}_{y}}\right\|$	$\displaystyle\lesssim\\|\phi_{n}^{x}\\|^{2}_{L^{\infty}_{T}\mathcal{H}^{1}_{x}}\\|g_{1}\\|_{\mathcal{H}^{\ell}_{y}}\\|g_{2}\\|_{\mathcal{H}^{\ell}_{y}},\qquad\forall g_{1},g_{2}\in\mathcal{H}_{y}^{\ell}.$

Taking the lemma for granted, (4.8) implies, since $H_{x}$ is self-adjoint,

	$\displaystyle\\|\phi_{n+1}^{x}(t)\\|_{\mathcal{H}^{\alpha}_{x}}^{2}$	$\displaystyle={\rm Re}{\langle}H_{x}^{\alpha}\phi^{x}_{n+1}(t),\phi^{x}_{n+1}(t){\rangle}_{L^{2}_{x}}$
		$\displaystyle=\\|H^{\alpha/2}_{x}\phi^{x}_{0}\\|_{L^{2}_{x}}^{2}+{\rm Re}\left(\int_{0}^{t}\frac{d}{ds}{\langle}H_{x}^{\alpha}\phi^{x}_{n+1}(s),\phi^{x}_{n+1}(s){\rangle}_{L^{2}_{x}}ds\right)$
		$\displaystyle=\\|H^{\alpha/2}_{x}\phi^{x}_{0}\\|_{L^{2}_{x}}^{2}-{\rm Re}\left(\int_{0}^{t}{\langle}i[H_{x}^{\alpha},\langle w_{n}\rangle_{y}]\phi^{x}_{n+1}(s),\phi^{x}_{n+1}(s){\rangle}_{L^{2}_{x}}ds\right)$
		$\displaystyle\leq\\|H_{x}^{\alpha/2}\phi_{0}^{x}\\|_{L^{2}_{x}}^{2}+CT\\|\phi^{y}_{n}\\|^{2}_{L^{\infty}_{T}\mathcal{H}^{1}_{y}}\sup_{t\in[0,T]}\\|\phi_{n+1}^{x}(t)\\|_{\mathcal{H}^{\alpha}_{x}}^{2}.$

We have a similar estimate for $\|H^{\beta/2}_{y}\phi^{y}_{n+1}(t)\|_{L^{2}_{x}}^{2}$ , and so if $\Phi_{n}\in B_{R,T}$ , then equations (4.6) and (4.7) imply

\|\Phi_{n+1}(t)\|_{X_{T}^{\alpha,\beta}}^{2}\leq\|\Phi_{0}\|_{\alpha,\beta}^{2}+CTR^{2}\|\Phi_{n+1}(t)\|_{X_{T}^{\alpha,\beta}}^{2}.

We infer that choosing $T>0$ sufficiently small in terms of $R$ , but independently of $n$ , $\Phi_{n}\in B_{R,T}$ implies $\Phi_{n+1}\in B_{R,T}$ . It thus remains to prove the lemma, which is the subject of the next subsection. ∎

Remark 4.3.

Lemma 4.1 holds as soon as $\alpha,\beta\geq 1$ , but this is not enough in order to conclude that the sequence $(\Phi_{n})_{n\in{\mathbb{N}}}$ converges to some solution of (1.2). Indeed, the mere boundedness in $X_{T}^{1,1}$ only implies the convergence of a subsequence in the weak-* topology: this is not enough to pass to the limit in (1.4), both because the subsequence need not retain consecutive indices, and because the topology considered is too large to pass to the limit in nonlinear terms. These issues are overcome by requiring $\alpha,\beta\geq 2$ in Sections 5 and 6.

4.3. Proof of Lemma 4.2

Of course, (4.5) implies the result when $k=1$ . Take $k\geq 1$ and assume that the result holds for all $m\leq k$ . We write

	$\displaystyle[H_{x}^{k+1},\langle w_{n}\rangle_{y}(t)]$	$\displaystyle=H_{x}^{k}[H_{x},\langle w_{n}\rangle_{y}(t)]+[H_{x}^{k},\langle w_{n}\rangle_{y}(t)]H_{x}$
		$\displaystyle=H_{x}[H^{k-1},\langle w_{n}\rangle_{y}(t)]H_{x}+H_{x}^{k}[H_{x},\langle w_{n}\rangle_{y}(t)]+[H_{x},\langle w_{n}\rangle_{y}(t)]H_{x}^{k}.$

We deduce from (4.5) and the recursive assumption that for $f_{1},f_{2}\in\mathcal{{\mathcal{S}}}({\mathbb{R}}^{d_{1}})$ , we have

	$\displaystyle\left\|{\langle}[H_{x}^{k+1},{\langle}w_{n}{\rangle}_{y}(t)]f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right\|$	$\displaystyle\leq\left\|{\langle}[H_{x}^{k-1},{\langle}w_{n}{\rangle}_{y}(t)]H_{x}f_{1},H_{x}f_{2}{\rangle}_{L^{2}_{x}}\right\|$
		$\displaystyle\quad+\left\|{\langle}H_{x}^{k}[H_{x},{\langle}w_{n}{\rangle}_{y}(t)]f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right\|+\left\|{\langle}[H_{x},{\langle}w_{n}{\rangle}_{y}(t)]H_{x}^{k}f,g{\rangle}_{L^{2}_{x}}\right\|.$

By the recursive assumption

	$\displaystyle\left\|{\langle}[H_{x}^{k-1},{\langle}w_{n}{\rangle}_{y}(t)]H_{x}f_{1},H_{x}f_{2}{\rangle}_{L^{2}_{x}}\right\|$	$\displaystyle\lesssim\\|\phi^{y}_{n}\\|_{L^{\infty}_{T}\mathcal{H}_{y}^{1}}^{2}\\|H_{x}f_{1}\\|_{\mathcal{H}^{k-1}_{x}}\\|H_{x}f_{2}\\|_{\mathcal{H}^{k-1}_{x}}$
		$\displaystyle\lesssim\\|\phi^{y}_{n}\\|_{L^{\infty}_{T}\mathcal{H}_{y}^{1}}^{2}\\|f_{1}\\|_{\mathcal{H}^{k+1}_{x}}\\|f_{2}\\|_{\mathcal{H}^{k+1}_{x}}.$

Finally, in view of (H3)_k,ℓ and Minkowski inequality, we have

	$\displaystyle\left\|{\langle}H_{x}^{k}[H_{x},{\langle}w_{n}{\rangle}_{y}(t)]f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right\|$	$\displaystyle\lesssim\\|\phi_{n}^{y}\\|_{L^{\infty}_{T}\mathcal{H}^{1}_{y}}^{2}\\|f_{1}\\|_{\mathcal{H}_{x}^{k+1}}\\|f_{2}\\|_{\mathcal{H}_{x}^{k+1}},$
	$\displaystyle\left\|{\langle}[H_{x},{\langle}w_{n}{\rangle}_{y}(t)]H_{x}^{k}f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right\|$	$\displaystyle\lesssim\\|\phi_{n}^{y}\\|_{L^{\infty}_{T}\mathcal{H}^{1}_{y}}^{2}\\|f_{1}\\|_{\mathcal{H}_{x}^{k+1}}\\|f_{2}\\|_{\mathcal{H}_{x}^{k+1}},$

which concludes the proof, after arguing similarly with $H_{y}$ .

5. Convergence in small norms

The second step of the proof of Theorem 3.11 consists in passing to the limit $n\rightarrow+\infty$ and prove the existence of a limit to the sequence $(\Phi_{n})_{n\in{\mathbb{N}}}$ of solutions to (1.4). The main result in this section is:

Lemma 5.1.

Assume that there exist $T>0$ and $R>0$ such that

\sup_{n\in{\mathbb{N}}}\|\Phi_{n}\|_{X_{T}^{2,2}}\leq R.

Then there exist $T_{1}\in]0,T]$ and $\Phi\in X_{T_{1}}^{2,2}$ such that

(5.1)

\sup_{0\leq t\leq T_{1}}\|\Phi_{n}(t)-\Phi(t)\|_{L^{2}_{x}\times L^{2}_{y}}=\|\Phi_{n}-\Phi\|_{X^{0,0}_{T_{1}}}\mathop{\longrightarrow}\limits_{n\rightarrow\infty}0.

If in addition $(\Phi_{n})_{n}$ is bounded in $X_{T}^{\alpha,\beta}$ for some integers $\alpha,\beta\geq 2$ , then $\Phi\in X_{T_{1}}^{\alpha,\beta}$ .

Proof.

