The Hartree-Fock equations in modulation spaces

TL;DR

This paper develops local and global well-posedness theories for nonlinear Hartree-Fock equations within modulation spaces, including cases with harmonic potentials, and proves boundedness of multilinear operators on these spaces.

Contribution

It introduces well-posedness results for Hartree-Fock equations in modulation spaces and extends these results to equations with harmonic potentials, along with proving multilinear operator boundedness.

Findings

Established local and global well-posedness in modulation spaces.

Proved well-posedness for equations with harmonic potentials.

Demonstrated boundedness of multilinear operators on modulation spaces.

Abstract

We establish both a local and a global well-posedness theories for the nonlinear Hartree-Fock equations and its reduced analog in the setting of the modulation spaces on $\mathbb R^d$. In addition, we prove similar results when a harmonic potential is added to the equations. In the process, we prove the boundedeness of certain multilinear operators on products of the modulation spaces which may be of independent interest.

Equations

$$\varphi (h(D))f={\mathcal{F}}^{-1}{e}^{it\varphi \circ h(\cdot )}\mathcal{F}f$$

$$\{\begin{array}{l}i{\partial }_t{\psi }_k=\varphi (h(D)){\psi }_k-{\sum }_{l=1}^N\left(\frac{\kappa }{|x{|}^{\gamma }}\ast |{\psi }_l{|}^2\right){\psi }_k+{\sum }_{l=1}^N{\psi }_l\left(\frac{\kappa }{|x{|}^{\gamma }}\ast \{\overline{{\psi }_l}{\psi }_k\}\right),\\ {\psi }_{k|t=0}={\psi }_{0k},\end{array}$$

$$H=\sum _{l=1}^N\left(\frac{\kappa }{|x{|}^{\gamma }}\ast |{\psi }_l{|}^2\right)$$

$$F({\psi }_k)=\sum _{l=1}^N{\psi }_l\left(\frac{\kappa }{|x{|}^{\gamma }}\ast \{\overline{{\psi }_l}{\psi }_k\}\right)$$

$$\{\begin{array}{l}i{\partial }_t{\psi }_k=\varphi (h(D)){\psi }_k-{\sum }_{l=1}^N\left(\frac{\kappa }{|x{|}^{\gamma }}\ast |{\psi }_l{|}^2\right){\psi }_k,\\ {\psi }_{k|t=0}={\psi }_{0k}.\end{array}$$

$$i{K}^{\prime }(t)=\left[G(t),K(t)\right]$$

$$K^{\ast }(t)=K(t),K(t)\le 1,\text{tr}K=N$$

$$X=\{\begin{array}{l}{M}^{p,q}({\mathbb{R}}^d)\text{if}1\le p\le 2,1\le q\le \frac{2d}{d+\gamma }\\ M_s^{p,1}({\mathbb{R}}^d)\text{if}1<p<\infty ,\frac{1}{p}+\frac{\gamma }{d}-1=\frac{1}{p+ϵ}\end{array}$$

$$\{\begin{array}{l}\left|{\varphi }^{(\mu )}(r)\right|\lesssim r^{m_1-\mu }\text{if}r\ge 1\\ \left|{\varphi }^{(\mu )}(r)\right|\lesssim r^{m_2-\mu }\text{if}0<r<1\end{array}$$

$$({\psi }_1,...,{\psi }_N)\in {\left(C([0,T],X)\right)}^N.$$

$$X=\{\begin{array}{l}{M}^{p,q}({\mathbb{R}}^d)\text{if}1\le p\le 2,1\le q\le \frac{2d}{d+\gamma }\\ M_s^{p,1}({\mathbb{R}}^d)\text{if}1<p<\infty ,\frac{1}{p}+\frac{\gamma }{d}-1=\frac{1}{p+ϵ}\end{array}$$

$$X=\{\begin{array}{l}{M}_{rad}^{p,q}({\mathbb{R}}^d)\text{if}1\le p\le 2,1\le q\le \frac{2d}{d+\gamma }\\ M_s^{p,1}({\mathbb{R}}^d)\cap L_{rad}^2({\mathbb{R}}^d)\text{if}2<p<\infty ,\frac{1}{p}+\frac{\gamma }{d}-1=\frac{1}{p+ϵ}\end{array}$$

$$({\psi }_1,...,{\psi }_N)\in {\left(C(\mathbb{R},X)\cap L_{loc}^{4\alpha /\gamma }(\mathbb{R},L^{4d/(2d-\gamma )}({\mathbb{R}}^d))\right)}^N.$$

$$\{\begin{array}{l}i{\partial }_t{\psi }_k-\left(-\Delta +|x{|}^2\right){\psi }_k={\sum }_{l=1}^N\left(\frac{\kappa }{|x{|}^{\gamma }}\ast |{\psi }_l{|}^2\right){\psi }_k+{\sum }_{l=1}^N{\psi }_l\left(\frac{\kappa }{|x{|}^{\gamma }}\ast \{\overline{{\psi }_l}{\psi }_k\}\right),\\ {\psi }_{k|t=0}={\psi }_{0k}\end{array}$$

$$\{\begin{array}{l}i{\partial }_t{\psi }_k-\left(-\Delta +|x{|}^2\right){\psi }_k={\sum }_{l=1}^N\left(\frac{\kappa }{|x{|}^{\gamma }}\ast |{\psi }_l{|}^2\right){\psi }_k,\\ {\psi }_{k|t=0}={\psi }_{0k},\end{array}$$

$$({\psi }_1,...,{\psi }_N)\in {\left(C([0,\infty ),M^{p,p}({\mathbb{R}}^d))\cap L_{loc}^{8/\gamma }([0,\infty ),L^{4d/(2d-\gamma )}({\mathbb{R}}^d))\right)}^N.$$

$$|\beta |=\sum _1^d{\beta }_j,\beta !=\prod _1^d{\beta }_j!,{\partial }^{\beta }=D^{\beta }={\left(\frac{\partial }{\partial x_1}\right)}^{{\beta }_1}\cdots {\left(\frac{\partial }{\partial x_d}\right)}^{{\beta }_d},$$

$$x^{\beta }=\prod _1^d{x}_j^{{\beta }_j}.$$

$$\Vert f{\Vert }_{L^p}={\left({\int }_{{\mathbb{R}}^d}|f(x){|}^{p}dx\right)}^{1/p}(1\le p<\infty ),$$

$$\Vert u{\Vert }_{L_t^r(I,X)}={\left({\int }_I\Vert u{\Vert }_X^{r}dt\right)}^{1/r},$$

$$\mathcal{F}f(w)=\hat{f}(w)={\int }_{{\mathbb{R}}^d}f(t)e^{-2\pi it\cdot w}dt,w\in {\mathbb{R}}^d.$$

$$\mathcal{F}{L}^p({\mathbb{R}}^d)=\left\{f\in {\mathcal{S}}^{\prime }({\mathbb{R}}^d):\Vert f{\Vert }_{\mathcal{F}{L}^p}:=\Vert \hat{f}{\Vert }_{L^p}<\infty \right\}.$$

$$L_s^p({\mathbb{R}}^d)=\left\{f\in {\mathcal{S}}^{\prime }({\mathbb{R}}^d):\Vert f{\Vert }_{L_s^p}:={\Vert {\mathcal{F}}^{-1}[⟨\cdot {⟩}^s\mathcal{F}(f)]\Vert }_{L^p}<\infty \right\},$$

$$V_{g}f(x,w)={\int }_{{\mathbb{R}}^d}f(t)\overline{g(t-x)}{e}^{-2\pi iw\cdot t}dt,(x,w)\in {\mathbb{R}}^{2d}$$

$$V_{g}f(x,w)=⟨f,M_w{T}_{x}g⟩$$

$$\Vert f{\Vert }_{M_s^{p,q}}={\left({\int }_{{\mathbb{R}}^d}{\left({\int }_{{\mathbb{R}}^d}|V_{g}f(x,w){|}^{p}dx\right)}^{q/p}(1+|w{|}^2{)}^{sq/2}dw\right)}^{1/q},$$

