###### Abstract

The success of Gross–Pitevskii and Bogoliubov theories in the description of
large systems of interacting bosons led to a substantial effort into
rigorously deriving these effective theories. In this work we shall review the
related existing literature in the context of dynamics of large bosonic
systems.

##  Chapter 0 Dynamics of interacting bosons: a compact review

Contents

toc

###  1 Introduction

####  1 Setup.

Bose–Einstein Condensation (BEC) is a phenomenon that occurs in systems of
bosonic particles where, at sufficiently low temperatures, a macroscopic
fraction of the particles starts to occupy a single quantum state. The idea of
BEC dates back to the works of Bose and Einstein in 1924 [26, 60] in which
they analysed non-interacting systems. Experimentally BEC has been produced
only in 1995 by Cornell, Wieman and Ketterle [55, 93]. Since then, fundamental
questions in the rigorous understanding of condensation and fluctuations
around the condensate in interacting systems have remained unanswered. Some of
them are essential for the understanding of interesting quantum effects that
can be observed even on macroscopic scales (such as, for example,
superfluidity).

In this work we will review the existing results concerning the dynamics of
Bose–Einstein condensates. The physical picture we have in mind is as follows.
Let $\Psi_{N,0}$ be the ground state of $H_{N}^{V}$ given by

$H_{N}^{V}=\sum\limits_{j=1}^{N}(-\Delta_{x_{j}}+V(x_{j}))+\frac{1}{N-1}\sum\limits_{1\leq
j<k\leq N}{w_{N}(x_{j}-x_{k})}.$ (1)

Here $V\in L^{\infty}_{\rm loc}(\mathbb{R}^{3},\mathbb{R})$ satisfying
$V(x)\to\infty$ as $|x|\to\infty$ is an external potential (which models the
trapping of the particles while they are cooled during the experiment) and
$w_{N}$ is the inter-particle potential that describes their interactions (and
could a priori depend on $N$). The Hamiltonian acts on the symmetric space
$\mathfrak{H}^{N}=\bigotimes_{\text{sym}}^{N}L^{2}(\mathbb{R}^{3})$. The
underlying physical space, if not specified otherwise, is assumed to be three
dimensional.

When the external potential $V$ is turned off, $\Psi_{N,0}$ is no longer a
ground state of the Hamiltonian $H_{N}\equiv H_{N}^{V=0}$ and the time
evolution

$\Psi_{N,t}=e^{-itH_{N}}\Psi_{N,0}$

is observed. Although the Schrödinger evolution is linear, its complexity
increases dramatically when $N$ becomes large. In typical experiments $N$ can
be of order $10^{5}$. Therefore, for computational purposes, it is important
to derive effective descriptions for collective behaviour of the quantum
system.

In this contribution we shall review existing results concerning the
approximation of the time-evolved bosonic many-body quantum system.

####  2 Scaling regimes.

From a mathematical point of view, the large number of particles involved will
be modelled by taking the limit $N\to\infty$. Note that the coupling constant
$1/(N-1)$ in front of the interaction is to ensure the kinetic energy is of
the same order (in $N$) as the interaction energy. We could choose $1/N$
instead of $1/(N-1)$ as well. The interaction potential $w_{N}$ given by

$w_{N}(x)=N^{3\beta}w(N^{\beta}x)$

for some $w\in L^{1}(\mathbb{R}^{3})$ is chosen to be of $O(1)$ (as
$N\to\infty$) in the sense that $\int w_{N}=\int w$.

The parameter $\beta$ in the definition of $w_{N}$ characterises different
scaling regimes that correspond to different physical situations. When
$\beta=0$ (the so-called Hartree or mean-field scaling) then the Hamiltonian
models a situation in which there are many but weak collisions between the
particles. This is because the size of the interaction potential is
$O(N^{-1})$ (because of the prefactor $1/(N-1)$ in front of the interaction
term), while the range of the interaction (which is $O(1)$) is much larger
than the mean inter-particle distance $N^{-1/3}$. Therefore, heuristically,
each particle ’sees’ all other particles.

When $\beta>0$, then $w_{N}$ converges formally to the Dirac-delta interaction

$\displaystyle\Big{(}\int_{\mathbb{R}^{3}}w\Big{)}\delta_{0}.$ (2)

As long as $0\leq\beta<1/3$, the range of the interaction potential (which is
$O(N^{-\beta})$) is much larger than the average distance between the
particles and there are many but weak collisions. Therefore, to the leading
order, the interaction potential experienced by each particle can be still
approximated by the effective mean-field potential $\rho*w_{N}$ where $\rho$
is the density of the system. If $\beta>1/3$, then the analysis is expected to
be more complicated due to strong correlations between particles. Despite the
physical difference between the case when $\beta\leq 1/3$ and $\beta>1/3$, the
formal limiting behaviour of the interaction in (2) is the same in both cases.
Therefore, we will call the regime when $\beta\in(0,1)$ the NLS regime as the
limiting effective description of the condensate will be in that case be given
by the nonlinear Schrödinger (NLS) equation.

The case when $\beta=1$ corresponds to the celebrated Gross–Pitaevskii regime
in which strong correlations occur on very short length scales. The
macroscopic properties of the system are well captured by the famous
Gross–Pitaevskii theory [85, 139]. In this theory, a quantum particle is
effectively felt by the others as a hard sphere whose radius is the scattering
length of the interaction potential. Recall that the scattering length ${\rm
Appl.~{}Opt.}$ of the potential $w$ is defined by the variational formula

$8\pi{\rm Appl.~{}Opt.}=\inf\left\\{\int_{\mathbb{R}^{3}}2|\nabla
f|^{2}+w|f|^{2},\quad\lim_{|x|\to\infty}f(x)=1\right\\}.$ (3)

When $w$ is sufficiently smooth, (3) has a minimizer $0\leq f\leq 1$ that
satisfies

$(-2\Delta+w)f=0.$ (4)

The scattering length can then be recovered from the formula

$8\pi{\rm Appl.~{}Opt.}=\int wf.$ (5)

By scaling, the scattering length of $w_{N}=N^{2}w(N\cdot)$ is ${\rm
Appl.~{}Opt.}N^{-1}$. If we formally replace the interaction potential
$w_{N}(x-y)$ in $H_{N}$ by $8\pi{\rm Appl.~{}Opt.}N^{-1}\delta_{0}(x-y)$, we
obtain a Hamiltonian with a Delta interaction. Such an object is (in three
dimensions) mathematically not well defined. Nevertheless, it is usually taken
as a starting point in the physics literature on cold gases.

####  3 Types of approximation.

Recall that our goal is to understand how the $N$-body wave function
$\Psi_{N,t}$ behaves when $N$ is very large. In the context of dynamics one
usually considers three possible effective descriptions. The first one,
usually called the leading order approximation, considers the approximation of
$\Psi_{N,t}$ in terms of reduced density matrices. Recall, that the (one-body)
reduced density matrix of a state $\Psi_{N}\in\mathfrak{H}^{N}$ (here we will
restrict ourselves to zero temperature) is the positive, trace class operator
$\gamma_{\Psi_{N}}:\mathfrak{H}\to\mathfrak{H}$ with kernel

$\displaystyle\gamma_{\Psi_{N}}(x,y)=N\int\Psi_{N}(x,x_{2},...,x_{N})\overline{\Psi_{N}(y,x_{2},...,x_{N})}dx_{2}...dx_{N}.$
(6)

The knowledge of $\gamma_{\Psi_{N}}$ allows to determine the expectation
values of one-body observables in the state $\Psi_{N}$. Indeed, let
$O:\mathfrak{H}\to\mathfrak{H}$ be an observable and let $O_{i}$ denote the
corresponding operator acting on the $i$-th particle in the $N$-body space.
Then

$\displaystyle\Big{\langle}\Psi_{N},\left(\sum_{i=1}^{N}O_{i}\right)\Psi_{N}\Big{\rangle}=\operatorname{Tr}(O\gamma_{\Psi_{N}}).$
(7)

We will say that the full many-body evolution $\Psi_{N,t}$ is to leading order
approximated by $\Phi_{N,t}$ if

$\displaystyle\lim_{N\to\infty}\frac{1}{N}\operatorname{Tr}|\gamma_{\Psi_{N,t}}-\gamma_{\Phi_{N,t}}|=0.$
(8)

Note that, due to (7), the trace norm topology is natural in this context. The
hope is that $\Phi_{N,t}$ can be determined in an easier way that
$\Psi_{N,t}$.