Consider (1.4) at steps $n+1$ and $n$ , respectively, and subtract the corresponding equations. We find, for $n\geq 1$ ,

	$\displaystyle\left(i{\partial}_{t}-H_{x}\right)\left(\phi_{n+1}^{x}-\phi_{n}^{x}\right)$	$\displaystyle={\langle}w_{n}{\rangle}_{y}\phi_{n+1}^{x}-{\langle}w_{n-1}{\rangle}_{y}\phi_{n}^{x}$
		$\displaystyle={\langle}w_{n}{\rangle}_{y}\left(\phi_{n+1}^{x}-\phi_{n}^{x}\right)+\left({\langle}w_{n}{\rangle}_{y}-{\langle}w_{n-1}{\rangle}_{y}\right)\phi_{n}^{x},$

and energy estimates yield, for $T_{1}\in]0,T]$ , since $\Phi_{n+1\mid t=0}=\Phi_{n\mid t=0}$ ,

(5.2)

\|\phi_{n+1}^{x}-\phi_{n}^{x}\|_{L^{\infty}_{T_{1}}L^{2}_{x}}\leq\int_{0}^{T_{1}}\left\|\left({\langle}w_{n}(s){\rangle}_{y}-{\langle}w_{n-1}(s){\rangle}_{y}\right)\phi_{n}^{x}(s)\right\|_{L^{2}_{x}}ds.

In view of (H2), the key term is estimated by

\left|{\langle}w_{n}(t){\rangle}_{y}-{\langle}w_{n-1}(t){\rangle}_{y}\right|\lesssim\int_{{\mathbb{R}}^{d_{2}}}(V_{1}(x)+V_{2}(y)+1)\left||\phi_{n}^{y}(t,y)|^{2}-|\phi^{y}_{n-1}(t,y)|^{2}\right|dy.

Writing $|\phi_{n}^{y}|^{2}-|\phi^{y}_{n-1}|^{2}=\text{\rm Re}\left((\phi_{n}^{y}-\phi^{y}_{n-1})(\overline{\phi_{n}^{y}}+\overline{\phi^{y}_{n-1}})\right)$ , and using Cauchy-Schwarz inequality,

	$\displaystyle\left\|{\langle}w_{n}(t){\rangle}_{y}-{\langle}w_{n-1}(t){\rangle}_{y}\right\|$	$\displaystyle\lesssim(V_{1}(x)+1)\left(\\|\phi_{n}^{y}\\|_{L^{2}_{y}}+\\|\phi_{n-1}^{y}\\|_{L^{2}_{y}}\right)\\|\phi_{n}^{y}-\phi_{n-1}^{y}\\|_{L^{2}_{y}}$
		$\displaystyle\quad+\left(\\|V_{2}\phi_{n}^{y}\\|_{L^{2}_{y}}+\\|V_{2}\phi_{n-1}^{y}\\|_{L^{2}_{y}}\right)\\|\phi_{n}^{y}-\phi_{n-1}^{y}\\|_{L^{2}_{y}}$
(5.3)			$\displaystyle\lesssim\left(V_{1}(x)+1)\right)\sup_{k\in{\mathbb{N}}}\\|\Phi_{k}\\|_{X^{2,2}_{T}}\\|\phi_{n}^{y}-\phi_{n-1}^{y}\\|_{L^{2}_{y}}.$

Plugging this estimate into (5.2), we infer, thanks to Minkowski inequality,

	$\displaystyle\\|\phi_{n+1}^{x}-\phi_{n}^{x}\\|_{L^{\infty}_{T_{1}}L^{2}_{x}}$	$\displaystyle\lesssim\sup_{k\in{\mathbb{N}}}\\|\Phi_{k}\\|_{X^{2,2}_{T}}\int_{0}^{T_{1}}\left\\|\left(V_{1}+1)\right)\phi_{n}^{x}(s)\right\\|_{L^{2}_{x}}\\|\phi_{n}^{y}(s)-\phi_{n-1}^{y}(s)\\|_{L^{2}_{y}}ds$
		$\displaystyle\lesssim\sup_{k\in{\mathbb{N}}}\\|\Phi_{k}\\|_{X^{2,2}_{T}}^{2}\int_{0}^{T_{1}}\\|\phi_{n}^{y}(s)-\phi_{n-1}^{y}(s)\\|_{L^{2}_{y}}ds$
		$\displaystyle\lesssim R^{2}T_{1}\sup_{t\in[0,T_{1}]}\\|\phi_{n}^{y}(t)-\phi_{n-1}^{y}(t)\\|_{L^{2}_{y}}.$

We obtain a similar estimate by exchanging the roles of $x$ and $y$ , and so

(5.4)

\|\Phi_{n+1}-\Phi_{n}\|_{X^{0,0}_{T_{1}}}\lesssim R^{2}T_{1}\|\Phi_{n}-\Phi_{n-1}\|_{X^{0,0}_{T_{1}}}.

Fixing $T_{1}\in]0,T]$ sufficiently small, the series

\sum_{n\in{\mathbb{N}}}\|\Phi_{n+1}-\Phi_{n}\|_{X^{0,0}_{T_{1}}}

converges geometrically, and $\Phi_{n}$ converges in $X_{T_{1}}^{0,0}$ , to some $\Phi\in X_{T_{1}}^{0,0}$ .

On the other hand, the boundedness of $(\Phi_{n})_{n}$ in $X_{T}^{2,2}$ implies that a subsequence is converging in the weak-* topology of $X_{T}^{2,2}$ . By uniqueness of limits in the sense of distributions, we infer $\Phi\in X_{T_{1}}^{2,2}$ . The same holds when $X_{T}^{2,2}$ is replaced by $X_{T}^{\alpha,\beta}$ for $\alpha,\beta\geq 2$ . ∎

6. Passing to the limit in the equation

We now have all the elements in hands for proving Theorem 3.11 by showing that the limit function $\Phi$ constructed in Lemma 5.1 is a solution to equation (1.2) with the properties stated in Theorem 3.11.

6.1. Existence of a local solution

Combining Lemmas 4.1 and 5.1, we infer that under the assumptions of Theorem 3.11, there exists $T_{1}>0$ such that $\Phi_{n}\to\Phi$ in $X_{T_{1}}^{0,0}$ . By uniqueness of the limit, we also have $\Phi_{n}\rightharpoonup\Phi$ in $X_{T_{1}}^{\alpha,\beta}$ (and no extraction of a subsequence is needed). Resuming the estimates from the proof of Lemma 5.1, we observe that for $n,m\in{\mathbb{N}}$ , $t\in[0,T_{1}]$ and $x\in{\mathbb{R}}^{d_{1}}$ ,

	$\displaystyle\|{\langle}w_{n}(t){\rangle}_{y}-{\langle}w_{m}(t){\rangle}_{y}\|$	$\displaystyle=\left\|\int_{{\mathbb{R}}^{d_{2}}}w(x,y)\left(\|\phi_{n}^{y}(t,y)\|^{2}-\|\phi_{m}^{y}(t,y)\|^{2}\right)dy\right\|$
		$\displaystyle\lesssim\left(V_{1}(x)+1\right)\sup_{k\in{\mathbb{N}}}\\|\Phi_{k}\\|_{X^{2,2}_{T}}\\|\phi_{n}^{y}(t)-\phi_{m}^{y}(t)\\|_{L^{2}_{y}}.$

Passing to the limit $m\rightarrow+\infty$ , we obtain that for $n\in{\mathbb{N}}$ , $t\in[0,T_{1}]$ and $x\in{\mathbb{R}}^{d_{1}}$ ,

\displaystyle|{\langle}w_{n}(t){\rangle}_{y}-{\langle}w(t){\rangle}_{y}|

\displaystyle\lesssim\left(V_{1}(x)+1\right)\sup_{k\in{\mathbb{N}}}\|\Phi_{k}\|_{X^{2,2}_{T}}\|\phi_{n}^{y}(t)-\phi^{y}(t)\|_{L^{2}_{y}}.