$${\rho }_k(\xi )=\rho (\xi -k)(k\in {\mathbb{Z}}^d).$$

Full text

1. 1 Introduction and Description of the Problem

2. 1.1 Motivation

3. 1.2 Hartree-Fock equations

4. 1.3 Hartree-Fock equation with harmonic potential

5. 2 Preliminaries

6. 2.1 Notations

7. 2.2 Modulation spaces

8. 2.3 Modulation space estimates for unimodular Fourier multipliers

9. 3 Trilinear $M^{p,q}$ estimates

10. 4 Proofs of main results

11. 4.1 Local well-posedness for Hartree-Fock equations

12. 4.2 Global Well-posedness for Hartree-Fock Equations

13. 5 Well-posedness for Hartree-Fock equations with harmonic potential

14. 5.1 Schrödinger propagator associated to harmonic oscillator

15. 5.2 Proof of Theorem 1.3

1. Introduction and Description of the Problem

1.1. Motivation

The Hartree equation, introduced by Hartree in the 1920s, arises as the mean-field limit of large systems of identical bosons, e.g., the Gross-Pitaevskii equation for Bose-Einstein condensates [25, 35], when taking into account the self-interactions of the bosons. A semirelativistic version of the Hartree equation was considered in [20, 31] for modeling boson stars. The Hartree-Fock equation, also developed by Fock [22] describes large systems of identical fermions by taking into account the self-interactions of charged fermions as well as an exchange term resulting from Pauli’s principle. A semirelativistic version of the Hartree-Fock equation was developed in [23] for modeling white dwarfs. The Hartree equation is also used for fermions as an approximation of the Hartree-Fock equation neglecting the impact of their fermionic nature. Hartree and Hartree-Fock equations are used for several applications in many-particle physics [34, Section 2.2].

In [10, 30] fractional Laplacians have been applied to model physical phenomena. It was formulated by Laskin [30] as a result of extending the Feynman path integral from the Brownian-like to Lévy-like quantum mechanical paths. The harmonic oscillator (Hermite operator) $-\Delta+|x|^{2}$ is a fundamental operator in quantum physics and in analysis [38]. Hartree-Fock equations with harmonic potential model Bose-Einstein condensates with attractive inter-particle interactions under a magnetic trap . The isotropic harmonic potential $|x|^{2}$ describes a magnetic field whose role is to confine the movement of particles. A class of nonlinear Schrödinger equations with a “nonlocal” nonlinearity that we call “Hartree type” also occurs in the modeling of quantum semiconductor devices (see [11] and the references therein).

1.2. Hartree-Fock equations

Before giving the exact form of the Hatree-Fock equations, we set some notations that will be used through the paper. For two functions $\phi$ and $h$ defined on $\mathbb{R}$ and $\mathbb{R}^{d}$ respectively, we set

[TABLE]

where $\mathcal{F}$ denotes the Fourier transform.

The Hartree-Fock equation of $N$ particles is given by

[TABLE]

where $t\in\mathbb{R},\psi_{k}:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{C},$ $k=1,2,...,N,0<\gamma<d,$ $\kappa$ is constant, and $\ast$ denotes the convolution in $\mathbb{R}^{d}.$

The Hartree factor

[TABLE]

describes the self-interaction between charged particles as a repulsive force if $\kappa>0$, and an attractive force if $\kappa<0.$ The last term on the right side of (1.1) is the so-called “exchange term (Fock term)”

[TABLE]

which is a consequence of the Pauli principle and thus applies to fermions. In the mean-field limit $(N\to\infty)$, this term is negligible compared to the Hatree factor. In this case, (1.1) is replaced by the $N$ coupled equations, the so-called reduced Hartree-Fock equations:

[TABLE]

The rigorous time-dependent Hartree-Fock theory has been developed first by Chadam-Glassey [15] for (1.2) with $\phi(h(D))=-\Delta$ in dimension $d=3.$ In this setting, (1.2) is equivalent to the von Neumann equation

[TABLE]

for $K(t)=\sum_{1}^{N}\left|\psi_{k}(t)\rangle\langle\psi_{k}(t)\right|$ and $G(t)=\phi(h(D))+H(x,t),$ see, e.g., [29, 32, 33]. In the above equation, we use Dirac’s notation $|u\rangle\langle v|$ for the operator $f\mapsto\langle v,f\rangle u$. The von Neumann equation (1.3) can also be considered for more general class of density matrices $K(t)$. For example, one can consider the class of nonnegative self-adjoint trace class operators, for which $K(t)$ satisfies the following conditions:

[TABLE]

where the condition $K(t)\leq 1$ corresponds to the Pauli exclusion principle, and $N$ is the “number of particles”.

The well-posedness for (1.3) was proved by Bove-Da Parto-Fano [8, 9] for a short-range paire-wise interaction potential $w(x-y)$ instead of Coulomb potential $\frac{1}{|x-y|}$ in the Hartree factor. The case of Coulomb potential was resolved by Chadam [14]. Lewin-Sabin [33] have established the well-posedness for (1.3) with density matrices of infinite trace for pair-wise interaction potentials $w\in L^{1}(\mathbb{R}^{3})$. However, their investigation did not include the Coulomb potential case. Moreover, Lewin-Sabin [32] prove the asymptotic stability for the ground state in dimension $d=2$. Recently, Fröhlich-Lenzmann [23] and Carles-Lucha-Moulay [13] studied the local and global well-posedness for (1.1) and (1.2) in $L^{2}-$based Sobolev spaces, when $d=3$. The existence of a global solution to (1.1) was established in [23, Theorem 2.2] assuming sufficiently small initial data. These results naturally raise two questions. First, could similar results be in other functions spaces? Second, is it possible to obtain the existence of global solutions to (1.1) and (1.2) with any initial data.

We investigate these two questions in the setting of the modulation spaces $M^{p,q}(\mathbb{R}^{d})$ (to be defined below), which have recently been considered as spaces of Cauchy data for certain nonlinear dispersive equations, see [1, 2, 3, 4, 5, 37, 40, 41, 42]. Generally modulation spaces are considered as low regularity spaces because they contain rougher functions than functions in any given fractional Bessel potential space (see Proposition 2.1 below). We refer to excellent survey [36] and the reference therein for details.

Taking these considerations into account, we initiate the study of (1.1) and (1.2) in modulation spaces. In particular, the two our main results can be stated as follows.

Theorem 1.1** (Local well-posedness).**

Let $N,d\in\mathbb{N}$, and $\gamma\in(0,d)$ be given. Let $X$ be given by

[TABLE]

for some $\epsilon>0$ and $s\geq 0$. Let $\phi:\mathbb{R}^{d}\to\mathbb{R}$ be such that there exist $m_{1},m_{2}>0$ with

[TABLE]

*for all $\mu\in\mathbb{N}_{0}.$ Furthermore, assume that $h\in C^{\infty}(\mathbb{R}^{d}\setminus\{0\})$ is either

(a) a positive function of homogeneous type of degree $\lambda>0$ with $m_{1}\lambda\leq 2$, or

(b) $\phi\circ h(\xi):=P(\xi)=\sum_{|\beta|\leq m}c_{\beta}\xi^{\beta}$ is a polynomial with order $m\leq 2.$*

Given initial data $\left(\psi_{01},...,\psi_{0N}\right)\in X^{N}$, the following statements hold.

1. (i)

There exists $T>0$ depending only on $\|\psi_{01}\|_{X},...,\|\psi_{0N}\|_{X},$ $d$ and $\gamma$ such that (1.1) has a unique local solution

[TABLE] 2. (ii)

There exists $T>0$ depending only on $\|\psi_{01}\|_{X},...,\|\psi_{0N}\|_{X},$ $d$ and $\gamma$ such that (1.2) has a unique local solution

[TABLE]

Our second main result deal with the global well-posedness of these equations. In the statement, we denote by $X_{rad}$, space of radial functions in the Banach space $X$.

Theorem 1.2** (Global well-posedness).**

Suppose that $\phi$ and $h$ are defined on $\mathbb{R}$ and $\mathbb{R}^{d}$ respectively such that $\phi\circ h(\xi)=|\xi|^{\alpha}$ for $\xi\in\mathbb{R}^{d}$, and where $\alpha>0.$ Assume that $0<\gamma<\min\{\alpha,d/2\}$, and that one of the following two statements holds:

• (a)

For $\alpha=2$ and $d\in\mathbb{N},$ let

[TABLE]

for some $\epsilon>0$ and $s\geq 0$.

• (b)

For $\alpha\in\left(\frac{2d}{2d-1},2\right)$ and $d\geq 2,$ let

[TABLE]

for some $\epsilon>0$ and $s\geq 0$.