The convergence (8) is closely related to the definition of Bose–Einstein
Condensation [133]. We will say that a system o bosons exhibits BEC in the
state $\Psi_{N}\in\mathfrak{H}^{N}$ if

$\displaystyle\lim_{N\to\infty}\operatorname{Tr}\big{|}\frac{1}{N}\gamma_{\Psi_{N}}-|\phi\rangle\langle\phi|\big{|}=0$
(9)

for some $\phi\in\mathfrak{H}$. One then often says that $\phi$ is the wave
function of the condensate. This terminology is related to the fact that if
one considers the so-called Hartree or product state, i.e. the uncorrelated
$N$-body wave function of the form $\phi^{\otimes
N}:=\phi(x_{1})\ldots\phi(x_{N})$ in which all particles occupy the same one-
particle state, then

$\gamma_{\phi^{\otimes N}}=N|\phi\rangle\langle\phi|.$

This is why one might sometimes run across the notation

$"\Psi_{N}\approx\phi^{\otimes N}\quad\text{to leading order}"$

which means asymptotic equality in terms of reduced density matrices in the
trace norm topology. Note, that BEC does not mean that all particles occupy
one single-particle state, but only that a macroscopic fraction does. In fact,
while a product state is a ground state of non-interacting system ($w_{N}=0$),
it can’t be one of an interacting system. In particular, if one considers the
state (let us skip the symmetry of the wave function for a moment)

$\Xi_{N}:=\prod_{i=1}^{N-1}\phi(x_{i})\phi^{\perp}(x_{N}),\quad\text{with}\quad\phi^{\perp}\perp\phi$

then obviously

$\Xi_{N}\perp\phi^{\otimes N}\quad\text{in}\,\,L^{2}(\mathbb{R}^{3N})$

but physically, for large $N$, both states describe a very similar situation.
In particular, both states exhibit BEC with the same condensate wave function.

Often, the knowledge about correlations in the system is crucial in order to
understand some of the physical properties of the system. As seen in the
example above, the leading order approximation is not enough for this purpose.
This is why one considers the most straightforward indicator of closeness: the
$N$-particle Hilbert space norm (or, shortly, norm approximation). More
precisely, the goal is to find a $N$-body wave function
$\Xi_{N,t}\in\mathfrak{H}^{N}$ that is easier to compute than $\Psi_{N,t}$ and
such that

$\displaystyle\lim_{N\to\infty}\|\Psi_{N,t}-\Xi_{N,t}\|_{\mathfrak{H}^{N}}=0.$
(10)

Clearly, since

$\displaystyle\operatorname{Tr}|O(\gamma_{\Psi_{N}}-\gamma_{\Xi_{N}})|\leq
2\|O\|\|\Psi_{N}-\Xi_{N}\|$ (11)

for any bounded operator $O$, the norm approximation implies the convergence
of reduced one-body density matrices.

Another possible way of approximating the wave function is given by the Fock
space approximation. In this approach, one considers the problem in the grand-
canonical setting, where the number of particles in the system is not fixed.
To this end one introduces the Fock space

$\mathcal{F}\equiv\mathcal{F}(\mathfrak{H})=\bigoplus_{n=0}^{\infty}\mathfrak{H}^{n}=\mathbb{C}\oplus\mathfrak{H}\oplus\mathfrak{H}^{2}\oplus\cdots.$

The wave function in the Fock space is denoted by
$\Psi\in\mathcal{F}(\mathfrak{H})$ and

$\Psi=\\{\Psi^{(0)},\Psi^{(1)},\ldots,\Psi^{(j)},\ldots\\}$

where $\Psi^{(0)}\in\mathbb{C}$ and $\Psi^{(i)}\in\mathfrak{H}^{i}$ for $i\geq
1$. The inner product on $\mathcal{F}$ is defined as

$\langle\Psi_{1},\Psi_{2}\rangle_{\mathcal{F}}=\sum_{i\geq
0}\langle\Psi_{1}^{(i)},\Psi_{2}^{(i)}\rangle_{\mathfrak{H}^{i}}.$

A state $\Psi_{N}$ with exactly $N$ particles is described on the Fock space
$\mathcal{F}$ by a sequence $\Psi=\\{\Psi^{(n)}\\}_{n\geq 0}$ where
$\Psi^{(n)}=0$ for all $n\neq N$ and $\Psi^{(N)}=\Psi_{N}$.

One can lift the many-body evolution to the Fock space. To this end we define
the Hamiltonian $\mathcal{H}_{N}$ on $\mathcal{F}$ by

$(\mathcal{H}_{N}\Psi)^{(n)}=\mathcal{H}_{N}^{(n)}\psi^{(n)}$ (12)

with the $n$-th sector operator

$\mathcal{H}_{N}^{(n)}=H_{N}^{V}=\sum\limits_{j=1}^{n}(-\Delta_{x_{j}}+V(x_{j}))+\frac{1}{N-1}\sum\limits_{1\leq
j<k\leq n}{w_{N}(x_{j}-x_{k})}$

where now the subscript $N$ is not related to the number of particles, but
only reflects the scaling in the interaction potential (of course, in the end,
$N$ will also be related with the number of particles in the initial Fock
state; otherwise, there would be no relation with the scaling regime).

In particular the $N$-particle evolution can be embedded into the Fock space
in the following way

$e^{-it\mathcal{H}_{N}}\\{0,0,\ldots,\Psi_{N},0,\ldots\\}=\\{0,0,\ldots,e^{-itH_{N}}\Psi_{N},0,\ldots\\}.$

This follows from the fact that the Hamiltonian $\mathcal{H}_{N}$ commutes
with the particle number operator $\mathcal{N}$ given by

$(\mathcal{N}\Psi)^{(n)}=n\Psi^{(n)}.$ (13)

Let $\Psi_{0}\in\mathcal{F}$ be a state in the Fock space. We will say that
$\Xi_{t}\in\mathcal{F}$ approximates the many-body evolution of $\Psi_{0}$ in
the Fock space if

$\lim_{N\to\infty}\|e^{-it\mathcal{H}_{N}}\Psi_{0}-\Xi_{t}\|_{\mathcal{F}}=0.$
(14)

Sometimes, it is possible to get some information on the $N$-particle space
convergence from convergence in the Fock space. This approach, however,
usually leads to worse estimates than direct methods on $N$-particle space and
often requires additional assumptions on the initial states.

####  4 Outline

The paper will organised as follows. In the next section we will provide a
brief overview about the ground state properties of (trapped) bosonic systems.

In Section 3 we will review the existing results on the leading order
approximation. In Section 4 we will review the literature on the norm
approximation. In Section 5 we will mention results on the Fock space
approximation.

###  2 Ground state properties

Recall that we want to consider initial states that are ground states of
Hamiltonians of the form (1). Therefore it makes sense to briefly review some
basics facts concerning this issue. For more details and references we refer
to the excellent review [144].

####  1 Leading order approxmation for the ground state

It is widely expected that ground states of trapped systems exhibit (complete)
Bose-Einstein condensation. In fact, when $0\leq\beta<1$ we have

$\displaystyle\lim_{N\to\infty}\left(\inf_{\|\Psi_{N}\|_{\mathfrak{H}^{N}}=1}\frac{\langle\Psi_{N},H_{N}^{V}\Psi_{N}\rangle}{N}-\inf_{\|u\|_{\mathfrak{H}}=1}\mathcal{E}^{V}_{\rm{H},N}(u)\right)=0$
(15)

where

$\mathcal{E}^{V}_{\rm{H},N}(u):=\frac{1}{N}\langle u^{\otimes
N},H_{N}^{V}u^{\otimes N}\rangle=\int_{\mathbb{R}^{3}}|\nabla
u|^{2}+V|u|^{2}+\frac{1}{2}|u|^{2}(w_{N}*|u^{2}|).$

Moreover, if the Hartree energy functional $\mathcal{E}^{V}_{\rm{H},N}(u)$ has
a unique minimizer $u_{\rm H}$, then the ground state $\Psi_{N}^{V}$ of
$H_{N}^{V}$ condensates on $u_{\rm H}$ in the sense that

$\displaystyle\lim_{N\to\infty}\operatorname{Tr}\big{|}\frac{1}{N}\gamma_{\Psi_{N}^{V}}-|u_{\rm
H}\rangle\langle u_{\rm H}|\big{|}=0.$ (16)

The rigorous justifications for (15) and (16) in various specific cases have
been given in [110, 70, 15, 114, 135, 141, 145]. Later, in a series of works
[105, 106, 104], Lewin, Nam and Rougerie provided proofs in a very general
setting. Most recently, the next order term in the expansion (16) has been
established in [124, 31].

