Rate of convergence towards equations of Hartree type for mixture condensates with factorized initial data

TL;DR

This paper proves that for a multi-component bosonic system with Coulomb interactions, the many-body Schrödinger evolution converges to the Hartree equations at a rate of O(N^{-1}) in the mean-field regime.

Contribution

It establishes a quantitative convergence rate of the many-body dynamics towards the Hartree equations for mixture condensates with factorized initial states.

Findings

Convergence rate of O(N^{-1}) for the mean-field limit.

Applicable to systems with Coulomb and generalized interactions.

Initial data as a tensor product of factorized states.

Abstract

We consider a system of $p$ components of bosons, each of which consists of $N_{1},N_{2},\dots,N_{p}$ particles, respectively. The bosons are in three dimensions with interactions via a generalized interaction potential which includes the Coulomb interaction. We set the initial condition to describe a mixture condensate, i.e., a tensor product of factorized states. We show that the difference between the many-body Schr\"odinger evolution in the mean-field regime and the corresponding $p$-particle dynamics due to a system of Hartree equation is $O(N^{-1})$ where $N=\sum_{q=1}^{p}N_{q}$.

Equations

$$N=\sum _{q=1}^p{N}_q.$$

$$\mathcal{H}=\underset{q=1}{\overset{p}{⨂}}{L}_s^2({\mathbb{R}}^{3N_q}),$$

$$E^{(q)}\mathcal{O}=(\underset{r<q}{⨂}I)\otimes \underset{}{\mathcal{O}}\otimes (\underset{r>q}{⨂}I)$$

$$E^{(1)}\mathcal{O}$$

$$E^{(2)}\mathcal{O}$$

$$⋮$$

$$E^{(q)}\mathcal{O}$$

$$⋮$$

$$E^{(p)}\mathcal{O}$$

$$H_p=\sum _{q=1}^p{E}^{(q)}{h}_q+\sum _{q<r}{\mathcal{V}}_{qr}$$

$$h_q=\sum _{j=1}^{N_q}(-{\Delta }_{x_{qj}})+\frac{1}{N_q}\sum _{1\le j<k\le N_p}{V}_{qq}(x_{qj}-x_{qk})$$

$${\mathcal{V}}_{qr}=\frac{1}{N}\sum _{j=1}^{N_q}\sum _{k=1}^{N_r}{V}_{qr}(x_{qj}-x_{rk}).$$

$$V_{qr}^2\le K(1-\Delta )$$

$$O(N)=\sum _{q=1}^p\frac{1}{N_q}\left(\genfrac{}{}{0ex}{}{N_q}{2}\right)+\frac{1}{N}\frac{N(N-1)}{2}.$$

$${\psi }_{\mathbf{N}}=\underset{q=1}{\overset{p}{⨂}}(\prod _{j=1}^{N_q}{u}_q)$$

$$e^{-\mathrm{i}{\mathcal{H}}_{\mathbf{N}}{\psi }_{\mathbf{N}}}:={\psi }_{\mathbf{N},t}\simeq \underset{q=1}{\overset{p}{⨂}}(\prod _{j=1}^{N_q}{u}_{q,t})$$

$$i{\partial }_t{u}_{q,t}=-\Delta u_{q,t}+(V_{qq}\ast |u_{q,t}{|}^2)u_{q,t}+\sum _{r=1}^p{c}_{qr}(V_{qr}\ast |u_{q,t}{|}^2)u_{q,t}$$

$${\gamma }_{\mathbf{N},t}^{\mathbf{I}}={\mathrm{Tr}}_{\mathbf{N}-\mathbf{I}}|{\psi }_{\mathbf{N},t}⟩⟨{\psi }_{\mathbf{N},t}|,$$

$$\begin{array}{rl}{\gamma }_{\mathbf{N},t}^{\mathbf{I}}(\mathbf{x};{\mathbf{x}}^{\prime })=& {\int }_{{\prod }_{q=1}^p{\mathbb{R}}^{3(N_q-1)}}\mathrm{d}\mu \overline{{\psi }_{\mathbf{N},t}}{\psi }_{\mathbf{N},t}\end{array}$$

$$\mathrm{d}\mu =\prod _{q=1}^p\prod _{n_q=2}^{N_q}\mathrm{d}{x}_{q{n}_q}.$$

$$\underset{N_q\to \infty }{\lim }\mathrm{Tr}\left|{\gamma }_{\mathbf{N},t}^{\mathbf{I}}-|{\mathbf{u}}_t⟩⟨{\mathbf{u}}_t|\right|=0$$

$${\mathbf{u}}_t=\underset{q=1}{\overset{p}{⨂}}{u}_{q,t}.$$

$$\mathrm{Tr}\left|{\gamma }_{N,t}^{(1)}-|{\phi }_t⟩⟨{\phi }_t|\right|$$

$$V_{qr}^2\le K(1-\Delta )$$

$$c_q=N_q/N$$

$${\psi }_t=e^{-it{\mathcal{H}}_{\mathbf{N}}}\psi$$

$$\operatorname{Tr}\,\Big{|}\,\gamma_{\mathbf{N},t}^{\mathbf{I}}-|\mathbf{u}_{t}\rangle\langle\mathbf{u}_{t}\,\Big{|}\leq C\left((\min_{q}c_{q})^{-1}p^{Cp}e^{Kt}\right)\cdot\frac{1}{N}$$

$$\operatorname{Tr}\,\Big{|}\,\gamma_{\mathbf{N},t}^{\mathbf{I}}-|\mathbf{u}_{t}\rangle\langle\mathbf{u}_{t}|\,\Big{|}\leq Ce^{Kt}\cdot\frac{(\log N)^{C}}{N}$$

$$\left|\frac{N_r}{N}-c_{qr}\right|\le \frac{C}{N^{2/p}}$$

$$h_q=\sum _{j=1}^{N_q}\left(-\frac{{\Delta }_{x_{qj}}}{m_q}\right)+\frac{1}{N_q}\sum _{1\le j<k\le N_p}{V}_{qq}(x_{qj}-x_{qk})$$

$$\sum _{j=1}^{N_q}\left(-\frac{{\Delta }_{x_{qj}}}{m_q}\right)$$

$$\frac{1}{N_q}\sum _{1\le j<k\le N_p}{V}_{qq}(x_{qj}-x_{qk}).$$

$$\mathbf{N}=(N_1,\dots ,N_p)\in {\mathbb{N}}^p\text{and}\mathbf{N}=\prod _{q=1}^p{N}_q.$$

Full text

1 Introduction

Bose-Einstein condensation (BEC), which shows macroscopic quantum phenomena, has been explored widely by both theorists and experimentalists. People have been observing the dynamical evolution of the mixture condensates [22, 31]. The first experimental realization of BEC was obtained with a single component element, typically ${}^{87}\mathrm{Rb}$ [23, 24, 33, 39]. Moreover, BEC may consist of different components of bosons. Two-component BEC, such as ${}^{41}\mathrm{K}$-${}^{87}\mathrm{Rb}$ [36], ${}^{41}\mathrm{K}$-${}^{85}\mathrm{Rb}$ [37], ${}^{39}\mathrm{K}$-${}^{85}\mathrm{Rb}$ [32] and ${}^{85}\mathrm{K}$-${}^{87}\mathrm{Rb}$ [41] ${}^{87}\mathrm{Rb}$-${}^{133}\mathrm{Cs}$[38], was also successfully reported. Furthermore, in [38], Gonzalez observed an absorption images of triply degenerate quantum gases of ${}^{41}\mathrm{K}$-${}^{40}\mathrm{K}$-${}^{6}\mathrm{Li}$ [49]. If multiple components of bosons are interacting, one may consider both intra-component and inter-component interactions.

We are interested in the time-evolution of a condensate of mixture of gases with $p$ components. Here, the number of components, $p$, can be any positive integer. We assume that the gas is in a state of complete condensation in each component, i.e., fully factorized. We claim that under suitable conditions on the density and the interactions between particles, under which the condensed system with $p$ components of bosons, is condensed at time $t\geq 0$. Additionally, the orbitals of each component are similar to the solutions of the system of equations (1.3).