Therefore, keeping the same notation $R$ as from Lemma 5.1,

	$\displaystyle\left\\|{\langle}w_{n}(t){\rangle}_{y}\phi_{n+1}^{x}(t)-{\langle}w(t){\rangle}_{y}\phi^{x}(t)\right\\|_{L^{2}_{x}}$	$\displaystyle\lesssim\left\\|\left({\langle}w_{n}(t){\rangle}_{y}-{\langle}w(t){\rangle}_{y}\right)\phi_{n+1}^{x}(t)\right\\|_{L^{2}_{x}}$
		$\displaystyle\quad+\left\\|{\langle}w(t){\rangle}_{y}\left(\phi_{n+1}^{x}(t)-\phi^{x}(t)\right)\right\\|_{L^{2}_{x}}$
		$\displaystyle\lesssim R\\|\phi_{n}^{y}(t)-\phi^{y}(t)\\|_{L^{2}_{y}}\left\\|(V_{1}+1)\phi_{n+1}^{x}(t)\right\\|_{L^{2}_{x}}$
		$\displaystyle\quad+\left\\|\left(V_{1}+1+\\|\phi^{y}\\|^{2}_{L^{\infty}_{T_{1}}\mathcal{H}^{1}_{y}}\right)\left(\phi_{n+1}^{x}(t)-\phi^{x}(t)\right)\right\\|_{L^{2}_{x}},$

where we have used (4.3) and the normalization (1.3). The first term on the right hand side goes to zero as $n\to\infty$ , uniformly in $t\in[0,T_{1}]$ . So does the last one in the case $\alpha,\beta\geq 3$ , since by interpolation $\Phi_{n}$ then converges to $\Phi$ strongly in $X^{2,2}_{T}$ . In the case where $\alpha$ or $\beta$ is equal to $2$ , we can only claim a weak convergence,

{\langle}w_{n}{\rangle}_{y}\phi_{n+1}^{x}\mathop{\rightharpoonup}\limits_{n\rightarrow\infty}{\langle}w{\rangle}_{y}\phi^{x}\quad\text{in }L^{\infty}([0,T_{1}];L^{2}_{x})\text{ weak-$*$}.

Similarly,

{\langle}w_{n}{\rangle}_{x}\phi_{n+1}^{y}\mathop{\rightharpoonup}\limits_{n\rightarrow\infty}{\langle}w{\rangle}_{x}\phi^{y}\quad\text{in }L^{\infty}([0,T_{1}];L^{2}_{y})\text{ weak-$*$},

and $\Phi$ solves (1.2) for $t\in[0,T_{1}]$ , in the sense of distributions. In view of the regularity $\Phi\in X_{T_{1}}^{\alpha,\beta}$ , Duhamel’s formula,

	$\displaystyle\phi^{x}(t)=e^{-itH_{x}}\phi_{0}^{x}-i\int_{0}^{t}e^{-i(t-s)H_{x}}\left({\langle}w{\rangle}_{y}\phi_{x}\right)(s)ds,$
	$\displaystyle\phi^{y}(t)=e^{-itH_{y}}\phi_{0}^{y}-i\int_{0}^{t}e^{-i(t-s)H_{y}}\left({\langle}w{\rangle}_{x}\phi_{y}\right)(s)ds,$

then shows the continuity in time $\Phi\in C([0,T_{1}];L^{2}_{x}\times L^{2}_{y})$ .

6.2. Uniqueness

At this stage, it is rather clear that uniqueness holds in $X^{2,2}_{T}$ , no matter how large $\alpha$ and $\beta$ are. Suppose that $\tilde{\Phi}\in X_{T}^{2,2}$ is another solution to (1.2) for $T>0$ : the system satisfied by $\Phi-\tilde{\Phi}$ is similar to the one satisfied by $\Phi_{n+1}-\Phi_{n}$ , and considered in the proof of Lemma 5.1. Since $\Phi,\tilde{\Phi}\in X_{T}^{2,2}$ , there exists $R>0$ such that

\|\Phi\|_{X_{T}^{2,2}}+\|\tilde{\Phi}\|_{X_{T}^{2,2}}\leq R,

and repeating the computations presented in the proof of Lemma 5.1, we obtain, for any $T_{1}\in]0,T]$ ,

\|\Phi-\tilde{\Phi}\|_{X_{T_{1}}^{0,0}}\leq CT_{1}R\|\Phi-\tilde{\Phi}\|_{X_{T_{1}}^{0,0}}.

Picking $T_{1}>0$ such that $CT_{1}R<1$ shows that $\Phi\equiv\tilde{\Phi}$ for $t\in[0,T_{1}]$ , and we infer that $\Phi\equiv\tilde{\Phi}$ on $[0,T]$ by covering $[0,T]$ by finitely many intervals of length at most $T_{1}$ .

6.3. Conservations

We now address the second point in Theorem 3.11: we assume that (1.2) has a unique solution $\Phi\in X^{2,2}_{T}$ for some $T>0$ . This implies in particular, in view of (1.2), that ${\partial}_{t}\phi^{x}\in L^{\infty}([0,T];L^{2}_{x})$ and ${\partial}_{t}\phi^{y}\in L^{\infty}([0,T];L^{2}_{y})$ , and the multiplier techniques evoked below are justified without using regularizing argument as in e.g. [10].

For the conservation of the $L^{2}$ -norms, multiply the first equation in (1.2) by $\overline{\phi^{x}}$ , integrate in space on ${\mathbb{R}}^{d_{1}}$ , and consider the imaginary part: we readily obtain

\frac{d}{dt}\|\phi^{x}(t)\|_{L^{2}_{x}}^{2}=0.

We proceed similarly for $\phi^{y}$ , and the conservation of the $L^{2}$ -norms follows.

For the energy, consider the multiplier ${\partial}_{t}\overline{\phi^{x}}$ in the equation for $\phi^{x}$ : as evoked above, all the products are well-defined, in the worst possible case as products of two $L^{2}$ functions. Integrate in space and consider the real part: we obtain

\frac{d}{dt}E(t)=0.

6.4. Globalization

In view of Lemmas 4.1 and 5.1, it suffices to prove a priori estimates on $\|\Phi\|_{X_{T}^{2,2}}$ , showing that this quantity is locally bounded in $T$ , to infer that $\Phi\in X_{T}^{2,2}$ for all $T>0$ , and then globalize the solution by the standard ODE alternative.

We use the conservation of the total energy, whose expression we develop:

	$\displaystyle E(t)$	$\displaystyle=\left(H_{x}\phi^{x}(t),\phi^{x}(t)\right)_{L^{2}_{x}}+\left(H_{y}\phi^{y}(t),\phi^{y}(t)\right)_{L^{2}_{y}}$
		$\displaystyle\quad+\iint_{{\mathbb{R}}^{d_{1}}\times{\mathbb{R}}^{d_{2}}}w(x,y)\|\phi^{x}(t,x)\|^{2}\|\phi^{y}(t,y)\|^{2}dxdy$
		$\displaystyle=\frac{1}{2}\\|\nabla_{x}\phi^{x}(t)\\|_{L^{2}({\mathbb{R}}^{d_{1}})}^{2}+\int_{{\mathbb{R}}^{d_{1}}}V_{1}(x)\|\phi^{x}(t,x)\|^{2}dx+\frac{1}{2}\\|\nabla_{y}\phi^{y}(t)\\|_{L^{2}({\mathbb{R}}^{d_{2}})}^{2}$
		$\displaystyle\quad+\int_{{\mathbb{R}}^{d_{2}}}V_{2}(y)\|\phi^{y}(t,y)\|^{2}dy+\iint_{{\mathbb{R}}^{d_{1}}\times{\mathbb{R}}^{d_{2}}}w(x,y)\|\phi^{x}(t,x)\|^{2}\|\phi^{y}(t,y)\|^{2}dxdy.$

Since $c_{0}<1$ in (H2), we infer

	$\displaystyle E(t)$	$\displaystyle\geq\frac{1}{2}\\|\nabla_{x}\phi^{x}(t)\\|_{L^{2}({\mathbb{R}}^{d_{1}})}^{2}+\frac{1}{2}\\|\nabla_{y}\phi^{y}(t)\\|_{L^{2}({\mathbb{R}}^{d_{2}})}^{2}+(1-c_{0})\int_{{\mathbb{R}}^{d_{1}}}V_{1}(x)\|\phi^{x}(t,x)\|^{2}dx$
		$\displaystyle\quad+(1-c_{0})\int_{{\mathbb{R}}^{d_{2}}}V_{2}(y)\|\phi^{y}(t,y)\|^{2}dy-c_{0}C\int_{{\mathbb{R}}^{d_{1}}}\|\phi^{x}(t,x)\|^{2}dx-c_{0}C\int_{{\mathbb{R}}^{d_{2}}}\|\phi^{y}(t,y)\|^{2}dy.$

The conservations established above yield

	$\displaystyle\frac{1}{2}\\|\nabla_{x}\phi^{x}(t)\\|_{L^{2}({\mathbb{R}}^{d_{1}})}^{2}$	$\displaystyle+\frac{1}{2}\\|\nabla_{y}\phi^{y}(t)\\|_{L^{2}({\mathbb{R}}^{d_{2}})}^{2}+(1-c_{0})\int_{{\mathbb{R}}^{d_{1}}}V_{1}(x)\|\phi^{x}(t,x)\|^{2}dx$
		$\displaystyle+(1-c_{0})\int_{{\mathbb{R}}^{d_{2}}}V_{2}(y)\|\phi^{y}(t,y)\|^{2}dy\leq E(0)+2c_{0}C.$

This is the coercivity property announced in the introduction, showing that there exists $M$ depending only on $\|\Phi_{0}\|_{1,1}$ such that

\|\Phi\|_{X_{T}^{1,1}}\leq M,

for any interval $[0,T]$ on which the solution is well-defined. Proceeding like in the proof of Lemma 4.1, we have

(6.1)

\sup_{t\in[0,T]}\|H_{x}\phi^{x}(t)\|_{L^{2}_{x}}^{2}\leq\|H_{x}\phi_{0}^{x}\|_{L^{2}_{x}}^{2}+2\int_{0}^{T}\left|\left([H_{x},\langle w\rangle_{y}(t)]\phi^{x}(t),H_{x}\phi^{x}(t)\right)_{L^{2}_{x}}\right|dt.