When $\beta=1$ (the Gross-Pitaevskii regime), the Hartree functional has to be
modified to capture the strong correlation between particles. In that case

$\displaystyle\lim_{N\to\infty}\left(\inf_{\|\Psi_{N}\|_{\mathfrak{H}^{N}}=1}\frac{\langle\Psi_{N},H_{N}^{V}\Psi_{N}\rangle}{N}-\inf_{\|u\|_{\mathfrak{H}}=1}\mathcal{E}^{V}_{\rm{GP}}(u)\right)=0$
(17)

where

$\displaystyle\mathcal{E}^{V}_{\rm{GP}}(u):=\int_{\mathbb{R}^{3}}|\nabla
u|^{2}+V|u|^{2}+4\pi{\rm Appl.~{}Opt.}|u|^{4}$ (18)

is the Gross–Pitaevskii functional. In that case one also has Bose–Einstein
Condensation on the Gross–Pitaevskii minimizer

$\displaystyle\lim_{N\to\infty}\frac{1}{N}\gamma_{\Psi_{N}^{V}}=|u_{\rm
GP}\rangle\langle u_{\rm GP}|$ (19)

in trace norm. The convergence (17) has been first proven by Lieb, Seiringer
and Yngvason in [113] while (19) has been first proven by Lieb and Seiringer
in [111, 112] (see also [127]). More recently, the optimal rates of
convergence for (17) and (19) have been given in [20, 22] (translation
invariant case) and [125] (trapped case with smallness condition on ${\rm
Appl.~{}Opt.}$, see also [86]).

####  2 Second order correction

The next order correction to the lower eigenvalues and eigenfunctions of
$H_{N}^{V}$ is predicted by Bogoliubov’s approximation [25]. In the mean-field
limit, this has been first derived rigorously by Seiringer in [145], and then
extended in various directions in [78, 108, 58, 129]. Bogoliubov theory is
formulated in the Fock space $\mathcal{F}$. At this point, let us briefly
recall the notion of second quantization.

We define the creation operator $a^{*}(f)$ and the annihilation operator
$a(f)$ that for every $f\in\mathfrak{H}$ is given by

$\displaystyle(a^{*}(f)\Psi)(x_{1},\dots,x_{n+1})$
$\displaystyle=\frac{1}{\sqrt{n+1}}\sum_{j=1}^{n+1}f(x_{j})\Psi(x_{1},\dots,x_{j-1},x_{j+1},\dots,x_{n+1})$
$\displaystyle(a(f)\Psi)(x_{1},\dots,x_{n-1})$
$\displaystyle=\sqrt{n}\int\overline{f(x_{n})}\Psi(x_{1},\dots,x_{n}){\rm
d}x_{n}$

for all $\Psi\in\mathfrak{H}^{n}$ and for all $n$. These operators satisfy the
canonical commutation relations (CCR)

$\displaystyle[a(f),a(g)]=[a^{*}(f),a^{*}(g)]=0,\quad[a(f),a^{*}(g)]=\langle
f,g\rangle$ (20)

for all $f,g\in\mathfrak{H}$. Creation and annihilation operators are used to
represent many-body states and operators on the Fock space. It is well-known
(see e.g. [19] or [148]) that for a symmetric operator $H$ on $\mathfrak{H}$
and an orthonormal basis $\\{f_{n}\\}_{n\geq 1}\subset D(h)$ of $\mathfrak{H}$
one has

$\displaystyle{\rm
d\Gamma}(H):=0\oplus\bigoplus_{N=1}^{\infty}\sum_{j=1}^{N}H_{j}=\sum_{m,n\geq
1}\langle f_{m},Hf_{n}\rangle a^{*}(f_{m})a(f_{n}).$ (21)

Similarly, for a symmetric operator $W$ on $\mathfrak{H}\otimes\mathfrak{H}$
such that

$\langle f_{m}\otimes f_{n},W\,f_{p}\otimes f_{q}\rangle=\langle f_{n}\otimes
f_{m},W\,f_{p}\otimes f_{q}\rangle$

for all $m,n,p,q\geq 1$ we have

$\displaystyle\quad 0\oplus 0\oplus\bigoplus_{N=2}^{\infty}\sum_{1\leq i<j\leq
N}W_{ij}$ $\displaystyle=\frac{1}{2}\sum_{m,n,p,q\geq 1}\langle f_{m}\otimes
f_{n},W\,f_{p}\otimes
f_{q}\rangle_{\mathfrak{H}^{2}}\,\,a^{*}(f_{m})a^{*}(f_{n})a(f_{p})a(f_{q}).$
(22)

If one does not want to work on a specific orthonormal basis, it is possible
to use the operator-valued distributions $a_{x}^{*}$ and $a_{x}$, with
$x\in\mathbb{R}^{3}$, defined by

$a^{*}(f)=\int_{\mathbb{R}^{3}}f(x)a_{x}^{*}{\rm d}x\qquad\text{and}\qquad
a(f)=\int_{\mathbb{R}^{3}}\overline{f(x)}a_{x}{\rm d}x$

for all $f\in\mathfrak{H}$. The canonical commutation relations (20) then
imply that

$\displaystyle[a^{*}_{x},a^{*}_{y}]=[a_{x},a_{y}]=0\qquad\text{and}\qquad[a_{x},a^{*}_{y}]=\delta(x-y).$
(23)

The second quantization formulas (21) and (2) can be rewritten as

$\displaystyle{\rm d\Gamma}(H)$ $\displaystyle=\iint
H(x,y)a^{*}_{x}a_{y}dxdy,$ (24) $\displaystyle\quad 0\oplus
0\oplus\bigoplus_{N=2}^{\infty}\sum_{1\leq i<j\leq N}W_{ij}$
$\displaystyle=\frac{1}{2}\iiiint
W(x,y;x^{\prime},y^{\prime})a^{*}_{x}a^{*}_{y}a_{x^{\prime}}a_{y^{\prime}}dxdydx^{\prime}dy^{\prime}$
(25)

where $H(x,y)$ and $W(x,y;x^{\prime},y^{\prime})$ are the kernels of $H$ and
$W$, respectively.

For example, the aforementioned particle number operator can be written as

$\mathcal{N}:={\rm
d\Gamma}(1)=\bigoplus_{n=0}^{\infty}n1_{\mathfrak{H}^{n}}=\int_{\mathbb{R}^{3}}a_{x}^{*}a_{x}\,dx$

and the $N$-body Hamiltonian $H_{N}$ can be extended to an operator on Fock
space $\mathcal{F}(\mathfrak{H})$ as

$\displaystyle\mathcal{H}_{N}={\rm
d\Gamma}(-\Delta)+\frac{1}{2(N-1)}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}w_{N}(x-y)a^{*}_{x}a^{*}_{y}a_{x}a_{y}dxdy.$
(26)

As already mentioned, Bogoliubov theory is formulated in the Fock space
$\mathcal{F}$ or, more precisely, the excited Fock space
$\mathcal{F}(\mathfrak{H}_{+})\equiv\mathcal{F}(\\{u_{\rm H}\\}^{\bot})$. Let
$\\{u_{m}\\}_{m\geq 0}$ be an orthonormal basis of $\mathfrak{H}$ such that
$u_{\rm H}\equiv u_{0}$. In the mean-field limit, the condensate is described
by the Hartree minimizer $u_{\rm H}$ and the excited particles are effectively
described by a quadratic Hamiltonian $\mathbb{H}^{V}$ of the form

$\displaystyle\mathbb{H}^{V}$ $\displaystyle=\sum_{m,n\geq 1}\langle
u_{m},(h+K_{1})u_{n}\rangle a^{*}_{m}a_{n}$ (27)
$\displaystyle\quad\quad+\sum_{m,n\geq 1}\frac{1}{2}\langle u_{m}\otimes
u_{n},K_{2}\rangle a^{*}_{m}a^{*}_{n}+\frac{1}{2}\langle K_{2},u_{m}\otimes
u_{n}\rangle a_{m}a_{n}$

acting on $\mathcal{F}(\\{u_{0}\\}^{\bot})$ and where
$K_{1}:\mathfrak{H}_{+}\to\mathfrak{H}_{+}$ and
$K_{2}:\overline{\mathfrak{H}_{+}}\to\mathfrak{H}_{+}$ are operators defined
by

$\displaystyle\langle u,K_{1}v\rangle$
$\displaystyle=\iint\overline{u(x)}v(y)u_{0}(x)\overline{u_{0}(y)}w(x-y)dxdy,$
(28) $\displaystyle\langle u,K_{2}\overline{v}\rangle$
$\displaystyle=\iint\overline{u(x)v(y)}u_{0}(x)u_{0}(y)w(x-y)dxdy$

for all $u,v\in\mathfrak{H}_{+}$. Finally, $h$ is the one-body operator given
by

$h=-\Delta+V+|u_{0}|^{2}\ast w-\mu$

which comes from the Hartree equation (the Euler-Lagrange equation for the
Hartree functional). Here $\mu$ is an appropriate constant to make $hu_{0}=0$.