In order to consider a mixture condensate with $N$ particles consisting of $p$ components of bosons in three dimensions, we first label $q=1,\dots,p$ for each component and let the number of particles of $q$-th type, $N_{q}$, for $q=1,\dots,p$. Thus we have

[TABLE]

The system is in the Hilbert space $\mathcal{H}$ such that

[TABLE]

where $L_{s}^{2}(\mathbb{R}^{3N_{q}})$ denotes the symmetric subspace of $L^{2}(\mathbb{R}^{3N_{q}})$.

To introduce our Hamiltonian, we define an embedding operator $E^{(q)}$ as follows: For an operator $\mathcal{O}$ acting on $L_{s}^{2}(\mathbb{R}^{3N_{q}})$, for $q=1,\dots,p$,

[TABLE]

where $I$ is the identity operator on $L_{s}^{2}(\mathbb{R}^{3N_{q}})$. Note that (i) if $q=1$, then $\bigotimes_{r<q}I=1$ and (ii) if $q=p$, then $\bigotimes_{r>q}I=1$. Thus, $E^{(q)}\mathcal{O}$ can be understood as an operator which is acting on the Hilbert space $\mathcal{H}$. Then $E^{(q)}\mathcal{O}$ acts only on the $q$-th component state with $\mathcal{O}$. For other sectors, it behaves like the identity operator. In other words, one can write

[TABLE]

It describes an operator that is only acting on the $q$-th component particle.

Now, the mean-field Hamiltonian of the system has the form

[TABLE]

where, for $q=1,2,\dots,p$, the $q$-th particle Hamiltonian $h_{p}$ is given by

[TABLE]

and the interaction between the $q$-th particle and the $r$-th particle $\mathcal{V}_{qr}$

[TABLE]

For each $q,r=1,2,\dots,p$, we assume that each interacting potential $V_{qr}$ satisfies the operator inequality

[TABLE]

on $L^{2}(\mathbb{R}^{3})$ for some constant $K>0$. We let $c_{qr}$ be the proportion of the particle number of $r$-th component among all particles, i.e. $N_{r}=c_{qr}N$ for each $q=1,2,\dots,p$.

Note that in front of the interaction potentials, we have different scaling limits $N_{p}^{-1}$ and $N^{-1}$ for intra- and inter- component interactions. We justify these scaling by the following: we have $N=\sum_{q=1}^{p}N_{q}$ kinetic terms and $\frac{1}{2}\binom{N}{2}=\frac{N(N-1)}{2}$ potential terms for the Hamiltonian. If we follow the sailings of the above, the potential energy is

[TABLE]

This is a generalization of the scalings given in [12, Section 1] of a mixture of two-component. A justification for our choice is provided in [35, Section 4].

We consider the $p$-component condensate state $\psi_{\mathbf{N}}\in\mathcal{H}$ of the form

[TABLE]

where $\mathbf{N}=(N_{1},\dots,N_{p})$ and $u_{q}\in H^{1}(\mathbb{R}^{3})$ with $\|u_{q}\|_{L^{2}}=1$.

We expect, as $N_{q}\to\infty$ for all $q=1,\dots,p$, the condensation remains approximately at time $t>0$. In other words, we hope to have

[TABLE]

where $(u_{q,t})_{q=1}^{p}$ is the solution of the system of equations of Hartree type

[TABLE]

with $u_{q,0}=u_{q}$ for all $q=1,\dots,p$.

We show that the difference between the many-body Schr dinger evolution in the mean-field regime and the corresponding equations of Hartree type is $O(N^{-1})$ where $N=\sum_{q=1}^{p}N_{q}$. To precisely describe the meaning of the approximate factorization, we let $\mathbf{I}=(1,1,\dots,1)$. Then, we introduce the following $\mathbf{I}$-particle density tensor

[TABLE]

where $|\psi_{\mathbf{N},t}\rangle\langle\psi_{\mathbf{N},t}|$ denotes the orthogonal projection onto the state $\psi_{\mathbf{N},t}$ and $\operatorname{Tr}_{\mathbf{N}-\mathbf{I}}$ denotes the trace over $(N_{q}-1)$ variables corresponding to the $q$-th factor of $\mathcal{H}$ for all $q=1,\dots,p$. The operator $\gamma_{\mathbf{N},t}^{\mathbf{I}}$ is a non-negative trace class operator on $\bigotimes_{q=1}^{p}L^{2}(\mathbb{R}^{3})$ with integral kernel

[TABLE]

where $\mathbf{x}=(x_{11},x_{21},\dots,x_{p1})$, $\mathbf{x}^{\prime}=(x_{11}^{\prime},x_{21}^{\prime},\dots,x_{p1}^{\prime})$, and $\mathrm{d}\mu$ denotes the infinitesimal product measure for all components of particle except the first particle of each components, i.e.

[TABLE]

Note that if $\|\psi_{\mathbf{N},t}\|_{2}=1$, we have $\operatorname{Tr}\gamma_{\mathbf{N},t}^{\mathbf{I}}=1$.

Thus, to show that the state is in approximately factorized state, it is to show that

[TABLE]

where

[TABLE]

It can be rephrased as $\gamma_{\mathbf{N},t}^{\mathbf{I}}\simeq|\mathbf{u}_{t}\rangle\langle\mathbf{u}_{t}|$ when $N_{q}\to\infty$ for all $q=1,\dots,p$. The limit above describes large systems, i.e. thermodynamic limit. In fact, all of the eigenvalues of $\gamma_{\mathbf{N},t}^{\mathbf{I}}$ are non-negative and $\operatorname{Tr}_{\mathbf{N}}\,\gamma_{\mathbf{N},t}^{\mathbf{I}}=1$.

Historically, the cases with single-component, $p=1$, were studied intensively. First, Spohn [50] proved the convergence, $\operatorname{Tr}|\gamma_{N,t}^{(1)}-|\varphi_{t}\rangle\langle\varphi_{t}||\to 0$ as $N\to\infty$, for bounded potential. Moreover, it was extended by Ginibre and Velo [16, 17]. Using the BBGKY hierarchy technique, Erdős and Yau [15] extended the result in [50] to singular potential (including the Coulomb case). Rodnianski and Schlein in [48], based on coherent state approach, obtained the trace norm difference

[TABLE]

which is bounded by $O(N^{-1/2})$ as well as it goes to zero. Knowles and Pickl [27] considered more singular interaction potentials and obtained similar estimates on the rate of convergence. The proof in [27] is based on the use of projection operators in the $N$-particle space $L_{s}^{2}(\mathbb{R}^{3N})$, and covers a large class of possibly time-dependent external potentials even though it does not gives the same rate of convergence. Grillakis, Machedon, and Margetis [20, 21] observed the second-order corrections. The optimal rate of convergence $O(N^{-1})$ was proved by Chen, Lee, and Schlein in [11] for the Coulomb case. It is known that the rate of convergence $O(N^{-1})$ is optimal in as shown in e.g. [18, 19]. It was later extended in [10] to cover the case $V\in L^{2}+L^{\infty}$. We also remark that Hott [26] pointed out that the same rate of convergence can be obtained in larger space than $H^{1}(\mathbb{R}^{3})$ for $V(x)=\lambda|x|^{-\gamma}$ with $1<\gamma<3/2$. Using time decay estimate and Strichartz estimate, the time dependence of the trace norm bound was investigated in [28]. A similar approach can also be applied to many-body semi-relativistic Schr dinger equations which describe a boson star. Lee [29] provided the optimal rate of convergence $O(N^{-1})$ for Coulomb interaction. Following the approach presented in this article, it is believed that one can obtain a corresponding bound for the semi-relativistic case by exploiting the properties of the mean-field solution. On the other hand, Gross-Pitaevskii dynamics for BEC has also been heavily studied, for example in [4, 5, 6, 13, 14].