In view of Lemma 4.2 with $k=1$ , $f=\phi^{x}$ and $g=H_{x}\phi^{x}$ , we infer

	$\displaystyle\sup_{t\in[0,T]}\\|H_{x}\phi^{x}(t)\\|_{L^{2}_{x}}^{2}$	$\displaystyle\leq\\|H_{x}\phi_{0}^{x}\\|_{L^{2}_{x}}^{2}+C\\|\phi^{y}\\|^{2}_{L^{\infty}_{T}\mathcal{H}^{1}_{y}}\int_{0}^{T}\\|\phi^{x}(t)\\|_{\mathcal{H}^{2}_{x}}\\|H_{x}\phi^{x}(t)\\|_{L^{2}_{x}}dt$
		$\displaystyle\leq\\|H_{x}\phi_{0}^{x}\\|_{L^{2}_{x}}^{2}+CM^{2}\int_{0}^{T}\\|\phi^{x}(t)\\|_{\mathcal{H}^{2}_{x}}^{2}dt.$

The conservation of the $L^{2}$ -norm of $\phi^{x}$ implies

\sup_{t\in[0,T]}\|\phi^{x}(t)\|_{\mathcal{H}^{2}_{x}}^{2}\leq\|\phi^{x}\|_{\mathcal{H}^{2}_{x}}^{2}+CM^{2}\int_{0}^{T}\|\phi^{x}(t)\|_{\mathcal{H}^{2}_{x}}^{2}dt,

hence an exponential a priori control of the $\mathcal{H}^{2}_{x}$ -norm of $\phi^{x}(t)$ by Gronwall lemma. The same holds for $\phi^{y}(t)$ , hence the conclusion of Theorem 3.11.

7. Proof of Lemma 3.9

We briefly explain why (3.3) implies (H3)_2,2, thanks to an integration by parts, in view of (3.2). Typically, for $f_{1},f_{2}\in{\mathcal{S}}({\mathbb{R}}^{d_{1}})$ ,

{\langle}[w(x,y),H_{x}]f_{1},f_{2}{\rangle}_{L^{2}_{x}}=\frac{1}{2}{\langle}\Delta_{x}w(\cdot,y)f_{1},f_{2}{\rangle}_{L^{2}_{x}}+{\langle}\nabla_{x}w(\cdot,y)\cdot\nabla f_{1},f_{2}{\rangle}_{L^{2}_{x}}.

Therefore, for almost all $y\in{\mathbb{R}}^{d_{2}}$ , Cauchy-Schwarz inequality yields

	$\displaystyle\left\|{\langle}[w(x,y),H_{x}]f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right\|$	$\displaystyle\leq\frac{1}{2}\\|\|\Delta_{x}w(\cdot,y)\|^{1/2}f_{1}\\|_{L^{2}_{x}}\\|\|\Delta_{x}w(\cdot,y)\|^{1/2}f_{2}\\|_{L^{2}_{x}}$
		$\displaystyle\quad+\\|\nabla_{x}f_{1}\\|_{L^{2}_{x}}\\|\nabla_{x}w(\cdot,y)f_{2}\\|_{L^{2}_{x}}.$

Using (3.3),

	$\displaystyle\\|\|\Delta_{x}w(\cdot,y)\|^{1/2}f\\|^{2}_{L^{2}_{x}}$	$\displaystyle\lesssim\\|\sqrt{V_{1}}f\\|^{2}_{L^{2}_{x}}+(1+V_{2}(y))\\|f\\|^{2}_{L^{2}_{x}}\lesssim\\|f\\|^{2}_{\mathcal{H}^{1}_{x}}+(1+V_{2}(y))\\|f\\|^{2}_{L^{2}_{x}},$
	$\displaystyle\\|\nabla_{x}w(\cdot,y)f\\|_{L^{2}_{x}}$	$\displaystyle\lesssim\\|(\sqrt{V_{1}}+V_{2}(y)+1)f\\|_{L^{2}_{x}}\lesssim\\|f\\|_{\mathcal{H}^{1}_{x}}+(1+V_{2}(y))\\|f\\|_{L^{2}_{x}}.$

We deduce the expected relation for $k=\ell=1$ :

\left|{\langle}[w(x,y),H_{x}]f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right|\lesssim(1+V_{2}(y))\|f_{1}\|_{\mathcal{H}^{1}_{x}}\|f_{2}\|_{\mathcal{H}^{1}_{x}}.

For $k=2$ , write

	$\displaystyle\left\|{\langle}H_{x}[w(x,y),H_{x}]f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right\|$	$\displaystyle=\left\|{\langle}[w(x,y),H_{x}]f_{1},H_{x}f_{2}{\rangle}_{L^{2}_{x}}\right\|$
		$\displaystyle\leq\frac{1}{2}\left\|{\langle}\Delta_{x}w(\cdot,y)f_{1},H_{x}f_{2}{\rangle}_{L^{2}_{x}}\right\|+\left\|{\langle}\nabla_{x}w(\cdot,y)\cdot\nabla f_{1},H_{x}f_{2}{\rangle}_{L^{2}_{x}}\right\|$
		$\displaystyle\lesssim\\|\left(1+V_{1}+V_{2}(y)\right)f_{1}\\|_{L^{2}_{x}}\\|H_{x}f_{2}\\|_{L^{2}_{x}}$
		$\displaystyle\quad+\\|\nabla_{x}w(\cdot,y)\cdot\nabla f_{1}\\|_{L^{2}_{x}}\\|H_{x}f_{2}\\|_{L^{2}_{x}}$
		$\displaystyle\lesssim\\|f_{1}\\|_{\mathcal{H}^{2}_{x}}\\|f_{2}\\|_{\mathcal{H}^{2}_{x}}+V_{2}(y)\\|f_{1}\\|_{L^{2}_{x}}\\|f_{2}\\|_{\mathcal{H}^{2}_{x}}$
		$\displaystyle\quad+\\|\nabla_{x}w(\cdot,y)\cdot\nabla f_{1}\\|_{L^{2}_{x}}\\|f_{2}\\|_{\mathcal{H}^{2}_{x}},$

where we have used the estimate $\|H_{x}f\|_{L^{2}_{x}}\leq\|f\|_{\mathcal{H}_{x}^{2}}$ . For the last term, (3.3) yields

	$\displaystyle\\|\nabla_{x}w(\cdot,y)\cdot\nabla f_{1}\\|_{L^{2}_{x}}$	$\displaystyle\lesssim\left\\|\left(\sqrt{V_{1}}+V_{2}(y)+1\right)\nabla f_{1}\right\\|_{L^{2}_{x}}$
		$\displaystyle\lesssim\left\\|\sqrt{V_{1}}\nabla f_{1}\right\\|_{L^{2}_{x}}+\left(V_{2}(y)+1\right)\\|\nabla f_{1}\\|_{L^{2}_{x}}$
		$\displaystyle\lesssim\left\\|\sqrt{V_{1}}\nabla f_{1}\right\\|_{L^{2}_{x}}+\left(V_{2}(y)+1\right)\\|f\\|_{L^{2}_{x}}^{1/2}\\|\Delta f\\|_{L^{2}_{x}}^{1/2}$
		$\displaystyle\lesssim\left\\|\sqrt{V_{1}}\nabla f_{1}\right\\|_{L^{2}_{x}}+\left(V_{2}(y)+1\right)\\|f_{1}\\|_{\mathcal{H}^{2}_{x}}.$

For the first term on the last right hand side, we use an integration by parts:

	$\displaystyle\left\\|\sqrt{V_{1}}\nabla f_{1}\right\\|_{L^{2}_{x}}^{2}$	$\displaystyle=\int_{{\mathbb{R}}^{d_{1}}}V_{1}(x)\nabla f_{1}(x)\cdot\nabla f_{1}(x)dx$
		$\displaystyle=-\int_{{\mathbb{R}}^{d_{1}}}V_{1}(x)f_{1}(x)\Delta f_{1}(x)dx-\int_{{\mathbb{R}}^{d_{1}}}f_{1}(x)\nabla V_{1}(x)\cdot\nabla f_{1}(x)dx.$

By Cauchy-Schwarz inequality, the first term on the right hand side is estimated by

\|V_{1}f_{1}\|_{L^{2}_{x}}\|\Delta f_{1}\|_{L^{2}_{x}}\leq 2\|H_{x}f_{1}\|_{L^{2}_{x}}^{2}.