It has been proven in [108] by Lewin, Nam, Serfaty, Solovej that if the
Hartree minimizer $u_{\rm H}$ is non-degenerate (in the sense that the Hessian
of $\mathcal{E}_{\rm H}^{V}(u)$ at $u_{\rm H}$ is bigger than a positive
constant), then the ground state $\Psi_{N}^{V}$ of $H_{N}^{V}$ admits the norm
approximation

$\displaystyle\lim_{N\to\infty}\left\|\Psi_{N}^{V}-\sum_{n=0}^{N}u_{\rm
H}^{\otimes(N-n)}\otimes_{s}\psi_{n}\right\|_{\mathfrak{H}^{N}}=0$ (29)

where $\Phi^{V}=(\psi_{n})_{n=0}^{\infty}\in\mathcal{F}(\\{u\\}^{\bot})$ is
the (unique) ground state of $\mathbb{H}^{V}$. Note that the norm convergence
(29) shows what we mentioned before, that is, the fact that if $w\not\equiv
0$, then $\Phi^{V}$ is not the vacuum $\Omega:=1\oplus 0\oplus 0\cdots$, and
hence $\Psi_{N}^{V}$ is never close to $u_{\rm H}^{\otimes N}$ in norm.

For $\beta>0$ Bogoliubov theory has been justified for translation invariant
systems in [23] ($\beta<1$) and in [21] ($\beta=1$).

###  3 Leading order approximation

In this section we shall review the results about the leading order
approximation for the Schrödinger evolution of a bosonic many-body wave
function in the sense of (8). From a physics perspective we want to answer the
following question: if the initial state of a trapped system exhibits
Bose–Einstein Condensation, does the condensate endure once the trap is
switched off and the system starts evolving in time?

Thus, one would like to show that if the initial many-body wave function
$\Psi_{N,0}$ satisfies

$\displaystyle\lim_{N\to\infty}\operatorname{Tr}\big{|}\frac{1}{N}\gamma_{\Psi_{N,0}}-|\phi_{0}\rangle\langle\phi_{0}|\big{|}=0$
(30)

for some $\phi\in L^{2}(\mathbb{R}^{3})$, then

$\displaystyle\lim_{N\to\infty}\operatorname{Tr}\big{|}\frac{1}{N}\gamma_{\Psi_{N,t}}-|\phi_{t}\rangle\langle\phi_{t}|\big{|}=0$
(31)

for some $\phi_{t}$ which can be found via an effective theory.

First results of that type have been obtained in the mean-field regime
($\beta=0$) by Hepp [87] (for differentiable $w$) and by Spohn in [149] (for
$w$ bounded) (although the setup there, especially in the work of Hepp, was a
priori quite different than the one presented here).

In the more familiar setup explained in the introduction the discussed
question has been raised in the literature again in the 2000’s. Since then, a
substantial effort of the community led to many interesting results which
often differ only slightly. Those differences might be difficult to spot for
non-specialists and one of the goals of this work is to clarify some of these
issues.

####  1 Results for different scaling regimes

##### Mean-field scaling.

As mentioned in the introduction, the simplest regime to consider is the
Hartree scaling. In this case, the general (and imprecise) form of the
statement describing the leading order approximation is given by the following

######  Theorem 3.1.

(Leading order approximation for mean-field dynamics) Let $\Psi_{N,0}$ be an
initial state satisfying

$\displaystyle\lim_{N\to\infty}\operatorname{Tr}\big{|}\frac{1}{N}\gamma_{\Psi_{N,0}}-|u_{0}\rangle\langle
u_{0}|\big{|}=0$ (32)

for a normalized wave function $u_{0}\in L^{2}(\mathbb{R}^{3})$. Then

$\displaystyle\lim_{N\to\infty}\operatorname{Tr}\big{|}\frac{1}{N}\gamma_{\Psi_{N,t}}-|u_{t}\rangle\langle
u_{t}|\big{|}=0$ (33)

where $\Psi_{N,t}=e^{-itH_{N}}\Psi_{N,0}$ is the many-body wave function
evolved by the mean-field (with $\beta=0$) Hamiltonian $H_{N}$ (with $V=0$)
and $u_{t}$ is the solution of the time-dependent Hartree equation

$i\partial_{t}u_{t}=\big{(}-\Delta+w*|u_{t}|^{2}-\mu_{t}\big{)}u_{t}$ (34)

with the initial datum $u_{0}$ and for some appropriate phase
$\mu_{t}\in\mathbb{R}$.

Note, that for the leading order the phase plays no role as it does not alter
the projection $|u_{t}\rangle\langle u_{t}|$.

The first result of the form of Theorem 3.1 was obtained by Bardos, Golse and
Mauser in [12] (with the additional condition
$\langle\Psi_{N,0},H_{N}\Psi_{N,0}\rangle\leq CN$). Shortly afterwards Erdös
and Yau obtained in [69] the same result for initial states that had to be a
product state. Clearly, the assumption (32) allows for more general initial
states. We refer to [13] for a recap and comparison of the two papers. We note
that the work of Erdös and Yau allowed to take $w(x)=1/|x|$, i.e. the Coulomb
potential. Both these works use the BBGKY (Bogoliubov-Born-Green-Kirkwood-
Yvon) hierarchy method (cf. Subsection 2). In particular, the BBGKY method
does not give any rates of convergence in (33).

The question of the convergence rate has been first answered by Rodnianski and
Schlein in [142]. Using the method of coherent states (cf. Subsection 2) they
showed that the convergence rate in (33) is of the form

$\frac{C}{\sqrt{N}}e^{Ct}$

for an initial state that is a product state. Their work included the Coulomb
interaction. This result has been extended in [97] by Knowles and Pickl to
cover more singular potentials (in the sense of the function $w$ rather than
the scaling which was still mean-field) and initial states that are not
necessarily product states but satisfy the more general condition (32). In
their work Knowles and Pickl used a method that was developed by Pickl in
[137] (which also provides a relatively simple, quantitative proof of (33) for
nice potentials in the mean-field setting, see also [72, 3, 109, 8]).

For nicer (bounded and integrable) interaction potentials Erdös and Schlein
proved in [62] an optimal convergence rate

$\frac{C}{N}e^{Ct}$

(again, they assumed factorized initial conditions). This result has been
extended in [36] by Chen and Lee to cover more general potentials and then,
together with Schlein, further improved to cover the Coulomb case [37] (see
also [98]).

In [61, 118] the convergence (33) was established for particles with a
relativistic dispersion relation (the kinetic energy $-\Delta$ is in that case
replaced by $\sqrt{1-\Delta}$) and with Coulomb type interaction $w(x)=\pm
1/|x|$ (this situation is physically interesting because it describes systems
of gravitating bosons, so called boson stars, and the related phenomenon of
stellar collapse). These systems have been further studied in [101] (optimal
convergence rate) and [9] (convergence in Sobolev trace norms). A detailed
analysis of the differences in various results regarding the leading order
convergence of mean-field bosonic systems can be found in [89].

Further developments in the study of the leading order behaviour of bosonic
systems include the analysis of the mean-field limit coupled to a semi-
classical limit [5, 6, 7, 71, 73, 76, 77, 100], compound mean-field system
systems [10, 116, 56, 102], systems with magnetic fields [115], systems with
three-body interactions [103], central limit type theorems for bosonic mean-
field dynamics [14, 35, 94].

Finally, let us mention that for bounded potentials a systematic, perturbative
way to compute higher order terms in the expansion of (33) has been developed
recently in [29] (see also [132]).