Given initial data $(\psi_{01},...,\psi_{0N})\in X^{N},$ the following statements hold.

1. (i)

There exists a unique global solution of (1.1) such that

[TABLE] 2. (ii)

There exists a unique global solution of (1.2) such that

[TABLE]

In the case $N=1$, first author in [4, Theorem 1.1] established the global well-posedness of  (1.2) in $M^{p,q}(\mathbb{R}^{d})$ when $1\leq p\leq 2,$ and $1\leq q<\frac{2d}{d+\gamma}.$ Part (ii) of Theorem 1.2 proves this result for the end point case for any $N\geq 1.$ We note that $M^{p,q}(\mathbb{R}^{d})\subset L^{p}(\mathbb{R}^{d})\ (q\leq\min\{p,p^{\prime}\})$ is sharp embedding and up to now we cannot get the global well-posedness of (1.1) in $L^{p}(\mathbb{R}^{d})(1\leq p<2)$ but in $M^{p,q}(\mathbb{R}^{d})$ (Theorem 1.2). Noticing for $s>\gamma/2,$ we have sharp embedding $H^{s}(\mathbb{R}^{d})\subset M^{2,\frac{2d}{d+\gamma}}(\mathbb{R}^{d})\subset L^{2}(\mathbb{R}^{d})$ (see Proposition 2.1 below), Theorem 1.2 reveals that we can solve (1.1) and (1.2) with Cauchy data beyond in $H^{s}(\mathbb{R}^{d})(s>\gamma/2).$

Remark 1.1**.**

The sign of $\kappa$ in Hartree and Fock terms determines the defocusing and focusing character of the nonlinearity, but, as we shall see, this character will play no role in our analysis on modulation spaces, as we do not use the conservation of energy of (1.1) and (1.2) to achieve global existence.

1.3. Hartree-Fock equation with harmonic potential

The Hartree-Fock equation with the harmonic potential of $N$ particles is given by

[TABLE]

and the corresponding reduced Hartree-Fock equation with the harmonic potential:

[TABLE]

where $t\in\mathbb{R},\psi_{k}:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{C},$ $k=1,2,...,N,0<\gamma<d,$ $\kappa$ is constant. In this context we establish the following result.

Theorem 1.3**.**

Let $0<\gamma<\text{min}\{2,d/2\},d\in\mathbb{N}$ and $1\leq p\leq\frac{2d}{d+\gamma}.$ Given initial data $(\psi_{01},...,\psi_{0N})\in\left(M^{p,p}(\mathbb{R}^{d})\right)^{N},$ the following statements hold.

1. (i)

There exists a unique global solution of (1.4) such that

[TABLE] 2. (ii)

There exists a unique global solution of (1.5) such that

[TABLE]

In the case $N=1,$ first author in [6, Theorem 1.1] proved that (1.5) is globally well-posed in $M^{p,p}(\mathbb{R}^{d})$ for $1\leq p<\frac{2d}{d+\gamma}.$ Part (ii) of Theorem 1.3 establishes this result for the end point case for any $N\geq 1.$

The rest of the paper is organized as follows. In Section 2, we introduce some notations and preliminary results which will be used in the sequel. In Section 3, we prove the boundedness for Hartree nonlinearity on modulation spaces. In Section 4 we establish two of our main results, namely Theorems 1.1, and 1.2. Finally, in Section 5 we prove Theorem 1.3.

2. Preliminaries

2.1. Notations

The notation $A\lesssim B$ means $A\leq cB$ for some constant $c>0$, whereas $A\asymp B$ means $c^{-1}A\leq B\leq cA$ for some $c\geq 1$. Given $a,b\in\mathbb{R}$ we let $a\wedge b=\text{min}\{a,b\}.$ The symbol $A_{1}\hookrightarrow A_{2}$ denotes the continuous embedding of the topological linear space $A_{1}$ into $A_{2}.$ Put $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}.$ If $\beta=(\beta_{1},\cdots,\beta_{d})\in\mathbb{N}_{0}^{d}$ is a multi-index, we set

[TABLE]

and if $x=(x_{1},\cdots x_{d})\in\mathbb{R}^{d},$

[TABLE]

The $L^{p}(\mathbb{R}^{d})$ norm is denoted by

[TABLE]

the $L^{\infty}(\mathbb{R}^{d})$ norm is $\|f\|_{L^{\infty}}=\text{ess.sup}_{x\in\mathbb{R}^{d}}|f(x)|$. For $1\leq p\leq\infty,$ $p^{\prime}$ denotes the Hölder conjugate of $p,$ that is, $1/p+1/p^{\prime}=1.$ We use $L_{t}^{r}(I,X)$ to denote the space-time norm

[TABLE]

where $I\subset\mathbb{R}$ is an interval and $X$ is a Banach space. The Schwartz space is denoted by $\mathcal{S}(\mathbb{R}^{d})$, and, its dual, the space of tempered distributions is denoted by $\mathcal{S^{\prime}}(\mathbb{R}^{d}).$ For $x=(x_{1},\cdots,x_{d}),y=(y_{1},\cdots,y_{d})\in\mathbb{R}^{d},$ we put $x\cdot y=\sum_{i=1}^{d}x_{i}y_{i}.$ Let $\mathcal{F}:\mathcal{S}(\mathbb{R}^{d})\to\mathcal{S}(\mathbb{R}^{d})$ be the Fourier transform defined by

[TABLE]

Then $\mathcal{F}$ is an isomorphism on $\mathcal{S}(\mathbb{R}^{d})$ which uniquely extends to an isomorphism on $\mathcal{S}^{\prime}(\mathbb{R}^{d}).$

The Fourier-Lebesgue spaces $\mathcal{F}L^{p}(\mathbb{R}^{d})$ is defined by

[TABLE]

For $p\in(1,\infty)$ and $s\geq 0$, $W^{s,p}(\mathbb{R}^{d})$ will denote standard Sobolev space. In particular, if $s$ is an integer, then $W^{s,p}$ consists of $L^{p}-$functions with derivatives in $L^{p}$ up to order $s$, hence coincides with the $L^{p}_{s}-$Sobolev space, also known as Bessel potential space, defined for $s\in\mathbb{R}$ by

[TABLE]

where $\langle\xi\rangle^{s}=(1+|\xi|^{2})^{s/2}\ (\xi\in\mathbb{R}^{d}).$ Note that $L^{p}_{s_{1}}(\mathbb{R}^{d})\hookrightarrow L^{p}_{s_{2}}(\mathbb{R}^{d})$ if $s_{2}\leq s_{1}.$

2.2. Modulation spaces

Feichtinger [21] introduced the modulation spaces by imposing integrability conditions on the short-time Fourier transform (STFT) of functions or distributions defined on $\mathbb{R}^{d}$. To be specific, the STFT of a function $f$ with respect to a window function $g\in{\mathcal{S}}(\mathbb{R}^{d})$ is defined by

[TABLE]

whenever the integral exists. For $x,w\in\mathbb{R}^{d}$ the translation operator $T_{x}$ and the modulation operator $M_{w}$ are defined by $T_{x}f(t)=f(t-x)$ and $M_{w}f(t)=e^{2\pi iw\cdot t}f(t).$ In terms of these operators the STFT may be expressed as

[TABLE]

where $\langle f,g\rangle$ denotes the inner product for $L^{2}$ functions, or the action of the tempered distribution $f$ on the Schwartz class function $g$. Thus $V:(f,g)\to V_{g}(f)$ extends to a bilinear form on $\mathcal{S}^{\prime}(\mathbb{R}^{d})\times\mathcal{S}(\mathbb{R}^{d})$ and $V_{g}(f)$ defines a uniformly continuous function on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ whenever $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and $g\in\mathcal{S}(\mathbb{R}^{d})$.

Definition 2.1** (Modulation spaces).**

Let $1\leq p,q\leq\infty,s\in\mathbb{R}$ and $0\neq g\in{\mathcal{S}}(\mathbb{R}^{d})$. The weighted modulation space $M_{s}^{p,q}(\mathbb{R}^{d})$ is defined to be the space of all tempered distributions $f$ for which the following norm is finite:

[TABLE]

for $1\leq p,q<\infty$. If $p$ or $q$ is infinite, $\|f\|_{M_{s}^{p,q}}$ is defined by replacing the corresponding integral by the essential supremum. For $s=0,$ we write $M^{p,q}_{0}(\mathbb{R}^{d})=M^{p,q}(\mathbb{R}^{d}).$

It is standard to show that this definition is independent of the choice of the particular window function, e.g., see, [24, Proposition 11.3.2(c)].