Invoking (3.2), and using Cauchy-Schwarz inequality again,

	$\displaystyle\left\|\int_{{\mathbb{R}}^{d_{1}}}f(x)\nabla V_{1}(x)\cdot\nabla f(x)dx\right\|$	$\displaystyle\lesssim\int_{{\mathbb{R}}^{d_{1}}}(1+V_{1}(x))\|f(x)\|\|\nabla f(x)\|dx\lesssim\\|(1+V_{1})f\\|_{L^{2}_{x}}\\|\nabla f\\|_{L^{2}_{x}}$
		$\displaystyle\lesssim\left(\\|f\\|_{L^{2}_{x}}+\\|H_{x}f\\|_{L^{2}_{x}}\right)\\|f\\|_{L^{2}_{x}}^{1/2}\\|\Delta f\\|_{L^{2}_{x}}^{1/2}$
		$\displaystyle\lesssim\\|H_{x}f\\|_{L^{2}_{x}}^{1/2}\\|f\\|_{L^{2}_{x}}^{3/2}+\\|H_{x}f\\|_{L^{2}_{x}}^{3/2}\\|f\\|_{L^{2}_{x}}^{1/2}\lesssim\\|f\\|_{L^{2}_{x}}^{2}+\\|H_{x}f\\|_{L^{2}_{x}}^{2},$

where we have used Young inequality for the last estimate.

\displaystyle\left|{\langle}[w(x,y),H_{x}]H_{x}f_{1},f_{2}{\rangle}\right|

\displaystyle=\left|{\langle}H_{x}f_{1},[w(x,y),H_{x}]f_{2}{\rangle}\right|\leq\|H_{x}f_{1}\|_{L^{2}_{x}}\|\Delta_{x}w(\cdot,y)f_{2}\|_{L^{2}_{x}}.

To estimate ${\langle}[w(\cdot,y),H_{x}]H_{x}f_{1},f_{2}{\rangle}_{L^{2}_{x}}$ , we use the self-adjointness of $H_{x}$ and write

{\langle}[w(\cdot,y),H_{x}]H_{x}f_{1},f_{2}{\rangle}_{L^{2}_{x}}={\langle}H_{x}f_{1},[w(\cdot,y),H_{x}]f_{2}{\rangle}_{L^{2}_{x}}.

We use the above estimate, where the roles of $f_{1}$ and $f_{2}$ have been swapped, to conclude that the first inequality in (H3)_2,2 holds. The proof of the second one is similar.

Appendix A Tangent space

For completeness, we give the elementary considerations for determining the tangent spaces of the Hartree manifold, Lemma 2.1.

Proof.

We consider a curve $\Gamma(s)=\varphi^{x}(s)\otimes\varphi^{y}(s)\in\mathcal{M}$ with $\Gamma(0)=u$ . Then,

\dot{\Gamma}(0)=\dot{\varphi}^{x}(0)\otimes\varphi^{y}+\varphi^{x}\otimes\dot{\varphi}^{y}(0),

which verifies the claimed representation of any tangent function as

v=v^{x}\otimes\varphi^{y}+\varphi^{x}\otimes v^{y}.

Let us consider $a=(a^{x},a^{y})\in{\mathbb{C}}^{2}$ with $a^{x}+a^{y}=0$ . We set $w^{x}=v^{x}+a^{x}\varphi^{x}$ and $w^{y}=v^{y}+a^{y}\varphi^{y}$ . Then, $w=w^{x}\otimes\varphi^{y}+\varphi^{x}\otimes w^{y}$ satisfies

w=v^{x}\otimes\varphi^{y}+\varphi^{x}\otimes v^{y}+(a^{x}+a^{y})\varphi^{x}\otimes\varphi^{y}=v.

Choosing $a^{x}=-\langle\varphi^{x},v^{x}\rangle/\langle\varphi^{x},\varphi^{x}\rangle$ and $a^{y}=-a^{x}$ , we obtain a representation of $v$ satisfying the claimed gauge condition. We verify that this condition implies uniqueness. We assume that $v=v^{x}\otimes\varphi^{y}+\varphi^{x}\otimes v^{y}=\tilde{v}^{x}\otimes\varphi^{y}+\varphi^{x}\otimes\tilde{v}^{y}$ with $\langle\varphi^{x},v^{x}\rangle=\langle\varphi^{x},\tilde{v}^{x}\rangle=0$ . Then, for any $\vartheta^{y}\in L^{2}_{y}$ ,

\langle\varphi^{x}\otimes\vartheta^{y},v\rangle=\langle\varphi^{x},\varphi^{x}\rangle\langle\vartheta^{y},v^{y}\rangle=\langle\varphi^{x},\varphi^{x}\rangle\langle\vartheta^{y},\tilde{v}^{y}\rangle,

which implies $v^{y}=\tilde{v}^{y}$ . Then, for any $\vartheta^{x}\in L^{2}_{y}$ ,

	$\displaystyle\langle\vartheta^{x}\otimes\varphi^{y},v\rangle_{L^{2}_{x,y}}$	$\displaystyle=\langle\vartheta^{x},v^{x}\rangle_{L^{2}_{x}}\langle\varphi^{y},\varphi^{y}\rangle_{L^{2}_{y}}+\langle\vartheta^{x},\varphi^{x}\rangle_{L^{2}_{x}}\langle\varphi^{y},v^{y}\rangle_{L^{2}_{y}}$
		$\displaystyle=\langle\vartheta^{x},\tilde{v}^{x}\rangle_{L^{2}_{x}}\langle\varphi^{y},\varphi^{y}\rangle_{L^{2}_{y}}+\langle\vartheta^{x},\varphi^{x}\rangle_{L^{2}_{x}}\langle\varphi^{y},\tilde{v}^{y}\rangle_{L^{2}_{y}},$

which implies $v^{x}=\tilde{v}^{x}$ . Choosing $v^{x}=0$ and $v^{y}=\varphi^{y}$ , we have $v=u$ so that $u\in\mathcal{T}_{u}\mathcal{M}$ . ∎

Appendix B Coulombic type coupling

We recall standard definition and results.

Definition B.1 (Admissible pairs in ${\mathbb{R}}^{3}$ ).

A pair $(q,r)$ is admissible if $q,r\geq 2$ , and

\frac{2}{q}=3\left(\frac{1}{2}-\frac{1}{r}\right).

As the range allowed for $(q,r)$ is compact, we set, for $I\subset{\mathbb{R}}$ a time interval,

\|u\|_{S(I)}=\sup_{(q,r)\text{ admissible}}\|u\|_{L^{q}(I;L^{r}({\mathbb{R}}^{3}))}.

In view of [11] and [13], we have:

Proposition B.2.

Let $d=3$ and $\mathbf{V}\in\mathcal{Q}$ . Denote $\mathbf{H}=-\frac{1}{2}\Delta+\mathbf{V}$ .
$(1)$ There exists $C_{\rm hom}$ such that for all interval $I$ such that $|I|\leq 1$ ,

\|e^{-it\mathbf{H}}\varphi\|_{S(I)}\leq C_{\rm hom}\|\varphi\|_{L^{2}},\quad\forall\varphi\in L^{2}({\mathbb{R}}^{3}).

$(2)$ Denote

D(F)(t,x)=\int_{0}^{t}e^{-i(t-\tau)\mathbf{H}}F(\tau,x)\mathrm{d}\tau.

There exists $C_{\rm inhom}$ such that for all interval $I\ni 0$ such that $|I|\leq 1$ ,

\left\lVert D(F)\right\rVert_{S(I)}\leq C_{\rm inhom}\left\lVert F\right\rVert_{S(I)^{*}}.