##### NLS regime.

The analysis of the dynamics becomes more complicated for positive $\beta$.
For $\beta\in(0,1)$ the typical result is of the form

######  Theorem 3.2.

(Leading order approximation in the NLS regime) Let $\Psi_{N,0}$ be an initial
state satisfying

$\displaystyle\lim_{N\to\infty}\operatorname{Tr}\big{|}\frac{1}{N}\gamma_{\Psi_{N,0}}-|u_{0}\rangle\langle
u_{0}|\big{|}=0$ (35)

for a normalized wave function $u_{0}\in H^{1}(\mathbb{R}^{3})$. Then

$\displaystyle\lim_{N\to\infty}\operatorname{Tr}\big{|}\frac{1}{N}\gamma_{\Psi_{N,t}}-|u_{t}\rangle\langle
u_{t}|\big{|}=0$ (36)

where $\Psi_{N}(t)=e^{-itH_{N}}\Psi_{N,0}$ is the many-body wave function
evolved by the many-body Hamiltonian $H_{N}$ (with $V=0$) in the NLS regime
($\beta\in(0,1)$) and $u_{t}$ is the solution of the time-dependent nonlinear
Schrödinger equation

$i\partial_{t}u_{t}=\big{(}-\Delta+b_{0}|u_{t}|^{2}-\mu_{t}\big{)}u_{t}$ (37)

with the initial data $u_{0}$. Here $b_{0}=\int w(x)dx$ and
$\mu_{t}\in\mathbb{R}$ is an appropriate phase.

The first, complete proof of Theorem 3.2 has been given by Erdös, Schlein and
Yau in [65]. It was valid for $\beta<1/2$ and initial states which are a
product state. Later, in [68], Erdös, Schlein and Yau extended this result to
all $\beta\in(0,1)$ and the general initial states (35). Both papers used the
BBGKY approach (cf. Subsection 2). For $\beta<1/6$, a similar result (for
general initial states and without the assumption on the positivity of the
interaction potential) has been obtained by Pickl in [136]. This work provided
also explicit bounds on the convergence rate. In one dimension, for
$\beta\in(0,1)$, the problem has been solved by Adami, Golse and Teta [1] (see
also [143, 2]). In two dimensions (on a torus) the problem has been studied
(for $\beta<3/4$) by Kirkpatrick, Schlein and Stafillani in [95] and more
recently by Jeblick and Pickl in [91] (without the positivity assumption on
the interaction).

Other results about the leading order approximation in the NLS regime include
lower dimensional systems with attractive interactions [45, 49], systems with
three-body interactions [40, 50], derivations of lower dimensional dynamics
from the three dimensional problem [44, 92, 48, 27, 146].

##### The GP regime.

The leading order approximation problem in the Gross-Pitaevski regime has been
first solved by Erdös, Schlein and Yau in [65] where they proved the following
theorem

######  Theorem 3.3.

(Leading order approximation in the Gross-Pitaevskii regime [67, Thm. 3.1])

Assume $w\geq 0$ is a smooth, even potential that decays sufficiently fast and
has scattering length ${\rm Appl.~{}Opt.}$. Let $\Psi_{N,0}$ be a family of
initial wave functions such that

$\langle\Psi_{N,0},H_{N}\Psi_{N,0}\rangle\leq CN,$

which exhibit Bose–Einstein Condensation

$\displaystyle\lim_{N\to\infty}\operatorname{Tr}\big{|}\frac{1}{N}\gamma_{\Psi_{N,0}}-|\varphi_{0}\rangle\langle\varphi_{0}|\big{|}=0$
(38)

for a normalized wave function $\varphi_{0}\in H^{1}(\mathbb{R}^{3})$. Then

$\displaystyle\lim_{N\to\infty}\operatorname{Tr}\big{|}\frac{1}{N}\gamma_{\Psi_{N,t}}-|\varphi_{t}\rangle\langle\varphi_{t}|\big{|}=0$
(39)

where $\Psi_{N,t}=e^{-itH_{N}}\Psi_{N,0}$ is the many-body wave function
evolved by many-body Gross–Pitaevskii Hamiltonian $H_{N}$ (with $V=0$ and
$\beta=1$) and $\varphi_{t}$ is the solution of the time-dependent Gross-
Pitaevskii equation

$i\partial_{t}\varphi_{t}=\big{(}-\Delta+8\pi{\rm
Appl.~{}Opt.}|\varphi_{t}|^{2})\big{)}\varphi_{t}$ (40)

with the initial condition $\varphi_{0}$.

Earlier, the same authors proved in [68] the same result under the additional
assumption that

$\displaystyle\sup_{r\geq 0}r^{2}w(r)+\int_{0}^{\infty}rw(r)dr$ (41)

is small enough. To remove this smallness condition Erdös, Schlein and Yau
used an intrinsic characterization of the correlation structure in terms of
the two-particle scattering wave operator. Genrally, however, both works [68,
67] were based on the BBGKY hierarchy method and did not provide any
quantitative estimates on the convergence.

Explicit bounds on the convergence rate in (39) have been later obtained by
Benedikter, de Oliveira and Schlein in [16], Pickl in [138] and by Brennecke
and Schlein in [33].

In the periodic setting on a unit torus partial results in the spirit of
Theorem 3.3 (with a modified many-body Hamiltonian which had a cut-off to
prevent pair interactions whenever at least three particles come into a region
with diameter much smaller than the typical inter-particle distance) have been
obtained by Erdös, Schlein and Yau in [64] and then the problem has been
solved by Sohinger in [147].

The two dimensional problem has been solved by Jeblick, Leopold and Pickl in
[90]. At this point let us stress that the Gross–Pitaevskii scaling in two
dimensions is characterized by the scaling $w_{N}(x)=e^{2N}w(e^{N}x)$ rather
than $w_{N}(x)=N^{2}w(N^{2}x)$. Other results in this regime include the
dimensionally reduced dynamics [28, 30], dynamics in magnetic fields [131],
dynamics of (pseudo-)spinor systems [117] and central limit type theorems for
dynamics [140].

####  2 Methods

In this section we shall very briefly explain the two main approaches to prove
the leading order convergence. For a more pedagogical introduction we refer to
the excellent lecture notes of Benedikter, Porta and Schlein [17].

##### The BBGKY hierarchy.

The BBGKY approach is based on the idea of investigating the $k$-body reduced
density matrices rather than the wave function itself. For a given $k$, the
$k$-body reduced density matrix of the state $\Psi_{N}$ is the generalization
of the one-body reduced density matrix and allows to compute expectation
values of $k$-body operators. It is defined as the operator
$\gamma^{(k)}_{\Psi_{N}}$ on $\mathfrak{H}^{k}$ whose kernel satisfies

$\displaystyle{N\choose
k}^{-1}\gamma^{(k)}_{\Psi_{N}}(x_{1},\ldots,x_{k};y_{1},\ldots,y_{k})=$ (42)
$\displaystyle\int\Psi_{N}(x_{1},\ldots,x_{k},x_{k+1},...,x_{N})\overline{\Psi_{N}(y_{1},\ldots,y_{k},x_{k+1},...,x_{N})}dx_{k+1}\ldots
dx_{N}.$

Note that by setting $k=1$ we recover (6). In other words,

$\gamma^{(k)}_{\Psi_{N}}={N\choose
k}\operatorname{Tr}_{k+1}|\Psi_{N}\rangle\langle\Psi_{N}|.$

Using the Schrödinger equation, or, more precisely, the von Neumann equation,
one can obtain a hierarchy of equations for
$\widetilde{\gamma}^{(k)}_{\Psi_{N}}={N\choose k}^{-1}\gamma^{(k)}_{\Psi_{N}}$
of the form111To be precise, the hierarchy (43) arises for the $H_{N}$ with
the coupling constant $N^{-1}$ rather than $(N-1)^{-1}$ in front of the
interaction term.