Using a uniform partition of the frequency domain, one can obtain an equivalent definition of the modulation spaces [41] as follows. Let $Q_{k}$ be the unit cube with the center at $k$, so $\{Q_{k}\}_{k\in\mathbb{Z}^{d}}$ constitutes a decomposition of $\mathbb{R}^{d},$ that is, $\mathbb{R}^{d}=\cup_{k\in\mathbb{Z}^{d}}Q_{k}.$ Let $\rho\in\mathcal{S}(\mathbb{R}^{d}),$ $\rho:\mathbb{R}^{d}\to[0,1]$ be a smooth function satisfying $\rho(\xi)=1\ \text{if}\ \ |\xi|_{\infty}\leq\frac{1}{2}$ and $\rho(\xi)=0\ \text{if}\ \ |\xi|_{\infty}\geq 1,$ where $|\xi|_{\infty}=\max_{k=1,\ldots,d}|\xi_{k}|$. Let $\rho_{k}$ be a translate of $\rho,$ that is,

[TABLE]

For each $0\neq k\in\mathbb{Z}$ let

[TABLE]

and when $k=0$, we simply write $\sigma_{0}=\sigma.$ Then $\{\sigma_{k}(\xi)\}_{k\in\mathbb{Z}^{d}}$ satisfies the following properties

[TABLE]

for some positive constant $c$.

The frequency-uniform decomposition operators can be defined by

[TABLE]

For $1\leq p,q\leq\infty,s\in\mathbb{R},$ it is known [21] that

[TABLE]

with natural modifications for $p,q=\infty.$ As observed in [42], the frequency-uniform decomposition operators obey an almost orthogonality relation: for each $k\in\mathbb{Z}$

[TABLE]

where $\|\ell\|_{\infty}=\max\{|\ell_{i}|:\ell_{i}\in\mathbb{Z},i=1,...,d\}.$

We now list some basic properties of the modulation spaces.

Lemma 2.1**.**

Let $p,q,p_{i},q_{i}\in[1,\infty]$ $(i=1,2),s,s_{1},s_{2}\in\mathbb{R}.$ Then

1. (1)

$M^{p_{1},q_{1}}_{s_{1}}(\mathbb{R}^{d})\hookrightarrow M^{p_{2},q_{2}}_{s_{2}}(\mathbb{R}^{d})$* whenever $p_{1}\leq p_{2}$ and $q_{1}\leq q_{2}$ and $s_{2}\leq s_{1}.$* 2. (2)

$M^{p,q_{1}}(\mathbb{R}^{d})\hookrightarrow L^{p}(\mathbb{R}^{d})\hookrightarrow M^{p,q_{2}}(\mathbb{R}^{d})$* holds for $q_{1}\leq\text{min}\{p,p^{\prime}\}$ and $q_{2}\geq\text{max}\{p,p^{\prime}\}$ with $\frac{1}{p}+\frac{1}{p^{\prime}}=1.$* 3. (3)

$M^{\min\{p^{\prime},2\},p}(\mathbb{R}^{d})\hookrightarrow\mathcal{F}L^{p}(\mathbb{R}^{d})\hookrightarrow M^{\max\{p^{\prime},2\},p}(\mathbb{R}^{d}),\frac{1}{p}+\frac{1}{p^{\prime}}=1.$** 4. (4)

$\mathcal{S}(\mathbb{R}^{d})$* is dense in $M^{p,q}(\mathbb{R}^{d})$ if $p$ and $q<\infty.$* 5. (5)

The Fourier transform $\mathcal{F}:M_{s}^{p,p}(\mathbb{R}^{d})\to M_{s}^{p,p}(\mathbb{R}^{d})$ is an isomorphism. 6. (6)

The space $M_{s}^{p,q}(\mathbb{R}^{d})$ is a Banach space. 7. (7)

The space $M_{s}^{p,q}(\mathbb{R}^{d})$ is invariant under complex conjugation.

Proof.

For the proof of parts (1), (2), (3), and (4) see [24, Theorem 12.2.2], [39, Proposition 1.7], [18, Corollary 1.1] and [24, Proposition 11.3.4] respectively. The proof of statement (5) can be derived from the fundamental identity of time-frequency analysis:

[TABLE]

which is easy to obtain. The proof of statement (6) is trivial, indeed, we have $\|f\|_{M^{p,q}}=\|\bar{f}\|_{M^{p,q}}.$ ∎

We can obtain examples of functions in the modulation spaces via embedding relations with certain classical functions spaces. For example the following result can be proved.

Proposition 2.1** (Examples).**

The following statements hold.

1. (i)

( **[28]**, **[36, Theorem 3.8]**) Let $1\leq p,q\leq\infty,$ $s_{1},s_{2}\in\mathbb{R},$ and

[TABLE]

Then $L^{p}_{s_{1}}(\mathbb{R}^{d})\subset M^{p,q}_{s_{2}}(\mathbb{R}^{d})$ if and only if one of the following conditions is satisfied:

[TABLE] 2. (ii)

(**[28]**, **[36, Theorem 3.8]**) Let $1\leq p,q\leq\infty,$ $s_{1},s_{2}\in\mathbb{R},$ and

[TABLE]

Then $M^{p,q}_{s_{1}}(\mathbb{R}^{d})\subset L^{p}_{s_{2}}(\mathbb{R}^{d})$ if and only if one of the following conditions is satisfied:

[TABLE] 3. (iii)

For $1\leq p<2,$ $M^{p,p}(\mathbb{R}^{d})\subset L^{p}(\mathbb{R}^{d})$ and there exists $f\in L^{p}(\mathbb{R}^{d})$ such that $f\notin M^{p,p}(\mathbb{R}^{d}).$ 4. (iv)

For $s>\frac{\gamma}{2}>0,$ $H^{s}(\mathbb{R}^{d})\subset M^{2,\frac{2d}{d+\gamma}}(\mathbb{R}^{d})$ and there exists $f\in M^{2,\frac{2d}{d+\gamma}}(\mathbb{R}^{d})$ such that $f\notin H^{s}(\mathbb{R}^{d}).$ 5. (v)

For $2<p<\infty$ and $s>d\left(1-\frac{1}{p}\right),$ $L^{p}_{s}(\mathbb{R}^{d})\subset M^{p,1}(\mathbb{R}^{d})$ and there exists $f\in M^{p,1}(\mathbb{R}^{d})$ such that $f\notin L^{p}_{s}(\mathbb{R}^{d}).$

Proof.

We only give proofs of the last three parts.

1. (iii)

For $1\leq p<2,$ by part (2) of Lemma 2.1, we have $M^{p,p}(\mathbb{R}^{d})\subset L^{p}(\mathbb{R}^{d}).$ We claim that $M^{p,p}(\mathbb{R}^{d})\subsetneq L^{p}(\mathbb{R}^{d}).$ If possible, suppose that claim is not true, that is, for all $f\in L^{p}(\mathbb{R}^{d}),$ we have $f\in M^{p,p}(\mathbb{R}^{d}).$ It follows that $L^{p}(\mathbb{R}^{d})=M^{p,p}(\mathbb{R}^{d}).$ But then by part (5) of Lemma 2.1, it follows that $L^{p}(\mathbb{R}^{d})$ invariant under the Fourier transform, which is a contradiction. Hence, the claim. Similarly, for $2<p\leq\infty,$ we have $M^{p,p}(\mathbb{R}^{d})\subsetneq L^{p}(\mathbb{R}^{d}).$ 2. (iv)

Noticing $\tau\left(2,\frac{2d}{d+\gamma}\right)=\frac{\gamma}{2},$ by part (i), we have $H^{s}(\mathbb{R}^{d})\subset M^{2,1}(\mathbb{R}^{d})$ for $s>\gamma/2.$ We claim that $H^{s}(\mathbb{R}^{d})\subsetneq M^{2,\frac{2d}{d+\gamma}}(\mathbb{R}^{d})$. If possible, suppose that claim is not true. Then we have $H^{s}(\mathbb{R}^{d})=M^{2,\frac{2d}{d+\gamma}}(\mathbb{R}^{d})$. But then, noticing $\sigma\left(2,\frac{2d}{d+\gamma}\right)=-\frac{\gamma}{2},$ part (ii) gives contradiction. Hence, the claim. 3. (v)