The existence of (local in time) Strichartz estimates of Proposition B.2 is the main ingredient of the proof of Theorem 3.1. Actually, as soon as such estimates are available for the operators $\mathbf{H}_{x}$ and $\mathbf{H}_{y}$ , then Theorem 3.1 remains valid. As mentioned in the introduction, such cases can be found in e.g. [20] or [5]. On the other hand, we emphasize that for superquadratic potentials, like $V_{1}$ in Example 3.5, Strichartz estimates suffer a loss of regularity; see [16, 21].

Remark B.3.

The case of a harmonic potential, $\mathbf{V}(x)=|x|^{2}$ , shows that $\mathbf{H}$ may have eigenvalues, and explains why the above time intervals $I$ are required to have finite length.

Remark B.4.

The potential $\mathbf{V}$ may also be time dependent, in view the original framework of [11]: $\mathbf{V}\in L^{\infty}_{\rm loc}({\mathbb{R}}_{t}\times{\mathbb{R}}_{x}^{3})$ is real-valued, and smooth with respect to the space variable: for (almost) all $t\in{\mathbb{R}}$ , $x\mapsto\mathbf{V}(t,x)$ is a $C^{\infty}$ map. Moreover, it is at most quadratic in space:

\forall T>0,\quad\forall\alpha\in{\mathbb{N}}^{d},\ |\alpha|\geq 2,\quad{\partial}_{x}^{\alpha}\mathbf{V}\in L^{\infty}([-T,T]\times{\mathbb{R}}_{x}^{3}).

Under these assumptions, suitable modifications of Proposition B.2 are needed, but they do not alter the conclusion of Theorem 3.1 (see [8]). See also [20] for another class of time dependent potentials.

Proof of Theorem 3.1.

We give the main technical steps of the proof, and refer to [10] for details. By Duhamel’s formula, we write (1.2) as

	$\displaystyle\phi^{x}(t)$	$\displaystyle=e^{-it\mathbf{H}_{x}}\phi_{0}^{x}-i\int_{0}^{t}e^{-i(t-\tau)\mathbf{H}_{x}}\left(v_{1}\phi^{x}+\left(W\ast\|\phi^{y}\|^{2}\right)\phi^{x}\right)(\tau)d\tau=:F_{1}(\phi^{x},\phi^{y}),$
	$\displaystyle\phi^{y}(t)$	$\displaystyle=e^{-it\mathbf{H}_{y}}\phi_{0}^{y}-i\int_{0}^{t}e^{-i(t-\tau)\mathbf{H}_{y}}\left(v_{2}\phi^{y}+\left(W\ast\|\phi^{x}\|^{2}\right)\phi^{y}\right)(\tau)d\tau=:F_{2}(\phi^{x},\phi^{y}).$

Theorem 3.1 follows from a standard fixed point argument based on Strichartz estimates. For $0<T\leq 1$ , we introduce

	$\displaystyle Y(T)$	$\displaystyle=\{(\phi^{x},\phi^{y})\in C([0,T];L^{2}({\mathbb{R}}^{3}))^{2}:\quad\\|\phi^{x}\\|_{S([0,T])}\leq 2C_{\rm hom}\\|\phi_{0}^{x}\\|_{L^{2}},$
		$\displaystyle\qquad\\|\phi^{y}\\|_{S([0,T])}\leq 2C_{\rm hom}\\|\phi_{0}^{y}\\|_{L^{2}}\},$

and the distance

d(\phi_{1},\phi_{2})=\|\phi_{1}-\phi_{2}\|_{S([0,T]},

where $C_{\rm hom}$ stems from Proposition B.2. Then $(Y(T),d)$ is a complete metric space.

By using Strichartz estimates and Hölder inequality, we have:

\displaystyle\|F_{1}(\phi^{x},\phi^{y})\|_{S([0,T])}

\displaystyle\leq C_{\rm hom}\|\phi_{0}^{x}\|_{L^{2}}+C_{\rm inhom}\left(\|v_{1}\phi^{x}\|_{S([0,T])^{*}}+\left\|\left(W\ast|\phi^{y}|^{2}\right)\phi^{x}\right\|_{S([0,T])^{*}}\right),

for any $(\phi^{x},\phi^{y})\in Y(T)$ . By assumption (see Theorem 3.1), we may write

v_{1}=v_{1}^{p}+v_{1}^{\infty},\quad v_{2}=v_{2}^{p}+v_{2}^{\infty},\quad W=W^{p}+W^{\infty},\quad v_{1}^{q},v_{2}^{q},W^{q}\in L^{q}({\mathbb{R}}^{3}),

and the value $p$ can obviously be the same for the three potentials, by taking the minimum between the three $p$ ’s if needed. Regarding $\|v_{1}\phi^{x}\|_{S([0,T])^{*}}$ , we write

	$\displaystyle\\|v_{1}^{\infty}\phi^{x}\\|_{S([0,T])^{*}}$	$\displaystyle\leq\\|v_{1}^{\infty}\phi^{x}\\|_{L^{1}([0,T];L^{2})}\leq\\|v_{1}^{\infty}\\|_{L^{\infty}}\\|\phi^{x}\\|_{L^{1}([0,T];L^{2})}$
		$\displaystyle\leq T\\|v_{1}^{\infty}\\|_{L^{\infty}}\\|\phi^{x}\\|_{L^{\infty}\leq([0,T];L^{2})}\leq T\\|v_{1}^{\infty}\\|_{L^{\infty}}\\|\phi^{x}\\|_{S([0,T])}.$

Let $r$ be such that

\frac{1}{r^{\prime}}=\frac{1}{r}+\frac{1}{p}\Longleftrightarrow 1=\frac{2}{r}+\frac{1}{p}.

Note that this exponent is the one introduced in the statement of Theorem 3.1. The assumption $p>3/2$ implies $2\leq r<6$ . Let $q$ be such that $(q,r)$ is admissible: $r<6$ implies $q>2$ . Hölder inequality yields

	$\displaystyle\\|v_{1}^{p}\phi^{x}\\|_{S([0,T])^{*}}$	$\displaystyle\leq\\|v_{1}^{p}\phi^{x}\\|_{L^{q^{\prime}}([0,T];L^{r^{\prime}})}\leq\\|v_{1}^{p}\\|_{L^{p}}\\|\phi^{x}\\|_{L^{q^{\prime}}([0,T];L^{r})}$
		$\displaystyle\leq T^{1/\theta}\\|v_{1}^{p}\\|_{L^{p}}\\|\phi^{x}\\|_{L^{q}([0,T];L^{r})}\leq T^{1/\theta}\\|v_{1}^{p}\\|_{L^{p}}\\|\phi^{x}\\|_{S([0,T])},$

where $\theta$ is such that

\frac{1}{q^{\prime}}=\frac{1}{q}+\frac{1}{\theta}.

Note that $\theta$ is finite, as $q>2$

For the convolution term, first write

	$\displaystyle\left\\|\left(W^{\infty}\ast\|\phi^{y}\|^{2}\right)\phi^{x}\right\\|_{S([0,T])^{*}}$	$\displaystyle\leq\left\\|\left(W^{\infty}\ast\|\phi^{y}\|^{2}\right)\phi^{x}\right\\|_{L^{1}([0,T];L^{2})}$
		$\displaystyle\leq\left\\|W^{\infty}\ast\|\phi^{y}\|^{2}\right\\|_{L^{1}([0,T];L^{\infty})}\\|\phi^{x}\\|_{L^{\infty}([0,T];L^{2})}$
		$\displaystyle\leq T\\|W^{\infty}\\|_{L^{\infty}}\\|\phi^{y}\\|_{L^{\infty}([0,T];L^{2})}^{2}\\|\phi^{x}\\|_{L^{\infty}([0,T];L^{2})}$
		$\displaystyle\leq T\\|W^{\infty}\\|_{L^{\infty}}\\|\phi^{y}\\|_{S([0,T])}^{2}\\|\phi^{x}\\|_{S([0,T])}.$

Introduce $r_{1}$ such that

(B.1)

\frac{1}{r_{1}^{\prime}}=\frac{1}{r_{1}}+\frac{1}{2p}\Longleftrightarrow 2=\frac{4}{r_{1}}+\frac{1}{p}\Longleftrightarrow 1+\frac{1}{2p}=\frac{1}{p}+\frac{2}{r_{1}}.