The corresponding result for two-component BEC, i.e., $p=2$, has also been established well, e.g. [1, 34, 35, 40]. In [34], Michelangeli and Olgiati derived effective evolution equations rigorously from the many-body Schr dinger equation using counting method developed in [27, 42, 43, 44]. A similar result was obtained by Anapolitanos, Hott, and Hundertmark [1]. Recently, de Oliveira and Michelangeli [12] obtained explicit bound $O(N^{-1/2})$ on the rate of the convergence in the the Coulomb type interaction, assuming a coherent initial state. In this paper, our goal is to improve the bound of the rate of convergence $O(N^{-1})$ for general $p\geq 2$, assuming not a coherent initial condition but a factorized initial condition. The following main theorem gives the desired goal:

Theorem 1.1**.**

Suppose that all the interaction potentials $V_{qr}$ satisfy the operator inequality

[TABLE]

on $L^{2}(\mathbb{R}^{3})$ for some constant $K>0$ for all $q,r=1,\dots,p$. Suppose that each $N_{q}$ is a positive integer, $N=\sum_{q=1}^{p}N_{q}$, and $c_{q}$ is real number that satisfies

[TABLE]

for some constant $D>0$ where $c_{q}\neq 0$ with $\sum_{q=1}^{p}c_{q}=1$. For $\mathbf{N}=(N_{1},\dots,N_{p})$ and $\mathbf{I}=(1,\dots,1)$, let $\gamma_{\mathbf{N},t}^{\mathbf{I}}$ be the reduced density operator associated to the solution

[TABLE]

with $\psi=\bigotimes_{q=1}^{p}u_{q}^{\otimes N_{q}}=:\mathbf{u}^{\mathbf{N}}$ to the Schr dinger equation. Then,

[TABLE]

for all $t\geq 0$, where $C$ and $K$ are positive constants that do not depend on $N_{q}$, and $u_{q,t}$ are the solutions of the system of equations (1.3) with initial datum $u_{q}$.

Remark 1.2*.*

The number of components $p$ may dependent on $N$. Letting $p=\left\lfloor\log\log N\right\rfloor$ where $\left\lfloor x\right\rfloor$ denotes the largest integer less than or equal to $x$, we get

[TABLE]

for some constant $C$. This result shows that BEC evolves asymptotically through the Hartree dynamics even when the number of components is sufficiently large (formally infinite as $N\to\infty$). This case, we need to have much more particles than the components of bosons. Here, if we have $p$-component BEC for large $p$, then we need to have the order of $\exp(\exp(p))$ particles for each component.

Remark 1.3*.*

Using result i [28], with appropriate initial data, one can generalize interaction potentials $V_{qr}$ appeared in (1.1) and (1.2) to $V(x)=\lambda e^{-\mu|x|}/|x|^{\gamma}$ with $\lambda\in\mathbb{R}$, $\mu\geq 0$, and $0<\gamma<3/2$. Moreover, if all the potentials are changed into the same component, the time dependence can be obtained similarly as in [28]. Using result in [29], one can have a similar result for semi-relativistic case. Thus we expect to have a similar result for boson star.

Remark 1.4*.*

For $p\geq 2$, we let $c_{q}=N_{q}/N$ in (1.9). Instead of this, we can let

[TABLE]

to have the same rate of convergence. Similar assumption was appeared in [12] for $p=2$ case. See Remark 3.2 for more detail.

Remark 1.5*.*

We omit the information about the mass $m_{q}$ of each component. If one have mass $m_{q}$ factors

[TABLE]

by scaling, by scaling to the solution $u_{q}$, one can move the information about mass $m_{q}$ from the kinetic terms

[TABLE]

to potential terms

[TABLE]

Hence through out this paper, we let $m_{q}=1/2$ for all $q=1,\dots,p$.

We derive a system of effective equations of Hartree type by generalizing the methods in Fock space developed by Hepp [25], Ginibre and Velo [16, 17], and Rodnianski and Schlein [48] (see also [2]). A review of these methods for single-component Bose gases can be found in [3]. The methods in Fock space have been studied for dynamical properties of single-component condensates [2, 6, 8, 9, 11, 30, 48]. By generalizing previous results, our result holds for factorized states as initial data (Theorem 1.1) with Coulomb interaction.

The main subject of this paper is $p$-component BEC for given integer $p$. Because of this extension, the analysis natually needs to be extended from the single component Fock space $\mathcal{F}$ to the $p$-tensor product Fock spaces $\mathcal{F}^{\otimes p}$. In this paper, we introduce new and compact notations which is appropriate to deal with generalized $p$-component BEC.

To deal with $p$-component BEC, an operator $J$, which is introduced to bound the trace norm, acting on $\mathcal{H}$ should be understood. Using tensor algebra, for finite $p$, an Hilbert-Schmidt operator $J$ can be interpreted as an infinite sum of simple tensors, i.e., $\sum_{\mathfrak{k}=1}^{\infty}(\bigotimes_{q=1}^{p}J_{q}^{\mathfrak{k}})$. Using this and the definition of Hilbert-Schmidt norm, one may recover the Hilbert-Schmidt norm of the operator $J$ such that $\|J\|_{\mathrm{HS}}^{2}=\sum_{\mathfrak{k}=1}^{\infty}\|\bigotimes_{q=1}^{p}J_{q}^{\mathfrak{k}}\|_{\mathrm{HS}}^{2}.$ The detailed explanation will be given in Section 2.1.

This paper is organized as follows: First, in Section 2, we introduce tensor algebra to describe operators on bosonic Fock space for mixture condensation. After introducing tensor algebra we introduce bosonic Fock space for mixture condensation, we omit the definitions and properties of Fock space for the non-mixture cases but we cite previous articles to avoid lengthy paper. Using tensor algebra introduced in Section 2.1, it is possible to generalize the definitions and properties of Fock space in Section 2.2. It also helps us to generalize the techniques provided in the previous works to the $p$-component case. We prove our theorem in Section 3. The main strategy is to embed our state into the Fock space and compare the time evolution of our stated and coherent state. For that, we argue that the creation and annihilation operator are similar to the square root of the number operator, which was suggested in [48] for non-mixture case. Using the argument, we provide the proof of Theorem 1.1; this crucial bound is stated in Proposition 3.3, which will be proved in Section 5. In Section 4, we prepare lemmas describing comparison dynamics to prove Proposition 3.3 in Section 5. The lemmas are similar to the lemmas appeared in previous works, e.g. see [9, 11, 29], for the non-mixture case. In Section 6, we provide lemmas to back up the lemmas in Sections 4 and 5.

Notional Remark

For two elements $a,b$ of an inner product space $A$, we denote by $\langle a,b\rangle_{A}$ the inner product of $a$ and $b$. Similarly, we denote the norm of $a$ by $\|a\|_{A}$. We denote the standard $L^{p}$-space in $d$-dimension as $L^{p}(\mathbb{R}^{d},\mathrm{d}x)$ or $L^{p}(\mathbb{R}^{d})$. Let $f\in L^{p}(\mathbb{R}^{d})$, we use $\|f\|_{p}=\|f\|_{L^{p}(\mathbb{R}^{d})}$ for the standard $L^{p}$-norm of $f$. We also let $\|f\|_{H^{p}}=\|f\|_{H^{p}(\mathbb{R}^{d})}$ for the standard Sobolev norm of $f\in H^{p}(\mathbb{R}^{d})$. We denote $\|J\|_{\mathrm{op}}$ as the operator norm of an operator $J$ and $\|J\|_{\mathrm{HS}}$ as the Hilbert-Schmidt norm of an operator $J$. In inequalities we have constants $C$ which may be different line by line. We basically use bold faced roman alphabet to denote vectors in $\mathbb{R}^{p}$ or elements in $\mathcal{F}^{\otimes p}$ and we use bold faced italic alphabet to denote the product of all component s of a vector in $\mathbb{R}^{p}$. For example, we denote

[TABLE]

The detailed description will be shown in the article.

2 Preliminaries

2.1 Review on Tensor Algebra

This section is to review tensor algebra to understand following sections which describes the Fock space of mixture states. Moreover, to apply the idea to bound the trace norm, we need to introduce the lemmas given in this section.

Let $H_{1}$ and $H_{2}$ be Hilbert spaces endowed with scalar products $\langle\cdot,\cdot\rangle_{H_{1}}$ and $\langle\cdot,\cdot\rangle_{H_{2}}$ respectively. Then inner product of tensor product is defined by

[TABLE]

for each $f_{1},f_{2}\in H_{1}$ and $g_{1},g_{2}\in H_{2}$.