Noticing $\tau(p,1)=d\left(1-\frac{1}{p}\right)$ and $\sigma(p,1)=-\frac{d}{p},$ parts (i) and (ii) give $L^{p}_{s}(\mathbb{R}^{d})\subsetneq M^{p,1}(\mathbb{R}^{d}).$

∎

Proposition 2.2**.**

(Algebra property, [39, Theorem 2.4]) Let $s\geq 0$, and $p,q,p_{i},q_{i}\in[1,\infty]$, where $i=0,1,2$. If $\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{0}}$ and $\frac{1}{q_{1}}+\frac{1}{q_{2}}=1+\frac{1}{q_{0}},$ then

[TABLE]

with norm inequality $\|fg\|_{M_{s}^{p_{0},q_{0}}}\lesssim\|f\|_{M_{s}^{p_{1},q_{1}}}\|g\|_{M_{s}^{p_{2},q_{2}}}.$ In particular, the space $M^{p,q}(\mathbb{R}^{d})$ is a poinwise $\mathcal{F}L^{1}(\mathbb{R}^{d})$-module, that is, it satisfies

[TABLE]

2.3. Modulation space estimates for unimodular Fourier multipliers

In this section, we consider the boundedness properties of a class of unimodular Fourier multipliers defined by

[TABLE]

for $f\in\mathcal{S}(\mathbb{R}^{d})$, where $\phi\circ h:\mathbb{R}^{d}\to\mathbb{R}$ is the composition function of $h:\mathbb{R}^{d}\to\mathbb{R}$ and $\phi:\mathbb{R}\to\mathbb{R}.$

Proposition 2.3**.**

Let $s\in\mathbb{R}$ and $1\leq p,q\leq\infty.$

1. (i)

(**[19, Theorem 1.1]**) Assume that there exist $m_{1},m_{2}>0$ such that $\phi$ satisfies

[TABLE]

for all $\mu\in\mathbb{N}_{0}$ and $h\in C^{\infty}(\mathbb{R}^{d}\setminus\{0\})$ is positive homogeneous function with degree $\lambda>0$. Then we have

[TABLE]

where $\gamma(m_{1},\lambda)=d(m_{1}\lambda-2)|1/2-1/p|$. 2. (ii)

(**[16, Theorems 1 and 2]**) Let $h(\xi)=|\xi|$ and $\phi(r)=r^{\alpha}$, with $1/2<\alpha\leq 2.$ Then

[TABLE] 3. (iii)

(**[41, Proposition 4.1]**) Let $2\leq p\leq\infty,1\leq q\leq\infty,$ $h(\xi)=|\xi|$ and $\phi(r)=r^{\alpha}\ (\alpha\geq 2).$ Then

[TABLE]

Another important class of unimodular Fourier multipliers that is not covered by Proposition 2.3, are the so-called Fourier multiplier with polynomial symbol. Specifically, for $f\in\mathcal{S}(\mathbb{R}^{d})$ let

[TABLE]

where $P(\xi)=\sum_{|\beta|\leq m}c_{\beta}\xi^{\beta}$ is a polynomial with order $m\geq 1.$ In this setting the following result was proved in [19].

Proposition 2.4**.**

([19, Theorems 4.3]) Let $s\in\mathbb{R},$ $1\leq p,q\leq\infty$ and $m\geq 2.$ Then

[TABLE]

where $\gamma(m)=d(m-2)|1/2-1/p|.$

To make the paper self content, we outline the proof of Proposition 2.4 in the particular case when $P(\xi,\eta)=|\xi|^{2}-|\eta|^{2},$ and note that the general case can be proved similarly. But first, we state a result that provides a criteria for the Fourier multiplier to be bounded on modulation spaces. In particular, it provides an application of the uniform decomposition operators given in (2.1).

Proposition 2.5**.**

Let $\square_{k}$ be defined as in (2.1) and $t\in\mathbb{R}.$ Suppose that there is an integer $M>0$ such that

[TABLE]

where $b_{1}\geq b_{2}\geq 0$ for all $f\in L^{1}(\mathbb{R}^{d})$. Then we have

[TABLE]

whenever $f\in M^{p,q}_{s}(\mathbb{R}^{d})$.

Proof.

By (2.1) and Plancherel theorem, we obtain

[TABLE]

for all $k\in\mathbb{Z}^{d}.$ By the Riesz-Thorin interpolation theorem, and for any $1\leq p\leq 2,$ we have

[TABLE]

Using a duality argument, we obtain the above two inequality for all $1\leq p\leq\infty.$ Using (2.3), for $f\in\mathcal{S}(\mathbb{R}^{d})$ and $|k|<M,$ we obtain

[TABLE]

Similarly, for $f\in\mathcal{S}(\mathbb{R}^{d})$ and $|k|\geq M,$ we obtain

[TABLE]

In view of (2.2), and the above two inequalities, we obtain

[TABLE]

This completes the proof.∎

Now to apply Proposition 2.5, we must have control on the $L^{1}-$norm of the projection operator $\|\square_{k}(e^{itP(D)}f)\|_{L^{1}(\mathbb{R}^{d})}.$ Since $\square_{k}(e^{itP(D)}f)=\mathcal{F}^{-1}(\sigma_{k}e^{itP})\ast f,$ in view of Young’s inequality, it suffices to control the norm $\|\mathcal{F}^{-1}(\sigma_{k}e^{itP})\|_{L^{1}}$, which we shall do in next two lemmas.

Lemma 2.2**.**

([19, Lemmas 4.1 and 4.2]) Let $t\in\mathbb{R},P(z)=P(\xi,\eta)=|\xi|^{2}-|\eta|^{2},z=(\xi,\eta)\in\mathbb{R}^{2d},k\in\mathbb{Z}^{2d},$ and $M>0.$ Then we have

[TABLE]

Proof.

Assume that $|k|>M+1.$ We introduce an auxiliary function defined by

[TABLE]

for all $k\in\mathbb{Z}^{2d}.$ Since $L^{1}-$norm is invariant under translation and modulation, we have

[TABLE]

where $g(z)=\sigma(z)e^{it\Lambda_{k}(z)}.$ Thus, to prove Lemma 2.2, it suffices to prove

[TABLE]

for $t>0.$ We consider

[TABLE]

By Cauchy-Schwarz inequality and Plancherel’s Theorem, we have

[TABLE]

Now we concentrate on $I_{2}.$ For $j\in\{1,2,\cdots,2d\},$ let

[TABLE]

[TABLE]

We note that

[TABLE]

Since $\sigma$ is compactly supported and $\Lambda_{k}$ is a smooth function, performing integration by parts and using Plancherel’s theorem, we obtain that for each $j\in\{1,2,...,2d\}$

[TABLE]

where we choose $L>d$ as an integer. Where we have used the fact that since $|x|^{2}=\sum_{j=1}^{2d}x_{j}^{2}\leq 2d|x_{j}|^{2}$ for $x\in E_{j,t},$ we have $|x_{j}|^{-2L}\lesssim|x|^{-2L}$. Consequently, we have

[TABLE]

Next, we claim that

[TABLE]

Once this claim is established, the proof of the lemma will follow from (2.4), (2.5), and (2.6).

We now give a proof of this claim. To this end, we note that by Taylor’s and Leibniz formula, we have

[TABLE]

and

[TABLE]

Since $P(z)$ is a polynomial of order 2, there exists $C_{\gamma}$ such that

[TABLE]

for all $\gamma\in\mathbb{N}_{0}^{2d}.$ We note that for $z\in\text{supp}\ \sigma,$ and $s\in[0,1],$ we have $|k+sz|\lesssim|k|,$ and in view of (2.7)-(2.9), we have that for all $|k|>M+1$

[TABLE]

which implies that

[TABLE]

for all $\gamma\in\mathbb{N}^{2d},$ where for each $l\in\{1,\cdots,|\gamma|\},$ $v_{l}=(v_{1},\cdots,v_{l})\in\mathbb{N}^{l}.$ For fixed $j,$ by Leibniz formula, we have

[TABLE]

Using this and (2.10), we obtain

[TABLE]

This proves the claim when $|k|>M+1.$ The case $|k|\leq M+1$ can be consider similarly (see e.g. [19, Lemma 4.2]). ∎

Sketch Proof of Proposition 2.4.