The assumption $p>3/2$ implies $2\leq r_{1}<3$ . Let $q_{1}$ be such that $(q_{1},r_{1})$ is admissible: $q_{1}>4$ . Hölder inequality yields

	$\displaystyle\left\\|\left(W^{p}\ast\|\phi^{y}\|^{2}\right)\phi^{x}\right\\|_{S([0,T])^{*}}$	$\displaystyle\leq\left\\|\left(W^{p}\ast\|\phi^{y}\|^{2}\right)\phi^{x}\right\\|_{L^{q_{1}^{\prime}}([0,T];L^{r_{1}^{\prime}})}$
		$\displaystyle\leq\left\\|W^{p}\ast\|\phi^{y}\|^{2}\right\\|_{L^{k}([0,T];L^{2p})}\\|\phi^{x}\\|_{L^{q_{1}}([0,T];L^{r_{1}})}$
		$\displaystyle\leq\left\\|W^{p}\ast\|\phi^{y}\|^{2}\right\\|_{L^{k}([0,T];L^{2p})}\\|\phi^{x}\\|_{S([0,T])},$

where $k$ is such that

\frac{1}{q_{1}^{\prime}}=\frac{1}{q_{1}}+\frac{1}{k}\Longleftrightarrow 1=\frac{2}{q_{1}}+\frac{1}{k}.

Note that since $q_{1}>4$ , we have $q_{1}>2k$ . In view of (B.1), Young inequality yields

	$\displaystyle\left\\|W^{p}\ast\|\phi^{y}\|^{2}\right\\|_{L^{k}([0,T];L^{2p})}$	$\displaystyle\leq\\|W^{p}\\|_{L^{p}}\left\\|\|\phi^{y}\|^{2}\right\\|_{L^{k}([0,T];L^{r_{1}/2})}=\\|W^{p}\\|_{L^{p}}\\|\phi^{y}\\|_{L^{2k}([0,T];L^{r_{1}})}^{2}$
		$\displaystyle\leq T^{\eta}\\|W^{p}\\|_{L^{p}}\\|\phi^{y}\\|_{L^{q_{1}}([0,T];L^{r_{1}})}^{2}\leq T^{\eta}\\|W^{p}\\|_{L^{p}}\\|\phi^{y}\\|_{S([0,T])}^{2},$

where $\eta>0$ is given by $\eta=1/(2k)-1/q_{1}$ .

The same inequalities obviously holds by switching $x$ and $y$ , and so for $T>0$ sufficiently small, $\Phi:=(\phi^{x},\phi^{y})\mapsto(F_{1}(\phi^{x},\phi^{y}),F_{2}(\phi^{x},\phi^{y}))=:\mathbf{F}(\Phi)$ leaves $Y(T)$ invariant.

Using similar estimates, again relying on Strichartz and Hölder inequalities involving the same Lebesgue exponents ( $\mathbf{F}$ is the sum of a linear and a trilinear term in $\Phi$ ), we infer that up to decreasing $T>0$ , $\mathbf{F}$ is a contraction on $Y(T)$ , and so there exists a unique $\Phi\in Y(T)$ solving (1.2). The global existence of the solution for (1.2) follows from the conservation of the $L^{2}$ -norms of $\phi^{x}$ and $\phi^{y}$ , respectively.

Uniqueness of such solutions follows once again from Strichartz and Hölder inequalities involving the same Lebesgue exponents as above, like for the contraction part of the argument. The main remark consists in noticing that the above Lebesgue indices satisfy $r>r_{1}$ , hence $q<q_{1}$ , and so $L_{\rm loc}^{q_{1}}L^{r_{1}}\subset L_{\rm loc}^{q}L^{r}\cap L^{\infty}L^{2}$ . ∎

Remark B.5 ( $H^{1}$ -regularity).

If in Theorem 3.1, we assume in addition that

\nabla v_{1},\nabla v_{2}\in L^{p}({\mathbb{R}}^{3})+L^{\infty}({\mathbb{R}}^{3})\quad\text{form some }p>3/2,

then for $\phi_{0}^{x},\phi_{0}^{y}\in H^{1}({\mathbb{R}}^{3})$ and $x\phi_{0}^{x},y\phi_{0}^{y}\in L^{2}({\mathbb{R}}^{3})$ (this last assumption may be removed when $\nabla V_{1},\nabla V_{2}\in L^{\infty}({\mathbb{R}}^{3})$ – the minimal assumption to work at the $H^{1}$ -level with $V_{1},V_{2}\in\mathcal{Q}$ is $\phi_{0}^{x}\nabla V_{1},\phi_{0}^{y}\nabla V_{2}\in L^{2}({\mathbb{R}}^{3})$ , see [9]), the global solution constructed in Theorem 3.1 satisfies

(\phi^{x},\phi^{y})\in C({\mathbb{R}};H^{1}({\mathbb{R}}^{3}))^{2}\cap L^{q}_{\rm loc}({\mathbb{R}};W^{1,r}({\mathbb{R}}^{3}))^{2},\quad(x\phi^{x},y\phi^{y})\in C({\mathbb{R}};L^{2}({\mathbb{R}}^{3})^{3})^{2}.

To see this, it suffices to resume the above proof, and check that $\nabla_{x}F_{1}(\phi^{x},\phi^{y})$ and $\nabla_{y}F_{2}(\phi^{x},\phi^{y})$ satisfy essentially the same estimates as $F_{1}(\phi^{x},\phi^{y}),F_{2}(\phi^{x},\phi^{y})$ in $S([0,T])$ . One first has to commute the gradient with $e^{-it\mathbf{H}_{x}}$ or $e^{-it\mathbf{H}_{y}}$ . Typically,

	$\displaystyle\nabla_{x}F_{1}(\phi^{x},\phi^{y})$	$\displaystyle=e^{-it\mathbf{H}_{x}}\nabla_{x}\phi_{0}^{x}-i\int_{0}^{t}e^{-i(t-\tau)\mathbf{H}_{x}}\nabla_{x}\left(v_{1}\phi^{x}+\left(W\ast\|\phi^{y}\|^{2}\right)\phi^{x}\right)(\tau)d\tau$
		$\displaystyle\quad-i\int_{0}^{t}e^{-i(t-\tau)\mathbf{H}_{x}}F_{1}(\phi^{x},\phi^{y})(\tau)\nabla_{x}V_{1}d\tau,$

where the last factor accounts for the possible lack of commutation between $\mathbf{H}_{x}$ and $\nabla_{x}$ , $[-i{\partial}_{t}-\mathbf{H}_{x},\nabla_{x}]=\nabla_{x}V_{1}$ . Since $V_{1}$ is at most quadratic, $\nabla V_{1}$ is at most linear, and we obtain a closed system of estimates by considering

	$\displaystyle xF_{1}(\phi^{x},\phi^{y})$	$\displaystyle=e^{-it\mathbf{H}_{x}}(x\phi_{0}^{x})-i\int_{0}^{t}e^{-i(t-\tau)\mathbf{H}_{x}}\left(x\left(v_{1}\phi^{x}+\left(W\ast\|\phi^{y}\|^{2}\right)\phi^{x}\right)\right)(\tau)d\tau$
		$\displaystyle\quad+i\int_{0}^{t}e^{-i(t-\tau)\mathbf{H}_{x}}\nabla_{x}F_{1}(\phi^{x},\phi^{y})(\tau)d\tau,$

where we have used $[-i{\partial}_{t}-\mathbf{H}_{x},x]=-\nabla_{x}$ . We omit the details, and refer to [10] (see also [8]). As pointed in Remark 3.3, the energy

	$\displaystyle E(t)$	$\displaystyle=\left(H_{x}\phi^{x}(t),\phi^{x}(t)\right)_{L^{2}_{x}}+\left(H_{y}\phi^{y}(t),\phi^{y}(t)\right)_{L^{2}_{y}}$
		$\displaystyle\quad+\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}W(x-y)\|\phi^{x}(t,x)\|^{2}\|\phi^{y}(t,y)\|^{2}dxdy,$

which is well defined with the above regularity, is independent of time. Formally, this can be seen by multiplying the first equation in (1.2) by ${\partial}_{t}\phi^{x}$ , the second by ${\partial}_{t}\phi^{y}$ , integrating in space, considering the real part, and summing the two identities. To make the argument rigorous (we may not have enough regularity to be allowed to proceed as described), one may use a regularization procedure as in [10], or rely on a clever use of the regularity provided by Strichartz estimates, as in [17].