This leads us to a standard norm $\|\cdot\|_{H_{1}\otimes H_{2}}$ derived by the inner product such that

[TABLE]

For Hilbert spaces $H_{q}$ for $q=1,\dots,p$, we generalize above properties by induction for $\bigotimes_{q=1}^{p}H_{p}$.

The following result can be understood by reading [7, Chapter II 4.4 Proposition 4] and [46, Section XIII.9]

Lemma 2.1**.**

Let $H$ be a tensor product of two Hilbert spaces $H_{1}$ and $H_{2}$, i.e. $H=H_{1}\otimes H_{2}$. Then given a Hilbert–Schmidt operator $J$ on $H$, it is a limit of linear combination of simple tensors $J_{1}^{\mathfrak{k}}\otimes J_{2}^{\mathfrak{k}}$ for $J_{1}^{\mathfrak{k}}\in L(H_{1})$ and $J_{2}^{\mathfrak{k}}\in L(H_{2})$ for $\mathfrak{k}\in\mathbb{N}$, i.e.,

[TABLE]

Proof.

We will show that there exists a sequence in $L(H_{1})\otimes L(H_{2})$ weakly convergent to $J$. Let $\{e_{i}\otimes e_{j}\}_{i,j=1}^{\infty}$ be an orthogonal basis of $H_{1}\otimes H_{2}$. For any $n$ define the truncated operator $J_{n}$ such that

[TABLE]

Then

[TABLE]

By rearranging $i,j,i^{\prime},$ and $j^{\prime}$ by $\mathfrak{k}$ and letting

[TABLE]

we get

[TABLE]

For more detail, see [7, Chapter II 4.4 Proposition 4].

For any $a\otimes b,c\otimes d\in H_{1}\otimes H_{2}$,

[TABLE]

Thus, as $n\to\infty,$ the boundedness of $J$ leads that

[TABLE]

For more detaeil, see [46, Section XIII.9].

Since the linear combination of $\{e_{i}\otimes e_{j}\}_{i,j=1}^{\infty}$ is dense in $H\otimes H$, we get the conclusion. ∎

Proposition 2.2**.**

A bounded linear Hilbert-Schmidt operator $J$ defined on $\mathcal{F}^{\otimes p}$ can be represented as

[TABLE]

such that $J\left(\bigotimes_{q=1}^{p}\psi_{q}\right)=\sum_{\mathfrak{k}=1}^{\infty}\left(\bigotimes_{q=1}^{p}J_{q}^{\mathfrak{k}}\right)\left(\bigotimes_{q=1}^{p}\psi_{q}\right)=\sum_{\mathfrak{k}=1}^{\infty}\left(\bigotimes_{q=1}^{p}J_{q}^{\mathfrak{k}}\psi_{q}\right)$ for $\psi_{q}\in\mathcal{F}$. Moreover,

[TABLE]

Proof.

First we consider the case with $p=2$. The proof for the linear combination is straight from Lemma 2.1. It is left to show the operator norm identity (2.2). Using (2.1), by defining

[TABLE]

it is possible to represent

[TABLE]

Then noting that $(J_{i^{\prime}i}\otimes J_{j^{\prime}j})^{*}$ and $(J_{i^{\prime}i}\otimes J_{j^{\prime}j})$ are positive operators,

[TABLE]

By induction we get the result for any positive integer $p\geq 2$. ∎

2.2 Fock space formalism for mixture condensation with $p$-component

of bosons

To avoid lengthy paper, we just cite the single-component Fock space formalism for previous works (for example [11, 48, 28, 10]).

We now consider $p$-copies of the bosonic Fock space $\mathcal{F}=\bigoplus_{n=0}^{\infty}L^{2}(\mathbb{R}^{3})^{\otimes_{s}n}$. In our model, the state space for the $p$-component BEC is in

[TABLE]

Here, we are using the standard tensor product of $p$-Hilbert spaces. We will often use the standard construction of tensor product of operators, as described in [45].

We introduce an (extended) embedding operator $\mathcal{E}^{(q)}$ as follows: For an operator $\mathcal{O}$ acting on $\mathcal{F}$,

[TABLE]

for $q=1,\dots,p$. Note that (i) if $q=1$, then $\bigotimes_{r<q}I=1$ and (ii) if $q=p$, then $\bigotimes_{r>q}I=1$. In other words,

[TABLE]

For example, one can embed an operator $\mathcal{O}$ on $\mathcal{F}$ to an operator on $\mathcal{F}^{\otimes p}$ by the following ways:

[TABLE]

[TABLE]

or

[TABLE]

Similar to $E^{(q)}$, the extended embedding operator $\mathcal{E}^{(q)}$ can be understood as it embeds an operator $\mathcal{O}$ on $\mathcal{F}$ to an operator on $\mathcal{F}^{\otimes p}$. It describes an operator that is only acting on the $q$-th component particle.

For $f\in L^{2}(\mathbb{R}^{3})$, we define creation and annihilation operators on each factor of $\mathcal{F}^{\otimes p}$ by using the embedding operator and the standard creation and annihilation operators ($a^{*}(f)$ and $a(f)$ respectively) on $\mathcal{F}$ defined in [10, 11, 28, 48],

[TABLE]

For $\mathbf{f}=(f_{1},f_{2},\dots,f_{p})$ with $f_{q}\in L^{2}(\mathbb{R}^{3})$ for $q=1,2,\dots,p$, we define the product of all creation and annihilation operators for all $q=1,\dots,p$,

[TABLE]

The corresponding operator valued distributions are given by

[TABLE]

It is useful to consider self-adjoint operators

[TABLE]

and

[TABLE]

for $Q\subset\{1,\dots,p\}$.

There are several commutation relations for the operators $a^{(q)*}(f)$ and $a^{(q)}(f)$. The only commutators that are not equal to zero are

[TABLE]

Consequently,

[TABLE]

Using the operator-valued distributions, we let $\mathcal{N}^{(q)}$ to denote the number of particles of component $q$, i.e.

[TABLE]

Then, we define

[TABLE]

It leads us to have such identity

[TABLE]

where $\mathcal{I}$ is the identity operator which can be written as

[TABLE]

Remark 2.3*.*

We want to remark that $\mathcal{N}_{\mathrm{total}}+p\mathcal{I}$ is a generalization of $\mathcal{N}+1$ given in previous works for example in [48] and $(\mathcal{N}+1)\otimes I+I\otimes(\mathcal{N}+1)$ in [12]. For each cases, if we put $p=1$ or $p=2$, it matches the definition.

We also generalize projection operator $P_{n}$. For each non-negative integers $n_{q}$, let $\mathbf{n}=(n_{1},\dots,n_{p})$. We introduce the projection operator $\mathcal{P}_{\mathbf{n}}$ onto the $\mathbf{n}$-particle sector of the Fock space $\mathcal{F}^{\otimes p}$,

[TABLE]

for $\psi=\bigotimes_{q=1}^{p}\psi_{q}=(\psi_{1}^{(0)},\psi_{1}^{(1)},\dots)\otimes(\psi_{2}^{(0)},\psi_{2}^{(1)},\dots)\otimes\dots\in\mathcal{F}$. For simplicity, with slight abuse of notation, we will use $\bigotimes_{q=1}^{p}\psi_{q}^{(n_{q})}$ (or more compactly $\psi^{(\mathbf{n})}$) to denote $\mathcal{P}_{\mathbf{n}}(\psi)$.

For this space we define inner product of $\psi=\bigotimes_{q=1}^{p}\psi_{q}$ and $\phi=\bigotimes_{q=1}^{p}\phi_{q}$ by

[TABLE]

This induces natural norm on $\mathcal{F}^{\otimes p}$ which will be denoted by $\|\cdot\|_{\mathcal{F}^{\otimes p}}$.