$\displaystyle i\partial_{t}\widetilde{\gamma}^{(k)}_{\Psi_{N,t}}$
$\displaystyle=\sum_{j=1}^{k}[-\Delta_{x_{j}},\widetilde{\gamma}^{(k)}_{\Psi_{N,t}}]+\frac{1}{N}\sum_{i<j}^{k}[w_{N}(x_{i}-x_{j}),\widetilde{\gamma}^{(k)}_{\Psi_{N,t}}]$
(43)
$\displaystyle+\frac{N-k}{N}\sum_{j=1}^{k}\operatorname{Tr}_{k+1}[w_{N}(x_{j}-x_{k+1}),\widetilde{\gamma}^{(k+1)}_{\Psi_{N,t}}]$

where we use the convention $\widetilde{\gamma}^{(N+1)}_{\Psi_{N,t}}=0$.
Consider the mean-field limit, i.e. $w_{N}=w$. Taking a formal limit
$N\to\infty$ one obtains

$\displaystyle i\partial_{t}\widetilde{\gamma}^{(k)}_{\infty,t}$
$\displaystyle=\sum_{j=1}^{k}[-\Delta_{x_{j}},\widetilde{\gamma}^{(k)}_{\infty,t}]+\sum_{j=1}^{k}\operatorname{Tr}_{k+1}[w(x_{j}-x_{k+1}),\widetilde{\gamma}^{(k+1)}_{\infty,t}].$
(44)

In the NLS/GP regime the limiting equation obtained from (43) is of the form

$\displaystyle i\partial_{t}\widetilde{\gamma}^{(k)}_{\infty,t}$
$\displaystyle=\sum_{j=1}^{k}[-\Delta_{x_{j}},\widetilde{\gamma}^{(k)}_{\infty,t}]+\sigma\sum_{j=1}^{k}\operatorname{Tr}_{k+1}[\delta(x_{j}-x_{k+1}),\widetilde{\gamma}^{(k+1)}_{\infty,t}].$
(45)

with $\sigma=\int w$ for $\beta\in(0,1)$ and $\sigma=8\pi{\rm Appl.~{}Opt.}$
for $\beta=1.$

One can check that (44)/(45) has a solution given by the Hartree/NLS(GP)
equation, i.e.

$\widetilde{\gamma}^{(k)}_{\infty,t}=|u_{t}\rangle\langle u_{t}|^{\otimes k}$

where $u_{t}$ solves the Hartree/NLS(GP) equation. This leads to the following
strategy of proving results like Theorems 3.1-3.3 which consists of three main
steps:

  1. 1.

Compactness: one needs to prove compactness of the sequence (in $N$) of
$\\{\widetilde{\gamma}^{(k)}_{\Psi_{N,t}}\\}_{k=1}^{N}$ with respect with an
appropriate (weak) topology.

  2. 2.

Convergence: one needs to characterize limit points of the sequence
$\\{\widetilde{\gamma}^{(k)}_{\Psi_{N,t}}\\}_{k=1}^{N}$ as solutions of
(44)/(45).

  3. 3.

Uniqueness: one has to prove the uniqueness of the solution of (44)/(45).

Proofs of all these steps can be accomplished in various ways depending on the
details of the model (like the regularity and sign of $w$, initial conditions
etc.). Compactness is usually achieved via a priori estimates. The larger
$\beta$, the more difficult it is to obtain those a priori estimates. The a
priori estimates will also determine the functional spaces where the solutions
can live in. In general, the most difficult step is to prove uniqueness. In
the NLS regime ($\beta<1/2$) Erdös, Schlein and Yau [65] proved uniqueness
using Feynman diagrams (for example, in their earlier work with Elgart [63]
they were not able to show uniqueness). In [96] Klainerman and Machedon
provided an alternative approach based on appropriate (conjectured) space-time
bounds on limit points of
$\\{\widetilde{\gamma}^{(k)}_{\Psi_{N,t}}\\}_{k=1}^{N}$. This set up a program
in which various research groups tried to establish these bounds. That was
first successfully done by Kirkptrick, Schlein and Staffilani on
$\mathbb{T}^{2}$ in [95]. In $\mathbb{R}^{3}$ the conjecture for $\beta<1$ has
been established by X. Chen and Holmer in [47] (see also [46] and the works by
Chen and Pavlović [39, 41, 42, 43]). For $\beta=1$ a new proof of uniqueness
of the hierarchy has been given by Chen, Hainzl, Pavlović and Seiringer in
[38]. Other recent, related works include [4, 88, 51].

##### Quantitative approaches.

As mentioned earlier, the BBGKY hierarchy approach does not, in general,
provide any convergence rate in (31). In 2009 Rodnianski and Schlein used the
coherent states approach to obtain a quantitative version of Theorem 3.1 for
the first time. Their method was inspired by the work of Hepp [87] and Ginibre
and Velo [74]. The idea is to consider the problem in the Fock space. The
initial state is a coherent state which is obtained by applying Weyl’s unitary
operator $W(f)=\exp\big{(}a^{*}(f)-a(f)\big{)}$ to the vacuum:

$\displaystyle W(f)\Omega=e^{-\left\lVert f\right\rVert^{2}/2}\sum_{n\geq
0}\frac{1}{\sqrt{n!}}f^{\otimes n}.$ (46)

In particular, each $n$-particle component of this state is a product state.
The Fock space evolution is then governed by $\mathcal{H}_{N}$ defined in
(12). More precisely, the initial state, in order to model a system of $N$
particles has to be scaled and is given by $W(\sqrt{N}\phi)\Omega$ where
$\phi$ corresponds to the initial data condition (30).

For states $\Psi\in\mathcal{F}$ in the Fock space we define the one-body
reduced density matrix of $\Psi$ to be the operator (on $\mathfrak{H}$) with
the kernel

$\Gamma_{\Psi}(x;y):=\frac{\langle\Psi,a_{x}^{*}a_{y}\Psi\rangle}{\langle\Psi,\mathcal{N}\Psi\rangle}.$

Clearly, this definition reduces (up to a normalization factor) to (6) for
states with exactly $N$ particles.

Rodnianski and Schlein proved in [142] that in the mean-field limit, the one-
body reduced density matrix $\Gamma_{\Psi_{t}}$ of the state

$\Psi_{t}=e^{it\mathcal{H}_{N}}W(\sqrt{N}u_{0})\Omega$

satisfies

$\operatorname{Tr}\big{|}\Gamma_{\Psi_{t}}-|u_{t}\rangle\langle
u_{t}\big{|}\leq\frac{Ce^{Kt}}{N}$

for some constants $C$ and $K$. Here $u_{t}$ is the solution of the Hartree
equation with initial condition $u_{0}$.

Let us stress again, that in this set-up the state $\Gamma_{\Psi_{t}}$ depends
on $N$ in the grand-canonical sense: $N$ is the expected number of particles
in the initial state. In particular, the result does not, a priori, cover
canonical initial conditions (which would be states coming from wave functions
in $\mathfrak{H}^{N}$). However, a nice property of coherent states allows to
project the result above to the $N$-particle sector. To do this, one uses the
following representation of a product state in terms of coherent states:

$\frac{(a^{*}(u))^{N}}{\sqrt{N!}}\Omega=d_{N}\int_{0}^{2\pi}\frac{d\theta}{2\pi}e^{i\theta
N}W(e^{-i\theta}\sqrt{N}u)\Omega$

where the constant $d_{N}=\frac{\sqrt{N!}}{N^{N/2}e^{-N/2}}$ satisfies
$d_{N}\approx N^{1/4}$ for large $N$. For the $N$-particle state

$\Psi_{N,t}=e^{it\mathcal{H}_{N}}\frac{(a^{*}(u_{0}))^{N}}{\sqrt{N!}}\Omega$

one then gets

$\displaystyle\begin{aligned}
\gamma_{\Psi_{N,t}}&=\frac{d_{N}^{2}}{N}\int_{0}^{2\pi}\frac{d\theta_{1}}{2\pi}\int_{0}^{2\pi}\frac{d\theta_{2}}{2\pi}e^{-i(\theta_{1}-\theta_{2})N}\times\\\
&\qquad\qquad\times\langle
e^{it\mathcal{H}_{N}}W(e^{-i\theta_{1}}\sqrt{N}u_{0})\Omega,a_{x}^{*}a_{y}e^{it\mathcal{H}_{N}}W(e^{-i\theta_{2}}\sqrt{N}u_{0})\Omega\rangle_{\mathcal{F}}.\end{aligned}$

For the inner product in the Fock space one can then use the results about
$\Gamma_{\Psi_{t}}$. We see that the price one pays by projecting the Fock
space result onto the $N$-particles sector is given by the constant
$d_{N}^{2}\approx N^{1/2}$. This is the reason why the rate of convergence in
the canonical ensembele is $N^{-1/2}$ rather than the $N^{-1}$ above. In the
Gross–Pitaevskii ($\beta=1$) regime a similar approach has been adopted by
Benedikter, De Oliveira and Schlein in [16] where they proved (by introducing
Bogoliubov transformations to track the correlations) that

$\operatorname{Tr}\big{|}\Gamma_{\Psi_{t}}-|\varphi_{t}\rangle\langle\varphi_{t}\big{|}\leq\frac{Ce^{Kt}}{N^{1/2}}$

where $\varphi_{t}$ solves the Gross–Pitaevski equation. As before, this
result was formulated in the Fock space but this time for correlated initial
states (cf. (68)). In this case the extension to initial $N$-particle states
can only be done under additional assumptions which also make the convergence
slower (see [66] for an earlier analysis on how correlations form).