Taking Proposition 2.5 and Lemma 2.2 into account, the proof follows when $P(\xi,\eta)=|\xi|^{2}-|\eta|^{2},(\xi,\eta)\in\mathbb{R}^{2d}$. The general case can be done similarly. ∎

3. Trilinear $M^{p,q}$ estimates

One of the main technical results needed to prove our main result is establishing a trilinear estimate for the following Hartree type trilinear operator. For $0<\gamma<d$, let

[TABLE]

where $f,g,h\in\mathcal{S}(\mathbb{R}^{d}).$

Proposition 3.1**.**

Let $0<\gamma<d,$ $1\leq p\leq 2,$ and $1\leq q\leq\frac{2d}{d+\gamma}$. Given $f,g,h\in M^{p,q}(\mathbb{R}^{d}),$ then $H_{\gamma}(f,g,h)\in M^{p,q}(\mathbb{R}^{d})$, and the following estimate holds

[TABLE]

Proof.

By Proposition 2.2, we have

[TABLE]

We note that

[TABLE]

and integrating with respect to $\xi,$ we get

[TABLE]

where $I^{\gamma}$ denotes the Riesz potential of order $\gamma$:

[TABLE]

By Hölder and Hardy-Littlewood Sobolev inequalities and Lemma 2.1, we have

[TABLE]

This completes the proof. ∎

We next prove a related result for weighted modulation spaces $M^{p,q}_{s}$.

Proposition 3.2**.**

Assume that $0<\gamma<d.$ The following statements hold

1. (i)

If $1<p_{1}<p_{2}<\infty$ with $\frac{1}{p_{1}}+\frac{\gamma}{d}-1=\frac{1}{p_{2}}$ and $1\leq q\leq\infty,s\geq 0.$ For any $f\in M^{p_{1},q}_{s}(\mathbb{R}^{d}),$ we have $\||\cdot|^{-\gamma}\ast f\|_{M^{p_{2},q}_{s}}\lesssim\|f\|_{M^{p_{1},q}_{s}}.$ 2. (ii)

Let $1<p<\infty$ and $\frac{1}{p}+\frac{\gamma}{d}-1=\frac{1}{p+\epsilon}$ for some $\epsilon>0.$ For any $f,g,h\in M^{p,1}_{s}(\mathbb{R}^{d}),$ we have

[TABLE]

Proof.

We may rewrite the STFT as $V_{g}(x,w)=e^{-2\pi ix\cdot w}(f\ast M_{w}g^{*})(x)$ where $g^{*}(y)=\overline{g(-y)}.$

1. (i)

Using Hardy-Littlewood-Sobolev inequality, we obtain

[TABLE]

This completes the proof of part (i). 2. (ii)

By Proposition 2.2 and part (1) of Lemma 2.1, we have

[TABLE]

for some $\epsilon>0.$ By part (i) and Proposition 2.2, we have $\|T_{\gamma}(fg)\|_{M_{s}^{p+\epsilon,1}}\lesssim\|fg\|_{M_{s}^{p,1}}\lesssim\|f\|_{M_{s}^{p,1}}\|g\|_{M_{s}^{p,1}}.$

∎

The following result immediately follows.

Proposition 3.3**.**

Let $1<p<\infty$ and $\frac{1}{p}+\frac{\gamma}{d}-1=\frac{1}{p+\epsilon}$ for some $\epsilon>0.$ For any $f,g,h\in M^{p,1}(\mathbb{R}^{d})\cap L^{2}(\mathbb{R}^{d}),$ we have

[TABLE]

Proof.

By part (2) of Lemma 2.1, we have

[TABLE]

This completes the proof. ∎

We will also need the following result.

Lemma 3.1**.**

Let $0<\gamma<d.$

1. (i)

Let $1\leq p\leq 2,1\leq q\leq\frac{2d}{d+\gamma}.$ For any $f,g\in M^{p,q}(\mathbb{R}^{d})$, we have

[TABLE] 2. (ii)

Let $1<p<\infty$ and $\frac{1}{p}+\frac{\gamma}{d}-1=\frac{1}{p+\epsilon}$ for some $\epsilon>0.$ For any $f,g\in M^{p,1}(\mathbb{R}^{d})\cap L^{2}(\mathbb{R}^{d}),$ we have

[TABLE]

Proof.

Notice that

[TABLE]

and

[TABLE]

This together with the following identity

[TABLE]

gives the desired inequality. ∎

4. Proofs of main results

4.1. Local well-posedness for Hartree-Fock equations

We can now prove our main results, beginning with Theorem 1.1.

Proof of Theorem 1.1.

By Duhamel’s principle, we rewrite the Cauchy problem (1.1) in an integral form: for $k=1,...,N,$

[TABLE]

We shall show that $\Psi$ has a unique fixed point in an appropriate function space, for small $t.$ For this, we consider Banach space $\left(C([0,T],X)\right)^{N},$ with the norm

[TABLE]

where $u=(u_{1},...,u_{N})\in\left(C([0,T],X)\right)^{N}.$ By Propositions 2.3 and 2.4, we have

[TABLE]

where $C_{T}=C(1+|t|)^{d\left|\frac{1}{p}-\frac{1}{2}\right|}.$ By Minkowski’s inequality for integrals, Propositions 2.3 and 2.4 and Propositions 3.1, and 3.2, we obtain

[TABLE]

Similarly,

[TABLE]

Thus, we have

[TABLE]

for some universal constant $c.$

For $M>0,$ put

[TABLE]

which is the closed ball of radius $M$ and centered at the origin in $\left(C([0,T],X)\right)^{N}$. Next, we show that the mapping $\Psi_{k}$ takes $B_{T,M}$ into itself for suitable choice of $M$ and small $T>0$. Indeed, if we take $M=2C_{T}\max\{\|\psi_{0k}\|_{X}:i=1,...,N\}$ and $\bar{\psi}=(\psi_{1},...,\psi_{N})\in B_{T,M},$ we obtain

[TABLE]

for all $k=1,...,N.$ We choose a $T$ such that $cC_{T}M^{2}\leq 1/2,$ that is, $T\leq\tilde{T}(\|\psi_{01}\|_{X},...,\|\psi_{0N}\|_{X},d,\gamma)$ and as a consequence we have

[TABLE]

So $B_{T,M}$ is invariant under the action of $\Psi$ provided that $T>0$ is sufficiently small. Up to diminishing $T,$ contraction follows readily, since $H_{\gamma}$ is a trilinear operator. So there exist a unique (in $B_{T,M}$) fixed point for $\Psi,$ that is, a solution to (1.1). This completes the proof of Theorem 1.1 part (i). Similarly, we can produce the proof of Theorem 1.1 part (ii) of which we omit the details. ∎

4.2. Global Well-posedness for Hartree-Fock Equations

In this section we prove Theorem 1.2.

Definition 4.1**.**

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

[TABLE]

We recall the following results. For details, see [27, 26].

Proposition 4.1** (Strichartz estimates).**

Denote

[TABLE]

1. (i)

Let $\phi\in L^{2}(\mathbb{R}^{d}),$ $d\in\mathbb{N}$ and $\alpha=2.$ Then for any time slab $I$ and admissible pairs $(p_{i},q_{i})$, $i=1,2,$ there exists a constant $C=C(|I|,q_{1})$ such that for all intervals $I\ni 0,$

[TABLE]

where $p_{i}^{\prime}$ and $q_{i}^{\prime}$ are Hölder conjugates of $p_{i}$ and $q_{i}$ respectively **[27]**. 2. (ii)

Let $d\geq 2$ and $\frac{2d}{2d-1}<\alpha<2.$ Assume that $\phi$ and $F$ are radial. Then for any time slab $I$ and admissible pairs $(p_{i},q_{i})$, $i=1,2,$ there exists a constant $C=C(|I|,q_{1})$ such that for all intervals $I\ni 0,$

[TABLE]

where $p_{i}^{\prime}$ and $q_{i}^{\prime}$ are Hölder conjugates of $p_{i}$ and $q_{i}$ **[26, Corollary 3.4]**.

We first establish the following preliminary results.