In general, for every Hilbert-Schmidt operator $J$ defined on the $\mathbf{I}$-particle sector, we have a representation

[TABLE]

The second quantization $d\Gamma(J)$ of $J$ is the operator on $\mathcal{F}^{\otimes p}$ whose action on the $\mathbf{n}$-particle sector is given by

[TABLE]

where $J_{qj}^{\mathfrak{n}}=1\otimes 1\otimes\cdots\otimes 1\otimes J_{q}^{\mathfrak{n}}\otimes 1\otimes\cdots$ are the operators acting only on the $j$-th particle of $q$-th component. If $J$ has a kernel $J(\mathbf{x};\mathbf{x}^{\prime})$, then $d\Gamma(J)$ can be expressed as $\sum_{\mathfrak{k}=1}^{\infty}\bigotimes_{q=1}^{p}d\Gamma(J_{q}^{\mathfrak{k}})$ due to the following:

Since $J$ is a bounded linear operator, it can be written in the tensor product form such that

[TABLE]

Then

[TABLE]

which also can be expressed

[TABLE]

for each $q=1,\dots,p$. We have

[TABLE]

The creation and annihilation operators for $q$-th component are bounded with respect to $\mathcal{E}^{(q)}\mathcal{N}^{1/2}$. The precise statement is given in the following lemma. The proof of the lemma follows from the corresponding well-known result for creation and annihilation operators on each factor of $\mathcal{F}\otimes\mathcal{F}$ (see [48] for a proof).

Lemma 2.4**.**

Let $\mathbf{f}:=\bigotimes_{q=1}^{p}f_{q}\in L^{2}(\mathbb{R}^{3})^{\otimes p}$. For any $\psi\in\mathcal{F}^{\otimes p}$, we have

[TABLE]

Lemma 2.5**.**

For $\alpha_{q}>0$, let $D\left(\mathcal{N}^{\mathbf{\alpha}}\right)=\{\mathcal{N}^{\mathbf{\alpha}}\psi\in\mathcal{F}^{\otimes p}:\sum_{n_{q}\geq 1}\prod_{q=1}^{p}n_{q}^{2\alpha_{q}}\|\psi_{q}^{(n_{q})}\|_{2}^{2}<\infty\}$ denote the domain of the operator $\mathcal{N}^{\mathbf{\alpha}}$. For any $\mathbf{f}=\bigotimes_{q=1}^{p}f_{q}\in\bigotimes_{q=1}^{p}L^{2}(\mathbb{R}^{3},\mathrm{d}x_{q})$ and any $\psi\in D(\bigotimes_{q=1}^{p}\mathcal{N}^{1/2})$, we have

[TABLE]

and

[TABLE]

Moreover, for any Hilbert–Schmidt $p$-particle operator $J$ on $L^{2}(\mathbb{R}^{3},dx)^{\otimes p}$ and for every $\psi\in D((\mathcal{N}^{1/2}\otimes\mathcal{N}^{1/2}))$, we find

[TABLE]

Proof.

These bounds are standard. A proof of (2.4) can be obtained using Lemma (2.4). As for (2.5), it is to observe that

[TABLE]

By the definition of $\mathcal{N}$ on $\mathcal{F}$,

[TABLE]

which completes the proof. ∎

For $Q\subset\{1,\dots,p\}$, let us denote

[TABLE]

We let $J\mathbf{u}_{t}=(J_{1}u_{1,t},\dots,J_{p}u_{p,t})$ and $J^{(Q)}=\prod_{q\in Q}\mathcal{E}^{(q)}J^{q}$. Then using the same idea proving Lemma 2.5, we have

Lemma 2.6**.**

Let $p$ be a positive integer and let $Q$ be a subset of positive integer such that $Q\subset\{1,\dots,p\}$. Then for $\psi\in\mathcal{F}^{\otimes p}$ and for any nonnegative integer $j$, we have

[TABLE]

and

[TABLE]

The Hamiltonian. We now define the extended Hamiltonian $\mathcal{H}_{\mathbf{N}}$ acting on $\mathcal{F}^{\otimes p}$. First, we observe that the set of all finite sums $\sum_{\mathbf{n}\in\mathbb{Z}_{\geq 0}^{p}}\psi_{q}^{(\mathbf{n})}$ is dense in $\mathcal{F}^{\otimes p}$. Note that the set of all finite linear combinations of products $\psi^{(\mathbf{n})}=\bigotimes_{q=1}^{p}\psi_{q}^{(n_{q})}$, denoted by $\mathcal{D}$, is dense in $\mathcal{F}^{\otimes p}$. We define the Hamiltonian $\mathcal{H}_{\mathbf{N}}$ acting on products $\psi^{(\mathbf{n})}$ by $\mathcal{H}_{\mathbf{N}}\,\psi^{(\mathbf{n})}=\mathcal{H}_{\mathbf{N}}^{\mathbf{n}}\,\psi^{(\mathbf{n})}$ where

[TABLE]

with

[TABLE]

for $q=1,\dots,p$, and

[TABLE]

for $q,r=1,\dots,p$.

We then extend $\mathcal{H}_{\mathbf{N}}$ to $\mathcal{D}$ by linearity. With the assumption/hypothesis on $V_{qr}$ in Theorem 1.1, the Hamiltonian $\mathcal{H}_{\mathbf{N}}$ on $\mathcal{D}$ gives rise to a self-adjoint operator [47].

Using the operator-valued distributions $a_{x}$ and $a_{x}^{*}$, the Hamiltonian $\mathcal{H}_{\mathbf{N}}$ can be written as

[TABLE]

where

[TABLE]

and

[TABLE]

The Hamiltonian $\mathcal{H}_{\mathbf{N}}$ conserves the number of particles in each factor of $\mathcal{F}^{\otimes p}$. In fact, it is simple to verify that, for $q=1,\dots,p$,

[TABLE]

Moreover we have

[TABLE]

Furthermore, for fixed $\mathbf{N}=(N_{1},\dots,N_{p})$, the subspace

[TABLE]

is invariant by $\mathcal{H}_{\mathbf{N}}$, and the Hamiltonian $\mathcal{H}_{\mathbf{N}}$ restricted to $\mathcal{S}_{\mathbf{N}}$ is equal to $H_{\mathbf{N}}$. Therefore, for initial data in $\mathcal{S}_{\mathbf{N}}$, the time evolution generated by $\mathcal{H}_{\mathbf{N}}$ reduces to the time evolution generated by $H_{\mathbf{N}}$.

The reduced density operator. For $\psi\in\mathcal{F}^{\otimes p}$, we define the reduced density operator $\gamma_{\psi}^{\mathbf{I}}$ as the operator on $L^{2}(\mathbb{R}^{3})^{\otimes p}$ determined by the kernel

[TABLE]

We observe that $\text{Tr}_{\mathbf{I}}\gamma_{\psi}^{\mathbf{I}}=1$. If $\psi$ is in the subspace $\mathcal{S}_{\mathbf{N}}$ of fixed number of particles, the above definition is reduced to the definition of $\gamma_{\mathbf{N},t}^{\mathbf{I}}$ given in (1.5).

For $f_{q}\in L^{2}(\mathbb{R}^{3})$, we define

[TABLE]

which is an operator on $\mathcal{F}^{\otimes p}$. We set a vacuum state $\omega=\Omega^{\otimes p}$ . Thus $\mathcal{W}(\mathbf{f})\,\omega$ is a tensor product of coherent states (or simply coherent state).

In the following lemma, we have some important properties of the operator $\mathcal{W}(\mathbf{f})$ and the coherent state $\mathcal{C}(\mathbf{f})=\mathcal{W}(\mathbf{f})\,\omega$. These properties follow easily from the corresponding well-known properties of Weyl operators (see [48], for example). The coherent state $\mathcal{C}(\mathbf{f})$ can also be written in terms of the Weyl operator as

[TABLE]

We collect the useful properties of the Weyl operator and the coherent states in the following lemma.

Lemma 2.7**.**

Let $\omega=\Omega^{\otimes p}$ and $f_{q}\in L^{2}(\mathbb{R}^{3})$ for all $q=1,\dots,p$.

The operator $\mathcal{W}(\mathbf{f})$ is unitary and

[TABLE] 2. 2.

We have

[TABLE] 3. 3.