Most recently, the method of [16] has been further extended in [33] by
Brennecke and Schlein to the case of $N$-particle initial states (with
convergence rate $O(N^{-1/2})$).

A different method has been introduced by Pickl in [136]. It has been used in
many works, in particular in [138] where the Gross–Pitaevskii regime was
analyzed. The Pickl method, as it now often called, is based on the analysis
of a certain functional, usually called $\alpha_{N}(\Psi_{N},\varphi)$, which
counts (in a weighted way) the number of particles of the $N$-body state
$\Psi_{N}$ that are in the one-particle state $\varphi$. The functional is
applied to the Schrödinger time evolved many-body wave function $\Psi_{N,t}$
and the relevant one-body state $u_{t}$ in the mean-field or NLS regime or
$\varphi_{t}$ in the GP regime. The crucial property of the functional
$\alpha_{N}$ is that

$\left(\alpha_{N}(\Psi_{N},\varphi)\to
0\right)\Rightarrow\left(N^{-1}\gamma_{\Psi_{N}}\to|\varphi\rangle\langle\varphi|\right)$

as $N\to\infty$. The analysis then concentrates on deriving an estimate on the
time derivative of $\alpha_{N}(\Psi_{N,t},\varphi_{t})$ in order to apply
Grönwall’s argument. In particular, this method is suited to cover initial
conditions of the (general) form (30).

###  4 Norm approximation

We shall now review results concerning the norm approximation of many-boson
dynamics. As mentioned in the introduction (recall (11)), this notion of
closeness is more precise than the leading order approximation discussed in
the previous section. The norm approximation is also well suited for initial
states that are $N$-particle states.

In the mean-field regime, this problem has been first analyzed by Lewin, Nam
and Schlein in [107]. They considered the $N$-particle initial states of the
form

$\displaystyle\Psi_{N,0}=\sum_{n=0}^{N}u_{0}^{\otimes(N-n)}\otimes_{s}\psi_{n,0}$
(47)

where
$\Phi_{0}:=(\psi_{n,0})_{n=0}^{\infty}\in\mathcal{F}(\\{u_{0}\\}^{\bot})$.
This form is motivated by the ground state property (29) of trapped systems.
It was proved in [107] that when $\beta=0$, the time evolution
$\Psi_{N,t}=e^{-itH_{N}}\Psi_{N,0}$ satisfies the norm approximation

$\displaystyle\lim_{N\to\infty}\left\|\Psi_{N,t}-\sum_{n=0}^{N}u_{t}^{\otimes(N-n)}\otimes_{s}\psi_{n,t}\right\|_{\mathfrak{H}^{N}}=0$
(48)

where $u_{t}$ is the Hartree evolution (34) with the right phase factor

$\mu_{t}=\frac{1}{2}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u_{t}(x)|^{2}w(x-y)|u_{t}(y)|^{2}{\rm
d}x{\rm d}y$

and the evolution of
$\Phi_{t}=(\psi_{n,t})_{n=0}^{\infty}\in\mathcal{F}(\\{u_{t}\\}^{\bot})$ is
generated by a quadratic Bogoliubov Hamiltonian. This approach was later
developed by Nam and the author of this article in [120] (for $\beta<1/3$) and
in [121] (for $\beta<1/2$).

The crucial ingerdient to pass from the $N$-body Hilbert space to the Fock
space (which is natural when describing correlations) is the mapping first
introduced in the static case by Lewin, Nam, Serfaty and Solovej in the
derivation of Bogoliubov theory [108]. The transformation allows to factor out
the condensate from the many-body wave function and is given by

$\begin{array}[]{cccl}U_{N}(t):&\mathfrak{H}^{N}&\to&\displaystyle\mathcal{F}_{+}^{\leq
N}(t):=\bigoplus_{n=0}^{N}\mathfrak{H}_{+}(t)^{n}\\\\[8.5359pt]
&\Psi&\mapsto&\psi_{0}\oplus\psi_{1}\oplus\cdots\oplus\psi_{N}.\end{array}$
(49)

where $\mathfrak{H}_{+}(t)=\\{u_{t}\\}^{\perp}$. The space
$\mathcal{F}_{+}^{\leq N}$ is often called the truncated (because the
$m$-particle sectors for $m>N$ are zero), excited (because it describes
excitations around the condensate) Fock space. The idea is to reformulate the
Schrödinger evolution $\Psi_{N,t}=e^{-itH_{N}}\Psi_{N,0}$ in terms of

$\Phi_{N,t}:=U_{N}(t)\Psi_{N,t}$

which belongs to $\mathcal{F}_{+}^{\leq N}(t)$ and satisfies the equation

$\displaystyle\left\\{\begin{aligned}
i\partial_{t}\Phi_{N,t}&=\widetilde{H}_{N}(t)\Phi_{N,t},\\\
\Phi_{N,0}&=1^{\leq N}\Phi_{0}.\end{aligned}\right.$ (50)

Here $1^{\leq N}$ is the projection onto $\mathcal{F}^{\leq
N}=\mathbb{C}\oplus\mathfrak{H}\oplus\cdots\oplus\mathfrak{H}^{N}$ and

$\widetilde{H}_{N}(t)=1^{\leq
N}\Big{[}\mathbb{H}(t)+\frac{1}{2}\sum_{j=0}^{4}(R_{j}+R_{j}^{*})\Big{]}1^{\leq
N}$

with

$\displaystyle\mathbb{H}(t):={\rm
d\Gamma}(h(t))+\frac{1}{2}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\Big{(}K_{2}(t,x,y)a^{*}_{x}a^{*}_{y}+\overline{K_{2}(t,x,y)}a_{x}a_{y}\Big{)}{\rm
d}x\,{\rm d}y,$ $\displaystyle h(t)=-\Delta+|u_{t}(\cdot)|^{2}\ast
w_{N}-\mu_{t}+Q(t)\widetilde{K}_{1}(t)Q(t),$ $\displaystyle
K_{2}(t,\cdot,\cdot)=Q(t)\otimes Q(t)\widetilde{K}_{2}(t,\cdot,\cdot).$

Here $\widetilde{K}_{1}(t)$ is the operator on $\mathfrak{H}$ with kernel
$\widetilde{K}_{1}(t,x,y)=u_{t}(x)w_{N}(x-y)\overline{u_{t}(y)}$, and
$\widetilde{K}_{2}(t,x,y)=u_{t}(x)w_{N}(x-y)u_{t}(y)$.

The state $\Phi_{N,t}$ describes the excitations around the condensate.
Bogoliubov theory assumes that the operators $R_{j}$ (which we did not write
out explicitly) are small in an appropriate sense. Thus one may expect that
the evolution $\Phi_{N,t}$ in (50) is close (in norm) to the solution of the
effective Bogoliubov equation

$\displaystyle\left\\{\begin{aligned}
i\partial_{t}\Phi_{t}&=\mathbb{H}(t)\Phi_{t},\\\
\Phi_{t=0}&=\Phi_{0}.\end{aligned}\right.$ (51)

Heuristically, the final steps consist of the proof that the number of
(excited) particles in the state $\Phi_{t}$ is uniformly bounded. To obtain
these kind of bounds one exploits the Bogoliubov equation (51). In particular,
to do so, one uses that appropriate norms of the solution of the Hartree
equation (which for $\beta>0$ is $N$-dependent) are uniform in $N$. This
approach, at least for $\beta<1/3$, turns out to work also in the case when
the interaction is attractive (up to times for which the effective equation is
well posed). This has been later exploited by Nam and the author of this note
in [123] where they derived the focusing NLS in dimensions one and two.

In regimes with $\beta>1/2$ the short scale correlation structure developed by
the solution of the many-body Schrödinger equation cannot be appropriately
described by a time-dependent Bogoliubov transformation satisfying an equation
of the form (51). A modifided approach is needed and this has been done by
Brennecke, Nam, Schlein an the author of this note in [34]. To take
correlations into account more precisely, it is useful to consider the ground
state of the Neumann problem

$\left[-\Delta+\frac{1}{2N}w_{N}\right]f_{N}=\lambda_{N}f_{N}$ (52)

on the ball $|x|\leq\ell$, for a fixed $\ell>0$. One fixes $f_{N}(x)=1$, for
$|x|=\ell$, and extends $f_{N}$ to $\mathbb{R}^{3}$ requiring that
$f_{N}(x)=1$ for all $|x|\geq\ell$. Because of the scaling of the potential
$w_{N}$, the scattering process takes place in the region $|x|\ll 1$; for this
reason, the precise choice of $\ell$ is not very important, as long as $\ell$
is of order one.