Proposition 4.2**.**

Let $\phi\circ h(\xi)=|\xi|^{\alpha}$ where $\xi\in\mathbb{R}^{d},\alpha>0$ and $0<\gamma<\min\{\alpha,d\}.$

1. (i)

Let $d\in\mathbb{N}$ and $\alpha=2.$ If $(\psi_{01},....,\psi_{0N})\in\left(L^{2}(\mathbb{R}^{d})\right)^{N}$ then (1.1) has a unique global solution

[TABLE]

*In addition, its *$L^{2}-$norm is conserved,

[TABLE]

and for all $\alpha-$ fractional admissible pairs $(p,q),$ and $(\psi_{1},...,\psi_{N})\in\left(L_{loc}^{p}(\mathbb{R},L^{q}(\mathbb{R}^{d}))\right)^{N}.$ 2. (ii)

Let $d\geq 2$ and $\frac{2d}{2d-1}<\alpha<2.$ If $(\psi_{01},....,\psi_{0N})\in\left(L_{rad}^{2}(\mathbb{R}^{d})\right)^{N}$ then (1.1) has a unique global solution

[TABLE]

*In addition, its *$L^{2}-$norm is conserved,

[TABLE]

and for all $\alpha-$ fractional admissible pairs $(p,q),$ and $(\psi_{1},...,\psi_{N})\in\left(L_{loc}^{p}(\mathbb{R},L^{q}(\mathbb{R}^{d}))\right)^{N}.$

Proof.

We first establish part (ii). By Duhamel’s formula, we write (1.1) as

[TABLE]

where Hartree factor $H=\sum_{l=1}^{N}\left(\frac{1}{|x|^{\gamma}}\ast|\psi_{l}|^{2}\right)$ and Fock term $F(\psi_{k})=\sum_{l=1}^{N}\psi_{l}\left(\frac{\kappa}{|x|^{\gamma}}\ast\{\overline{\psi_{l}}\psi_{k}\}\right)$. Put $s=\frac{\alpha}{2}$. We introduce the space

[TABLE]

and the distance

[TABLE]

where $\phi_{1}=(f_{1},...,f_{N})$ and $\phi_{2}=(g_{1},...,g_{N}).$ Then $(Y,d)$ is a complete metric space. Now we show that $\Phi$ takes $Y(T)$ to $Y(T)$ for some $T>0.$ We put

[TABLE]

Note that $(q,r)$ is $\alpha-$fractional admissible and

[TABLE]

Let $(\bar{q},\bar{r})\in\{(q,r),(\infty,2)\}.$ By part (ii) of Proposition 4.1 and Hölder’s inequality, we have

[TABLE]

Since $0<\gamma<\min\{\alpha,d\}$, by the Hardy-Littlewood-Sobolev lemma, we have

[TABLE]

Observe that in the last inequality we use the inclusion relation for the finite measure space $L^{p}([0,T])$. Thus, we have

[TABLE]

This shows that $\Phi$ maps $Y(T)$ to $Y(T).$ Next, we show $\Phi$ is a contraction. To this end, we notice the following identity: for fixed $j\in\{1,...,N\}$ and $K(x)=|x|^{-\gamma},$ we have

[TABLE]

and

[TABLE]

It follows that

[TABLE]

Put $\delta=\frac{8s}{4s-\gamma}.$ Notice that $\frac{1}{q^{\prime}}=\frac{1}{2}+\frac{1}{\delta},\frac{1}{2}=\frac{1}{\delta}+\frac{1}{q},$ and thus by Hölder’sinequality, we obtain

[TABLE]

Similarly,

[TABLE]

Let $u=(u_{1},...,u_{N})$ and $v=(v_{1},....,v_{N}).$ Now in view of (4.1), (4.3), and (4.4), we have

[TABLE]

Thus $\Phi$ is a contraction from $Y(T)$ to $Y(T)$ provided that $T$ is sufficiently small. Then there exists a unique $(\psi_{1},...,\psi_{N})\in Y(T)$ solving (1.1). The global existence of the solution (1.1) follows from the conservation of the $L^{2}-$norm of $\psi_{k}.$ The last property of the proposition then follows from the Strichartz estimates applied with an arbitrary $\alpha-$fractional admissible pair on the left hand side and the same pairs as above on the right hand side. This completes the proof of part (ii).

The proof of part (i) follows by setting $\alpha=2$ and using Proposition 4.1 part (i). ∎

Proposition 4.3**.**

Let $\phi\circ h(\xi)=|\xi|^{\alpha}$ for $\xi\in\mathbb{R}^{d},\alpha>0$ and $0<\gamma<\min\{\alpha,d\}.$

1. (i)

Let $d\in\mathbb{N}$ and $\alpha=2.$ If $(\psi_{01},....,\psi_{0N})\in\left(L^{2}(\mathbb{R}^{d})\right)^{N}$ then (1.2) has a unique global solution

[TABLE]

*In addition, its *$L^{2}-$norm is conserved,

[TABLE]

and for all $\alpha-$ fractional admissible pairs $(p,q),$ and $(\psi_{1},...,\psi_{N})\in\left(L_{loc}^{p}(\mathbb{R},L^{q}(\mathbb{R}^{d}))\right)^{N}.$ 2. (ii)

Let $d\geq 2$ and $\frac{2d}{2d-1}<\alpha<2.$ If $(\psi_{01},....,\psi_{0N})\in\left(L_{rad}^{2}(\mathbb{R}^{d})\right)^{N}$ then (1.2) has a unique global solution

[TABLE]

*In addition, its *$L^{2}-$norm is conserved,

[TABLE]

and for all $\alpha-$ fractional admissible pairs $(p,q),$ and $(\psi_{1},...,\psi_{N})\in\left(L_{loc}^{p}(\mathbb{R},L^{q}(\mathbb{R}^{d}))\right)^{N}.$

Proof.

Since the proof is similar to that of Proposition 4.2, we omit its details. ∎

Let $\Psi=(\psi_{1},...,\psi_{N}):(\mathbb{R}\times\mathbb{R}^{d})^{N}\to\mathbb{C}$ be a global $L^{2}-$ solution given by Proposition 4.2. Let $T_{+}$ denotes the maximal time of existence:

[TABLE]

Theorem 1.1 tells us that $T_{+}(\Psi)>0$ if initial data $(\psi_{01},...,\psi_{0N})\in\left(C([0,T],X\cap L^{2}(\mathbb{R}^{d}))\right)^{N}.$

Lemma 4.1**.**

Assume that $0<T_{+}<\infty.$ Then

[TABLE]

Proof.

We proceed by contradiction and assume that there exist $M>0$ and $\{t_{n}\}_{n=1}^{\infty}$ such that

[TABLE]

Recall that the life span of the local solution in Theorem 1.1 depends on the norm of the initial data. Therefore, there is $T=T(M)>0$ such that for each $n\in\mathbb{N},$ the solution $\Psi(t)=(\psi_{1}(t),...,\psi_{N}(t))$ of (1.1) can be established on the time interval $[t_{n},t_{n}+T(M)].$ By uniqueness, $\psi_{k}(t)$ coincides with standard global $L^{2}-$solution on this interval, which implies

[TABLE]

for some $\epsilon\in(0,T(M))$ and for $k=1,2...,N$ but this is a contradiction. ∎

Now we shall see that the solution constructed before is global in time. In fact, in view of Proposition 4.2, to prove Theorem 1.2, it suffices to prove that the modulation space norm of $\psi_{k}$, that is, $\|\psi_{k}\|_{M^{p,q}}$ cannot become unbounded in finite time for all $k=1,...,N.$ To this end, let $T_{0}>0$ and $\psi_{k}:[0,T_{0}]\times\mathbb{R}^{d}\to\mathbb{C}$ be a local solution to (1.1) such that

[TABLE]

for any $T\in(0,T_{0})$ and for $k=1,...,N.$

Lemma 4.2**.**

Assume that $0<\gamma<\min\{\alpha,d/2\}.$ Then

[TABLE]

Proof.