References

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[5] N. Burq, F. Planchon, J. G. Stalker, and A. S. Tahvildar-Zadeh. Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Indiana Univ. Math. J., 53(6):1665–1680, 2004.
[6] F. Cacciafesta, A.-S. de Suzzoni, and D. Noja. A Dirac field interacting with point nuclear dynamics. Math. Ann., 376(3-4):1261–1301, 2020.
[7] E. Cancès and C. Le Bris. On the time-dependent Hartree-Fock equations coupled with a classical nuclear dynamics. Math. Models Methods Appl. Sci., 9(7):963–990, 1999.
[8] R. Carles. Nonlinear Schrödinger equation with time dependent potential. Commun. Math. Sci., 9(4):937–964, 2011.
[9] R. Carles. Sharp weights in the Cauchy problem for nonlinear Schrödinger equations with potential. Z. Angew. Math. Phys., 66(4):2087–2094, 2015.
[10] T. Cazenave. Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics. New York University Courant Institute of Mathematical Sciences, New York, 2003.
[11] D. Fujiwara. Remarks on the convergence of the Feynman path integrals. Duke Math. J., 47(3):559–600, 1980.
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	$\displaystyle\left\|{\langle}H_{x}^{k-1}[w(\cdot,y),H_{x}]f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right\|$	$\displaystyle+\left\|{\langle}[w(\cdot,y),H_{x}]H_{x}^{k-1}f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right\|$
		$\displaystyle\leq c_{1}\left(1+V_{2}(y)\right)\\|f_{1}\\|_{\mathcal{H}_{x}^{k}}\\|f_{2}\\|_{\mathcal{H}_{x}^{k}},\quad\text{for a.a. }y\in{\mathbb{R}}^{d_{2}},$
	$\displaystyle\left\|{\langle}H_{y}^{\ell-1}[w(x,\cdot),H_{y}]g_{1},g_{2}{\rangle}_{L^{2}_{y}}\right\|$	$\displaystyle+\left\|{\langle}[w(x,\cdot),H_{y}]H_{y}^{\ell-1}g_{1},g_{2}{\rangle}_{L^{2}_{y}}\right\|$
		$\displaystyle\leq c_{2}\left(1+V_{1}(x)\right)\\|g_{1}\\|_{\mathcal{H}_{y}^{\ell}}\\|g_{2}\\|_{\mathcal{H}_{y}^{\ell}},\quad\text{for a.a. }x\in{\mathbb{R}}^{d_{1}}.$

	$\displaystyle\left\|{\langle}[H_{x}^{k},\langle w_{n}\rangle_{y}(t)]f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right\|$	$\displaystyle\lesssim\\|\phi_{n}^{y}\\|^{2}_{L^{\infty}_{T}\mathcal{H}^{1}_{y}}\\|f_{1}\\|_{\mathcal{H}^{k}_{x}}\\|f_{2}\\|_{\mathcal{H}^{k}_{x}},\qquad\forall f_{1},f_{2}\in\mathcal{H}_{x}^{k}$
	$\displaystyle\left\|{\langle}[H_{y}^{\ell},\langle w_{n}\rangle_{x}(t)]g_{1},g_{2}{\rangle}_{L^{2}_{y}}\right\|$	$\displaystyle\lesssim\\|\phi_{n}^{x}\\|^{2}_{L^{\infty}_{T}\mathcal{H}^{1}_{x}}\\|g_{1}\\|_{\mathcal{H}^{\ell}_{y}}\\|g_{2}\\|_{\mathcal{H}^{\ell}_{y}},\qquad\forall g_{1},g_{2}\in\mathcal{H}_{y}^{\ell}.$

	$\displaystyle\\|\phi_{n+1}^{x}(t)\\|_{\mathcal{H}^{\alpha}_{x}}^{2}$	$\displaystyle={\rm Re}{\langle}H_{x}^{\alpha}\phi^{x}_{n+1}(t),\phi^{x}_{n+1}(t){\rangle}_{L^{2}_{x}}$
		$\displaystyle=\\|H^{\alpha/2}_{x}\phi^{x}_{0}\\|_{L^{2}_{x}}^{2}+{\rm Re}\left(\int_{0}^{t}\frac{d}{ds}{\langle}H_{x}^{\alpha}\phi^{x}_{n+1}(s),\phi^{x}_{n+1}(s){\rangle}_{L^{2}_{x}}ds\right)$
		$\displaystyle=\\|H^{\alpha/2}_{x}\phi^{x}_{0}\\|_{L^{2}_{x}}^{2}-{\rm Re}\left(\int_{0}^{t}{\langle}i[H_{x}^{\alpha},\langle w_{n}\rangle_{y}]\phi^{x}_{n+1}(s),\phi^{x}_{n+1}(s){\rangle}_{L^{2}_{x}}ds\right)$
		$\displaystyle\leq\\|H_{x}^{\alpha/2}\phi_{0}^{x}\\|_{L^{2}_{x}}^{2}+CT\\|\phi^{y}_{n}\\|^{2}_{L^{\infty}_{T}\mathcal{H}^{1}_{y}}\sup_{t\in[0,T]}\\|\phi_{n+1}^{x}(t)\\|_{\mathcal{H}^{\alpha}_{x}}^{2}.$

	$\displaystyle\left\|{\langle}H_{x}^{k}[H_{x},{\langle}w_{n}{\rangle}_{y}(t)]f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right\|$	$\displaystyle\lesssim\\|\phi_{n}^{y}\\|_{L^{\infty}_{T}\mathcal{H}^{1}_{y}}^{2}\\|f_{1}\\|_{\mathcal{H}_{x}^{k+1}}\\|f_{2}\\|_{\mathcal{H}_{x}^{k+1}},$
	$\displaystyle\left\|{\langle}[H_{x},{\langle}w_{n}{\rangle}_{y}(t)]H_{x}^{k}f_{1},f_{2}{\rangle}_{L^{2}_{x}}\right\|$	$\displaystyle\lesssim\\|\phi_{n}^{y}\\|_{L^{\infty}_{T}\mathcal{H}^{1}_{y}}^{2}\\|f_{1}\\|_{\mathcal{H}_{x}^{k+1}}\\|f_{2}\\|_{\mathcal{H}_{x}^{k+1}},$

	$\displaystyle\left\|{\langle}w_{n}(t){\rangle}_{y}-{\langle}w_{n-1}(t){\rangle}_{y}\right\|$	$\displaystyle\lesssim(V_{1}(x)+1)\left(\\|\phi_{n}^{y}\\|_{L^{2}_{y}}+\\|\phi_{n-1}^{y}\\|_{L^{2}_{y}}\right)\\|\phi_{n}^{y}-\phi_{n-1}^{y}\\|_{L^{2}_{y}}$
		$\displaystyle\quad+\left(\\|V_{2}\phi_{n}^{y}\\|_{L^{2}_{y}}+\\|V_{2}\phi_{n-1}^{y}\\|_{L^{2}_{y}}\right)\\|\phi_{n}^{y}-\phi_{n-1}^{y}\\|_{L^{2}_{y}}$
(5.3)			$\displaystyle\lesssim\left(V_{1}(x)+1)\right)\sup_{k\in{\mathbb{N}}}\\|\Phi_{k}\\|_{X^{2,2}_{T}}\\|\phi_{n}^{y}-\phi_{n-1}^{y}\\|_{L^{2}_{y}}.$

On the Cauchy problem for the Hartree approximation in quantum dynamics

Abstract.

1. Introduction

1.1. Outline

1.2. Notations

2. Variational principle

Lemma 2.1 (Tangent space).

3. Main result

3.1. Coupling potentials of Coulombic form

Theorem 3.1.

Remark 3.2.

Remark 3.3.

Remark 3.4.

3.2. Coupling potentials with polynomial growth

Example 3.5.

3.2.1. Restriction to non-negative potentials

3.2.2. Functional setting

3.2.3. Main assumptions

Remark 3.6.

Remark 3.7.

Remark 3.8.

Lemma 3.9 (Sufficient conditions).

Remark 3.10.

3.3. Main result and comments

Theorem 3.11.

4. Analysis of the iterative scheme: existence and uniform bounds

Lemma 4.1.

4.1. Well-posedness of the scheme

4.2. Uniform bounds

Proof of Lemma 4.1.

Lemma 4.2.

Remark 4.3.

4.3. Proof of Lemma 4.2

5. Convergence in small norms

Lemma 5.1.

Proof.

6. Passing to the limit in the equation

6.1. Existence of a local solution

6.2. Uniqueness

6.3. Conservations

6.4. Globalization

7. Proof of Lemma 3.9

Appendix A Tangent space

Proof.

Appendix B Coulombic type coupling

Definition B.1 (Admissible pairs in ℝ3{\mathbb{R}}^{3}).

Proposition B.2.

Remark B.3.

Remark B.4.

Proof of Theorem 3.1.

Remark B.5 (H1H^{1}-regularity).

References

Definition B.1 (Admissible pairs in ${\mathbb{R}}^{3}$ ).

Remark B.5 ( $H^{1}$ -regularity).