For $\bm{\alpha}=(\alpha_{1},\dots,\alpha_{p})\in\left\{0,1\right\}^{p}$, we have

[TABLE]

Schr dinger dynamics. We introduce Hardamard product $\odot:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ such that for $\mathbf{u},\mathbf{v}\in\mathbb{R}^{d}$ with $\mathbf{u}=(u_{1},\dots,u_{d})$ and $\mathbf{v}=(v_{1},\dots,v_{d})$, the Hardamard product of $\mathbf{u}$ and $\mathbf{v}$ is given by

[TABLE]

We denote for $r\in\mathbb{R}$

[TABLE]

and for $\mathbf{r}=(r_{1},\dots,r_{d})\in\mathbb{R}^{d}$

[TABLE]

For $u_{q}\in H^{1}(\mathbb{R}^{3})$ with $\|u_{q}\|_{L^{2}}=1$ , we have

[TABLE]

and denote by $t\mapsto\mathbf{u}_{t}=e^{-\mathrm{i}t\mathcal{H}_{\mathbf{N}}}\mathbf{u}$ the solution to the Schr dinger equation $\mathrm{i}\partial_{t}\mathbf{u}_{t}=\mathcal{H}_{\mathbf{N}}\mathbf{u}_{t}$ with initial condition $\mathbf{u}_{t=0}=\mathbf{u}$. We will study the family of solutions $\{\mathbf{u}_{t}=(u_{1,t},\dots,u_{p,t})\}$ as $N_{q}$ go to infinity.

Finally, we collect lemmas on the Weyl operator acting on a state with fixed number of particles, which will be used in Section 6. Define

[TABLE]

We note that $C^{-1}N^{1/4}\leq d_{N}\leq CN^{1/4}$ for some constant $C>0$ independent of $N$, which can be easily checked by using Stirling’s formula. We denote

[TABLE]

and

[TABLE]

3 Proof of the Main Result

We now turn to the proof of our main theorem (Theorem 1.1). We first introduce unitary operators and their generators to give the proof.

Remark 3.1*.*

Similar to the argument given in [48, Remark 1.4], we will prove the theorem not for the trace norm but for the Hilbert-Schmidt norm, i.e.,

[TABLE]

Note that $\operatorname{Tr}\gamma_{\mathbf{N},t}^{\mathbf{I}}=1$, all the eigenvalues of $\gamma_{\mathbf{N},t}^{\mathbf{I}}$ are non-negative, and $|\mathbf{u}_{t}\rangle\langle\mathbf{u}_{t}|$ is a rank $p$ projection. Then the operator $\gamma_{\mathbf{N},t}^{\mathbf{I}}-|\mathbf{u}_{t}\rangle\langle\mathbf{u}_{t}|$ can have at most $p$ negative eigenvalues so that the set of negative eigenvalues $\Lambda_{\mathrm{neg}}$ have at most $p$ elements. Using

[TABLE]

we have

[TABLE]

Thus one may see that

[TABLE]

Using Cauchy-Schwarz inequality,

[TABLE]

3.1 Unitary operators and their generators

Fluctuation dynamics. We define $\alpha(t)=\int_{0}^{t}\mathrm{d}s\,\varphi_{\mathbf{N}}(s)$, where $\varphi_{\mathbf{N}}$ is a real-valued function that we will choose later. Let $\mathbf{u}_{t}=(u_{1,t},\dots,u_{p,t})$ be the solutions to the equations of Hartree type (1.3). For $t,s\in\mathbb{R}$, we set

[TABLE]

We refer to the operator $\mathcal{U}(t,s)$ as the fluctuation dynamics. We abbreviate

[TABLE]

Thus, we may write $\mathcal{U}_{t,s}=e^{\mathrm{i}(\alpha(t)-\alpha(s))}\mathcal{W}_{t}e^{-\mathrm{i}(t-s)\mathcal{H}_{\mathbf{N}}}\mathcal{W}_{s}$. We also define

[TABLE]

Using the above definitions, it is simple to verify that

[TABLE]

For each $s\in\mathbb{R}$, the fluctuation dynamics satisfies the equation

[TABLE]

where

[TABLE]

with

[TABLE]

and

[TABLE]

Then

[TABLE]

and

[TABLE]

Remark 3.2*.*

Note that $\mathcal{L}_{0}(t)=0$ because we define $c_{qr}=N_{r}/N$. Here, however, we put generalized definition of it to leave a little room for changing condition on $c_{qr}$. For example, one can assume such that

[TABLE]

for some $C$ for all $q,r=1,\dots,p$.

Let $u_{q,t}$ be the solution of the equations of Hartree type (1.3). If we perturb $c_{qr}$ by $O(N^{-1})$ and denote it $\widetilde{c}_{qr}$, and let $\widetilde{u}_{q,t}$ be the solution of the perturbed equations of Hartree type.

[TABLE]

where, the last line we used that

[TABLE]

Then, using Cauchy-Schwarz inequality, our assumption about $V_{qr}$ in (1.8), and the fact that both $u_{q,t}$ and $\widetilde{u}_{q,t}$ is bounded $H^{1}$-norm independent to time $t$,

[TABLE]

This implies from Gr nwall inequality that

[TABLE]

Thus the $L^{2}$-norm difference is of $O(N^{-1})$. Hence we have

[TABLE]

Let $\widetilde{\mathcal{L}}=\mathcal{L}_{0}(t)+\mathcal{L}_{2}(t)+\mathcal{L}_{4}(t)$ and define the unitary operator $\widetilde{\mathcal{U}}\left(t;s\right)$ by

[TABLE]

Since $\widetilde{\mathcal{L}}$ does not change the parity of the number of particles,

[TABLE]

We refer to [29, Lemma 8.2] for a rigorous proof of (3.3).

3.2 Proof of Theorem 1.1

To have compact notation, we denote the products over indices from $1$ to $p$ such that

[TABLE]

[TABLE]

and

[TABLE]

For any nonempty subset $Q$ of positive integers, we let the products over all indices in the set $Q$ using the following notation

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

If $Q=\emptyset$, we define all the products above as $1$.

Proposition 3.3**.**

Suppose that the assumptions in Theorem 1.1 hold. For a Hilbert–Schmidt operator $J$ on $L^{2}(\mathbb{R}^{3})$, let

[TABLE]

Then, there exist constants $C$ and $K$, depending only on $\sup_{|s|\leq t}\|u_{q,s}\|_{H^{1}}$, such that

[TABLE]

for all subsets $Q$ and $R$ of $\{1,\dots,p\}$ such that $|Q|+|R|\geq 1$.

A proof of Propositions 3.3 will be given later in Section 5. With Propositions 3.3, we now prove Theorem 1.1.

Proof of Theorem 1.1.

Recall that

[TABLE]

From the definition of the creation operator in (2.3), we can easily find that

[TABLE]

where the $u_{q}^{\otimes N_{q}}$ on the left-hand side is in the $N_{q}$-th sector of the Fock space $\mathcal{F}$ of the $q$-th component particle. Recall that $P_{N}$ is the projection onto the $N$-particle sector of the Fock space $\mathcal{F}$. From the property of Weyl operator, we find that

[TABLE]

Since $\mathcal{H}_{\mathbf{N}}$ does not change the number of particles, we also have that

[TABLE]

To simplify it further, we use the relation

[TABLE]

(and an analogous result for the creation operator) to obtain that

[TABLE]

Thus,

[TABLE]

with

[TABLE]

for $Q$ and $R$ be subsets of $\{1,\dots,p\}$ such that $Q\cup R\neq\emptyset$.

Recall the definition of $E_{t}^{Q,R}(J)$ in Proposition 3.3 and the definition of $\mathbf{I}:=(1,\dots,1)\in\mathbb{Z}^{p}$. For any $\mathbf{I}$-particle Hilbert-Schmidt operator $J$ on $\bigotimes_{q=1}^{p}L^{2}(\mathbb{R}^{3})$, we have

[TABLE]

Thus, from Propositions 3.3, we find that

[TABLE]

Since the space of compact operators is the dual to that of the trace class operators, and since $\gamma_{\mathbf{N},t}^{\mathbf{I}}$ and $\left|\mathbf{u}_{t}\right\rangle\left\langle\mathbf{u}_{t}\right|$ are Hermitian, using Remark 3.1

[TABLE]

which concludes the proof of Theorem 1.1. ∎

4 Comparison dynamics

This section is devoted to giving and proving lemmas.