There exists $C=C(d,\gamma)$ such that the Fourier transform of $K(x)=\kappa|x|^{-\gamma}$ is

[TABLE]

We can decompose the Fourier transform of Hartree potential into Lebesgue spaces: indeed, we have

[TABLE]

where $k_{1}:=\chi_{\{|\xi|\leq 1\}}\widehat{K}\in L^{p}(\mathbb{R}^{d})$ for all $p\in[1,\frac{d}{d-\gamma})$ and $k_{2}:=\chi_{\{|\xi|>1\}}\widehat{K}\in L^{q}(\mathbb{R}^{d})$ for all $q\in(\frac{d}{d-\gamma},\infty].$

In view of (4.5) and to use the Hausdorff-Young inequality we let $1<\frac{d}{d-\gamma}<q\leq 2,$ and we obtain

[TABLE]

where we have used Proposition 2.2, Hölder’s inequality, and the conservation of the $L^{2}-$norm of $\psi_{k}$ ($k=1,...,N$) and $C_{T}$ is defined as in the proof of Theorem 1.1. We note that the requirement on $q$ can be fulfilled if and only if $0<\gamma<d/2.$ To apply Proposition 4.3, we let $\beta>1$ and $(2\beta,2q)$ is $\alpha-$fractional admissible, that is, $\frac{\alpha}{2\beta}=d\left(\frac{1}{2}-\frac{1}{2q}\right)$ such that $\frac{1}{\beta}=\frac{d}{\alpha}\left(1-\frac{1}{q}\right)<1.$ This is possible provided $\frac{q-1}{q}<\frac{\alpha}{d}:$ this condition is compatible with the requirement $q>\frac{d}{d-\gamma}$ if and only if $\gamma<\alpha.$ Using Hölder’s inequality for the last integral, we obtain

[TABLE]

where $\beta^{\prime}$ is the Hölder conjugate exponent of $\beta.$ Let

[TABLE]

For a given $T>0,$ $h$ satisfies an estimate of the form,

[TABLE]

provided that $0\leq t\leq T,$ and where we have used the fact that $\beta^{\prime}$ is finite. Using Hölder’s inequality we infer that,

[TABLE]

Raising the above estimate to the power $\beta^{\prime}$, we find that

[TABLE]

In view of Gronwall inequality, one may conclude that $h\in L^{\infty}([0,T]).$ Since $T>0$ is arbitrary, $h\in L^{\infty}_{loc}(\mathbb{R}).$ This completes the proof. ∎

We can now prove Theorem 1.2.

Proof of Theorem 1.2.

Taking Theorem 1.1 into account and combining Lemmas 4.2 and 4.1, the proof of Theorem 1.2 part (i) follows. Similarly, we can produced the proof of Theorem 1.2 part (ii), we shall omit the details. ∎

5. Well-posedness for Hartree-Fock equations with harmonic potential

In this final section we consider the Hatree-Fock and reduced Hartree-Fock equations with a harmonic potential as given by (1.4) and (1.5).

5.1. Schrödinger propagator associated to harmonic oscillator

We start by recalling the spectral decomposition of $H=-\Delta+|x|^{2}$ by the Hermite expansion. Let $\Phi_{\alpha}(x),\ \alpha\in\mathbb{N}^{d}$ be the normalized Hermite functions which are products of one dimensional Hermite functions. More precisely, $\Phi_{\alpha}(x)=\Pi_{j=1}^{d}h_{\alpha_{j}}(x_{j})$ where

[TABLE]

The Hermite functions $\Phi_{\alpha}$ are eigenfunctions of $H$ with eigenvalues $(2|\alpha|+d)$ where $|\alpha|=\alpha_{1}+...+\alpha_{d}.$ Moreover, they form an orthonormal basis for $L^{2}(\mathbb{R}^{d}).$ The spectral decomposition of $H$ is then written as

[TABLE]

where $\langle\cdot,\cdot\rangle$ is the inner product in $L^{2}(\mathbb{R}^{d})$. Given a function $m$ defined and bounded on the set of all natural numbers we can use the spectral theorem to define $m(H).$ The action of $m(H)$ on a function $f$ is given by

[TABLE]

This operator $m(H)$ is bounded on $L^{2}(\mathbb{R}^{d}).$ This follows immediately from the Plancherel theorem for the Hermite expansions as $m$ is bounded. On the other hand, the mere boundedness of $m$ is not sufficient to imply the $L^{p}$ boundedness of $m(H)$ for $p\neq 2$ (see [38]). We define Schrödinger propagator associated to harmonic oscillator

[TABLE]

with $m(n)=e^{itn}$ for $n\in\mathbb{N},t\in\mathbb{R}.$ The next result proves that $e^{it(-\Delta+|x|^{2})}$ is uniformly bounded on $M^{p,p}(\mathbb{R}^{d}).$ More specifically, we have.

Theorem 5.1**.**

([7, Theorem 5], cf. [17]) The Schrödinger propagator associated to harmonic oscillator $e^{it(-\Delta+|x|^{2})}$ is bounded on $M^{p,p}(\mathbb{R}^{d})$ for each $t\in\mathbb{R}$, and all $1\leq p<\infty.$ Moreover, we have

[TABLE]

5.2. Proof of Theorem 1.3

In this section we give a proof of Theorem 1.3. But first, we state the following definition and some preliminary results.

Definition 5.1**.**

A pair $(q,r)$ is admissible if $2\leq r<\frac{2d}{d-2}$ with $2\leq r\leq\infty$ if $d=1$, and $2\leq r<\infty$ if $d=2$, whenever

[TABLE]

Proposition 5.1**.**

([12, Proposition 2.2]) Let $\phi\in L^{2}(\mathbb{R}^{d})$ and

[TABLE]

Then for any time slab $I$ and admissible pairs $(p_{i},q_{i})$, $i=1,2,$ there exists a constant $C=C(|I|,q_{1})$ such that for all intervals $I\ni 0,$

[TABLE]

where $p_{i}^{\prime}$ and $q_{i}^{\prime}$ are Hölder conjugates of $p_{i}$ and $q_{i}$ respectively.

Proposition 5.2**.**

Let $0<\gamma<\text{min}\{2,d\},d\in\mathbb{N}$. Assume that $(\psi_{01},...,\psi_{0N})\in\left(L^{2}(\mathbb{R}^{d})\right)^{N}.$ Then

1. (i)

There exists a unique global solution of (1.4) such that

[TABLE]

*In addition, its *$L^{2}-$norm is conserved,

[TABLE]

and for all admissible pairs $(p,q),$ and $(\psi_{1},...,\psi_{N})\in\left(L_{loc}^{p}(\mathbb{R},L^{q}(\mathbb{R}^{d}))\right)^{N}.$ 2. (ii)

There exists a unique global solution of (1.5) such that

[TABLE]

*In addition, its *$L^{2}-$norm is conserved,

[TABLE]

and for all admissible pairs $(p,q),$ and $(\psi_{1},...,\psi_{N})\in\left(L_{loc}^{p}(\mathbb{R},L^{q}(\mathbb{R}^{d}))\right)^{N}.$

Proof.

The proof follows from Proposition 5.1 and using ideas similar to the proof of Proposition 4.2. ∎

We can now establish local well-posedness results for (1.4) and (1.5).

Theorem 5.2** (Local well-posedness).**

Let $1\leq p\leq\frac{2d}{d+\gamma}$ and $0<\gamma<d.$ Assume that $\left(\psi_{01},...,\psi_{0N}\right)\in\left(M^{p,p}(\mathbb{R}^{d})\right)^{N}$. Then

1. (i)

There exists $T>0$ depending only on $\|\psi_{01}\|_{M^{p,p}},...,\|\psi_{0N}\|_{M^{p,p}},$ $d$ and $\gamma$ such that (1.4) has a unique local solution

[TABLE] 2. (ii)

There exists $T>0$ depending only on $\|\psi_{01}\|_{M^{p,p}},...,\|\psi_{0N}\|_{M^{p,p}},$ $d$ and $\gamma$ such that (1.5) has a unique local solution

[TABLE]

Proof.

The results are established by applying a standard contraction mapping argument and using Theorem 5.1 and Proposition 3.1. ∎

Sketch proof of Theorem 1.3.

The proof is similar to that of Theorem 1.2 using Proposition 5.1 and Theorem 5.2. ∎

Acknowledgment: D.G. B is very grateful to Professor Kasso Okoudjou for hosting and arranging research facilities at the University of Maryland. D.G. B is thankful to SERB Indo-US Postdoctoral Fellowship (2017/142-Divyang G Bhimani) for the financial support. D.G.B is also thankful to DST-INSPIRE and TIFR CAM for the academic leave. K. A. O. was partially supported by a grant from the Simons Foundation $\#319197$, the U. S. Army Research Office grant W911NF1610008, the National Science Foundation grant DMS 1814253, and an MLK visiting professorship.