4.1 Comparison dynamics for $\mathcal{U}$

Lemma 4.1**.**

Suppose that the assumptions in Theorem 1.1 hold. Let $\mathcal{U}\left(t;s\right)$ be the unitary evolution defined in (3.1). Then for any $\psi\in\mathcal{F}$ and $j\in\mathbb{N}$, there exist constants $C\equiv C(j)$ and $K\equiv K(j)$ such that

[TABLE]

The proof of Lemma 4.1 will be followed by the following argument. The argument for single component case is developed in [48]. For the two-component BEC, see [12, Proposition 3.2]. For $p\geq 2$, $p$-component case will be proved by the following argument which is a generalization of previous works [12, 48].

We introduce a truncated time-dependent generator with fixed $\mathbf{M}>0$ as follows:

[TABLE]

with

[TABLE]

Here $\chi_{q}=\mathcal{E}^{(q)}\chi(\mathcal{N}\leq M_{q})$ where $\chi$ denotes a characteristic function.

We remark that $\mathbf{M}$ will be chosen later such that $\mathbf{M}=\mathbf{N}$. Define a unitary operator $\mathcal{U}^{(\mathbf{M})}$ such that

[TABLE]

The detailed proof will be omitted. The main idea is to follow the steps using the argument in [48] and generalized in [12]. One can easily generalize further for $p$-component case.

Step 1. Truncation with respect to $\mathcal{N}$ with $\mathbf{M}>0$.

Lemma 4.2**.**

Let $t>s$ and $\psi\in\mathcal{F}^{\otimes p}$. Then

[TABLE]

To prove this lemma, keep in mind that $N_{r}/N_{q}$, $N/N_{q}$, and $\sqrt{N_{q}N_{r}}/N$ are bounded for any $q,r=1,\dots,p$ because

[TABLE]

Proof.

Fix any $j\in\mathbb{N}$. Then we want to bound

[TABLE]

by

[TABLE]

for some $C_{j}$ to use Gr nwall inequality.

[TABLE]

where $\mathbf{k}=(k_{1},\dots,k_{p})$, $\mathrm{add}\mathbf{k}=\sum_{q=1}^{p}k_{q}$, and the multinomial coefficient

[TABLE]

and

[TABLE]

Here, we denote

[TABLE]

for $\mathbf{k}=(k_{1},\dots,k_{p})$.

We now want to prove that for all $\ell=1,2,3$,

[TABLE]

Because it implied form Gr nwall inequality, we will have the conclusion.

To have the bound (4.1), we need to note that we have commutators for single component case,

[TABLE]

Consequently,

[TABLE]

and

[TABLE]

Using these and denoting $\mathbf{j}=(j_{1},\dots,j_{p})$ and$\mathbf{k}_{q}$ a vector such that the $q$-th sector is $k$ and all the other $r$-th sector is $j_{r}$, we find that

[TABLE]

Consequently,

[TABLE]

and suppose $q<q^{\prime}$

[TABLE]

We now want to estimate

[TABLE]

for $\ell=1,2,3$. To have a bound of this, we need to bound each $|\mathcal{Q}_{\ell,q,r}|$. Using the above and denoting $\boldsymbol{\ell}_{q}$ a vector such that the $q$-th sector is $\ell$ and all the other $r$-th sector is $j_{r}$

[TABLE]

Then, using the Cauchy-Schwarz inequality and Lemma (2.6),

[TABLE]

Thus, using the assumption on $V_{qr}$ in (1.8), the first sum is bounded by

[TABLE]

The second term is bounded by, also using the assumption on $V_{qr}$ in (1.8),

[TABLE]

Applying the Cauchy-Schwarz inequality again and rewriting with multinomial coefficient, we obtain

[TABLE]

Similar bound can be obtained for

[TABLE]

This upper bound contributes to the first term on the right hand side of (4.1).

Now it remains to bound

[TABLE]

Noting that $[\chi_{q},a_{x}^{(r)}]=0$,

[TABLE]

and

[TABLE]

with similar approach to bound $|\mathcal{Q}_{1,q,q^{\prime}}|$, we can bound that

[TABLE]

∎

Step 2. Weak bounds on the $\mathcal{U}$ dynamics

Lemma 4.3**.**

For arbitrary $t,s\in\mathbb{R}$ and $\psi\in\mathcal{F}^{\otimes p}$, we have

[TABLE]

Moreover, for every $j\in\mathbb{N}$,

[TABLE]

for appropriate constant $C_{j}$.

Proof.

Using Lemma 2.7, one may calculate

[TABLE]

Then using the Cauchy-Schwarz inequality, Lemma 2.6, and Lemma 2.7, we have

[TABLE]

Therefore, we have

[TABLE]

which is (4.2).

To show (4.3), we use induction on $j$. Define

[TABLE]

Then

[TABLE]

We argue that

[TABLE]

for some constant $C$. By Cauchy-Schwarz inequality, we have

[TABLE]

with

[TABLE]

Using the Cauchy-Schwarz inequality and Lemma 2.6, we obtain

[TABLE]

These estimates for $A$ and $B$ prove our argument.

[TABLE]

for all $m\in\mathbb{N}$ by following proof is given by performing straightforward calculation.

By induction, it follows that, for all $j\in\mathbb{N}$, there are constants $C_{j}$ and $D_{j}$ such that

[TABLE]

[TABLE]

In fact, this is already proved for $j=1$ by (4.5) and (4.6). Suppose that (4.7) and (4.8) hold true for all $j<k$. We will prove the estimates for $j=k$.

For (4.7), we have

[TABLE]

Thus

[TABLE]

as desired. Here, we used the operator inequality

[TABLE]

which follows using the commutator expansion

[TABLE]

For (4.8), using (4.5) and (4.7), we obtain

[TABLE]

as desired. Therefore, we have proved (4.7) and (4.8).

Let us finish the proof of Lemma (4.3). First we observe that, similarly as we estimated $Z_{t,s}^{2j}$, we can prove that

[TABLE]

for all $j\in\mathbb{N}$. Hence

[TABLE]

This proves (4.3).

Then, using Cauchy-Schwarz inequality, we have

[TABLE]

Thus we get (4.4). ∎

Step 3. Comparison between $\mathcal{U}$ and $\mathcal{U}^{(M)}$

dynamics.

Lemma 4.4**.**

For every $j\in\mathbb{N}$, there exists constant $C_{j}$ and $K_{j}$ such that

[TABLE]

and

[TABLE]

for all $\psi\in\mathcal{F}^{\otimes p}$ and $t>s$.

Proof.

The proof of this lemma is also followed by Lemma 5.4 in [12]. To prove (4.9), we use

[TABLE]

Note that

[TABLE]

where $\chi_{q}^{c}=\mathcal{E}^{(q)}\chi(\mathcal{N}>M_{q}).$

Using these formulae, we write

[TABLE]

where

[TABLE]

Now we bound each $w_{1,q}$, $w_{2,q}$, $w_{3,q,r}$, and $w_{4,q.r}$. First,

[TABLE]

Using Lemma 4.1, 4.2, and 4.3, one may have

[TABLE]

Similarly, one can estimate $w_{2,q,r}$, $w_{3,q,r}$, and $w_{4,q,r}$ and obtain similar bounds. This proves (4.9). One can also obtain (4.10) using similar technique as in the proof of (4.9). ∎

Proof of Lemma 4.1.

From Lemmas 4.2-4.4 with the choice $\mathbf{M}=\mathbf{N}$,

[TABLE]

This proves the desired lemma. ∎

4.2 Comparison dynamics for others

Lemma 4.5**.**

Suppose that the assumptions in Theorem 1.1 hold. Let $\widetilde{\mathcal{U}}$ be the unitary operator defined in (3.2). Then, for any $\psi\in\mathcal{F}$ and $j\in\mathbb{N}$, there exist constants $C\equiv C(j)$ and $K\equiv K(j)$ such that

[TABLE]

Proof.

This will follow and use the proof of [9].

Let $\widetilde{\psi}=\widetilde{\mathcal{U}}(t;s)\psi$. We have

[TABLE]

To bound the list three terms, we investigate it further. It is enough to bound

[TABLE]

Due to the symmetric structure, it is enough to show for the $q$-th term for $q\leq p$ and the last term of (4.11).

Then since $\left\langle\widetilde{\psi},[\mathrm{i}(\mathcal{L}_{2}^{q}(t)+\mathcal{L}_{4}^{q}(t)),\left(\mathcal{E}^{(r)}\left(\mathcal{N}+1\right)^{j}\right)]\widetilde{\psi}\right\rangle=0$ and $\mathcal{L}_{2}^{q}(t)+\mathcal{L}_{4}^{q}(t)$ does not change the number of the particles of the second component,

[TABLE]

For the last term,

[TABLE]

Applying Gr nwall lemma, we conclude that

[TABLE]

Since the proof was symmetric up to the component of particle,

[TABLE]

which proves the desired lemma. ∎

The main difference between the unitary operators $\mathcal{U}$ and $\widetilde{\mathcal{U}}$ comes from the generator $\mathcal{L}_{3}$. In the following lemma, we find an estimate on $\mathcal{L}_{3}$.

Lemma 4.6**.**

Suppose that the assumptions in Theorem 1.1 hold. Then, for any $\psi\in\mathcal{F}$ and $j\in\mathbb{N}$, there exist a constant $C\equiv C(j)$ such that

[TABLE]

Proof.

We basically follow the proof in [29, Lemma 5.3]. For $q,r=1,\dots,p$, let

[TABLE]

Then, by definition, we have

[TABLE]

Then we get

[TABLE]

The first term in (4.12), $\left(\mathcal{N}_{\mathrm{total}}+p\mathcal{I}\right)^{j/2}A_{3}^{(q,q)}(t)$, satisfies for any $\xi\in\mathcal{F}^{\otimes p}$ that

[TABLE]

This leads that

[TABLE]

Hence,

[TABLE]

Since $\xi$ was arbitrary, by applying the assumption on interaction potentials $V_{qr}$ in (1.8), we obtain that

[TABLE]

For the second term of (4.12), using similar argument,

[TABLE]

For (4.13), similarly we get

[TABLE]

and

[TABLE]

Hence, from (4.14), (4.15), (4.16), and (4.17), we get

[TABLE]

which was to be proved. ∎

Finally, we prove the following lemma on the difference between $\mathcal{U}$ and $\widetilde{\mathcal{U}}$.

Lemma 4.7**.**

Suppose that the assumptions in Theorem 1.1 hold. Then, for all $j\in\mathbb{N}$, there exist constants $C\equiv C(j)$ and $K\equiv K(t)$ such that, for any $\mathbf{f}=(f_{1},\dots,f_{p})$ with $f_{q}\in L^{2}\left(\mathbb{R}^{3}\right)$ for all $q=1,\dots,p$,

[TABLE]

Proof.

We follow the proof in [29, Lemma 5.4]. Let

[TABLE]

and

[TABLE]

so that

[TABLE]

We begin by estimating the first term in the right-hand side of (4.18). From Lemma 4.1,

[TABLE]

From Lemma 4.6 and the assumption on interaction potentials $V_{qr}$ in (1.8), we get

[TABLE]

By Lemma 4.5,

[TABLE]

Using the fact that the right hand side does not depend on time $s$ and Lemma 2.5,

[TABLE]

Thus, from Lemma 4.5, we obtain for $\mathcal{R}_{1}(\mathbf{f})$ that

[TABLE]

The study of $\mathcal{R}_{2}(\mathbf{f})$ is similar and we can again obtain that

[TABLE]

Noting that $N_{q}$ are of order $N$,

[TABLE]

for some $c,C>0$. Hence we have both

[TABLE]

and

[TABLE]

This completes the proof of the desired lemma. ∎

5 Proof of Propositions

5.1 Proof of Propositions 3.3

Noting that for disjoint subsets $Q$ and $R$ of $\{1,\dots,p\}$,

[TABLE]

in this section, we prove proposition 3.3 by applying the lemmas proved in the following section.

Proof of Proposition 3.3.

For the proof of Proposition 3.3, we use strategy provided in [29, Lemma 4.2]. Recall that

[TABLE]

Then also recall the definitions of $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ in the proof of Lemma 4.7. Let $\mathcal{R}(\mathbf{f})=\mathcal{R}_{1}(\mathbf{f})+\mathcal{R}_{2}(\mathbf{f})$ so that

[TABLE]

From the parity conservation (3.3),

[TABLE]

for all $p$-dimensional vector $\mathbf{k}\in\mathbb{Z}_{\geq 0}^{d}$ with nonnegative integer components. (See [29, Lemma 8.2] for more detail.) Thus,

[TABLE]

Then

[TABLE]

The first factor is easily bounded by using

[TABLE]

that

[TABLE]

Case 1. $|Q|+|R|\geq 2$

The second factor is bounded by using Lemma 2.6,

[TABLE]

By Lemma 2.6,

[TABLE]

Thus, using $|Q|+|R|\geq 2$,

[TABLE]

This case can be easily bounded. First, using Cauchy-Schwarz inequality,

[TABLE]

Using (5.1), Lemma 6.1, 2.6, and 4.7, we have

[TABLE]

It follows from Lemma Lemma 2.6 that

[TABLE]

which is the concludes the case for $|Q|+|R|\geq 2$.

Case 2. $|Q|+|R|=1$, i.e., one of $Q$ or $R$ is empty and the other has only one element.

For this case either $Q$ or $R$ is empty and the other set has only one element. Hence, $Q\cap R=\emptyset$ and $Q\Delta R$ has also only one element. Let us denote the element $q$, i.e., $\{q\}=Q\Delta R$. Using Lemma 6.2, letting

[TABLE]

with $\ell_{q}=2k_{q}-1$ and the other sectors $\ell_{r}=0$ for all $r\neq q$. Due to the parity conservation of $\widetilde{\mathcal{U}}$, only the $q$-th component particle occupies odd number sectors. The other components are only occupying the zeroth sector because $\phi^{(q)}(J_{q}u_{q,t})$ is acting as an identity for the $r$-th sector for $r\neq q$ and $\widetilde{\mathcal{U}}(t)$ is unitary. This leads that

[TABLE]

The projection operator has come from parity conservation of $\widetilde{\mathcal{U}}$ for each components.

Let $K=\frac{1}{2}N^{\odot 1/3}$so that Lemma 6.2 implies that

[TABLE]

Since

[TABLE]

we get

[TABLE]

For the last factor, from (4.7) and Lemma 2.6, we get

[TABLE]

Then we get

[TABLE]

Similarly, we ontain

[TABLE]

can be bounded using the same argument for the case $|Q|+|R|\geq 2$.

After all, we have

[TABLE]

which is desired conclusion. ∎

6 Lemmas

Lemma 6.1**.**

There exists a constant $C>0$ independent of $N$ such that, for any $\varphi\in L^{2}(\mathbb{R}^{3})$ with $\|\varphi\|=1$, we have

[TABLE]

Proof.

We follow the one particle result in [9, Lemma 6.3]. Let

[TABLE]

for $n_{q}<N_{q}$ for $q=1,\dots,p$, then noting that $\mathbf{n}=(n_{1},\dots,n_{p})$ and $\mathbf{N}=(N_{1},\dots,N_{p}$), we get

[TABLE]

According to [9, Lemma 6.3], we have

[TABLE]

Moreover, for $n_{q}\geq N_{q}$, using (6.16) of [9, Lemma 6.3] the sum of its squares from $N_{q}$ to infinity is bounded by $1$, i.e.,

[TABLE]

Thus,

[TABLE]

This proves the desired lemma. ∎

Lemma 6.2**.**

Let $P_{m}$ be the projection onto the $m$-particle sector of the Fock space $\mathcal{F}$ for a non-negative integer $m$. Then, for any non-negative integer $k_{q}\leq(1/2)N^{1/3}$,

[TABLE]

[TABLE]

Proof.

Proof can be found in [29, Lemma 7.2]. ∎

Acknowledgments

The author would like to thank gratefully Ji Oon Lee for many motivational and helpful discussions.