The solution of (52) can be used, first of all, to give a better approximation
of the evolution of the condensate wave function, replacing the solution of
the limiting nonlinear Schrödinger equation (37) with the solution of the
modified, $N$-dependent, Hartree equation

$i\partial\varphi_{N,t}=-\Delta\varphi_{N,t}+(w_{N}f_{N}*|\varphi_{N,t}|^{2})\varphi_{N,t}$
(53)

with initial data $\varphi_{N,0}=\varphi_{0}$ describing the condensate at
time $t=0$.

Furthermore, (52) can be used to describe correlations among particles. To
this end, let

$T_{N,t}=\exp\left(\frac{1}{2}\int
dxdy\,\left[k_{N,t}(x,y)a_{x}a_{y}-\text{h.c.}\right]\right)$ (54)

with the integral kernel

$k_{N,t}(x;y)=(Q_{N,t}\otimes
Q_{N,t})\left[-N(1-f_{N})(x-y)\varphi^{2}_{N,t}((x+y)/2)\right]$ (55)

where $Q_{N,t}=1-|\varphi_{N,t}\rangle\langle\varphi_{N,t}|$ is the orthogonal
projection onto the orthogonal complement of the solution of the modified
Hartree equation (53). In particular, in this context the operator $U_{N}(t)$
will now project onto the orthogonal compliment of $\varphi_{N,t}$ and will be
denoted by $U_{\varphi_{N,t}}$. Since $T_{N,t}$ aims at generating
correlations, it is natural to define its kernel $k_{N,t}$ through the
solution of (52). In particular, the choice (54) guarantees a crucial
cancellation in the generator of the fluctuation dynamics. The final result
can be formulated as follows

######  Theorem 4.1.

(Norm approximation in the NLS regime [34, Theorem 3])

Consider the initial state $\Psi_{N,0}\in L^{2}_{s}(\mathbb{R}^{3N})$ with the
reduced one-particle density matrix $\gamma_{\Psi_{N,0}}$ such that

$N-\langle\varphi_{0},\gamma_{\Psi_{N,0}}\varphi_{0}\rangle\leq C$ (56)

and

$\left|\frac{1}{N}\langle\Psi_{N,0},H_{N}^{V}\Psi_{N,0}\rangle-\mathcal{E}_{\rm
GP}^{V}(\varphi_{0})\right|\leq CN^{-1}$ (57)

with $\mathcal{E}_{\rm GP}^{V}$ defined in (18). Let $\Psi_{N,t}$ be the
solution of the Schrödinger equation with initial data $\Psi_{N,0}$. Then, for
all $\alpha<\min(\beta/2,(1-\beta)/2)$, there exists a constant $C>0$ such
that

$\begin{split}\big{\|}\Psi_{N,t}-U_{\varphi_{N,t}}^{*}T_{N,t}^{*}e^{-i\int_{0}^{t}d\tau\;\eta_{N}(\tau)}\,\mathcal{U}_{2}(t;0)\,T_{N,0}\,&U_{\varphi_{N,0}}\Psi_{N,0}\big{\|}^{2}\\\
&\leq CN^{-\alpha}\exp(C\exp(C|t|))\end{split}$ (58)

for all $N$ sufficiently large and all $t\in\mathbb{R}$. Here $\eta_{N}(t)$ is
a phase factor and $\mathcal{U}_{2}(t;0)$ is a unitary dynamics on
$\mathcal{F}$ with an appropriate quadratic generator that can be defined
using $\varphi_{N,t},w_{N},k_{N,t}$ (see [34, eq. (41)]).

Notice, that the conditions (56) and (57) have been recently jutified in
[125]. Other results on the norm approximation (involving a slightly different
approach which avoids using second quantization) in the mean-field regime have
been obtained by Mitrouskas, Petrat and Pickl in [119]. Petrat, Pickl and
Soffer extended this result to a mean-field analysis coupled to a large volume
in [134]. A perturbative expansion has been analyzed by Bossmann, Petrat,
Pickl and Soffer in [29]. In [128] Nam and Salzmann provided a norm
approximation for systems with three-body interactions in the NLS regime.

Let us present an outline of the proof of Theorem 4.1, as the strategy is
slightly different than the one for $\beta<1/2$ and does not involve the
analysis of the Bogoliubov equations. The start is similar and involves the
action of the map $U_{\varphi_{N,t}}$ on $\Psi_{N,t}$. This allows us to
remove the condensate described at time $t$ by $\varphi_{N,t}$ and to focus on
the orthogonal fluctuations. We set

$\Phi_{N,t}=U_{\varphi_{N,t}}\Psi_{N,t},$ (59)

and we observe that $\Phi_{N,t}\in\mathcal{F}^{\leq N}_{\perp\varphi_{N,t}}$
satisfies the equation

$i\partial_{t}\Phi_{N,t}=\mathcal{L}_{N,t}\Phi_{N,t}$ (60)

with the generator

$\mathcal{L}_{N,t}=(i\partial_{t}U_{\varphi_{N,t}})U^{*}_{N,t}+U_{\varphi_{N,t}}H_{N}U_{\varphi_{N,t}}^{*}.$
(61)

A tedious but straightforward computation shows that one can write

$\displaystyle\mathcal{L}_{N,t}=\sum_{j=0}^{4}\mathcal{L}_{N,t}^{(j)}$ (62)

where

$\displaystyle\mathcal{L}_{N,t}^{(0)}$
$\displaystyle=\frac{N+1}{2}\left\langle\varphi_{N,t},[w_{N}(1-2f_{N})\ast|\varphi_{N,t}|^{2}]\varphi_{N,t}\right\rangle-\mu_{N}(t),$
$\displaystyle\mathcal{L}_{N,t}^{(1)}$
$\displaystyle=\frac{1}{2}\left\langle\varphi_{N,t},[w_{N}\ast|\varphi_{N,t}|^{2}]\varphi_{N,t}\right\rangle\frac{\mathcal{N}(\mathcal{N}+1)}{N}$
$\displaystyle\quad+\left[\sqrt{N}\left[a^{*}(Q_{N,t}[(w_{N}(1-f_{N}))\ast|\varphi_{N,t}|^{2}]\varphi_{N,t})\right.\right.$
$\displaystyle\left.\left.\quad-a^{*}(Q_{N,t}[w_{N}\ast|\varphi_{N,t}|^{2}]\varphi_{N,t})\frac{\mathcal{N}}{N}\right]\sqrt{\frac{N-\mathcal{N}}{N}}+{\rm
h.c.}\right],$ $\displaystyle\mathcal{L}_{N,t}^{(2)}$ $\displaystyle={\rm
d\Gamma}\Big{(}-\Delta+(w_{N}f_{N})*|\varphi_{N,t}|^{2}+K_{1,N,t}-\mu_{N,t}\Big{)}$
$\displaystyle\quad+{\rm
d\Gamma}\Big{(}Q_{N,t}(w_{N}(1-f_{N})*|\varphi_{N,t}|^{2})Q_{N,t}\Big{)}$
$\displaystyle\quad-{\rm
d\Gamma}\Big{(}Q_{N,t}(w_{N}*|\varphi_{N,t}|^{2})Q_{N,t}+K_{1,N,t}\Big{)}\frac{\mathcal{N}}{N}$
$\displaystyle\quad+\bigg{[}\frac{1}{2}\int{\rm d}x{\rm
d}y\,K_{2,N,t}(x,y)a^{*}_{x}a^{*}_{y}\frac{\sqrt{(N-\mathcal{N})(N-\mathcal{N}-1)}}{N}+{\rm
h.c.}\bigg{]},$ $\displaystyle\mathcal{L}_{N,t}^{(3)}$
$\displaystyle=\left[\frac{1}{\sqrt{N}}\int dxdydx^{\prime}dy^{\prime}\
\,(Q_{N,t}\otimes Q_{N,t}w_{N}Q_{N,t}\otimes
1)(x,y;x^{\prime},y^{\prime})\times\right.$
$\displaystyle\left.\qquad\qquad\times\varphi_{N,t}(y^{\prime})a_{x}^{*}a_{y}^{*}a_{x^{\prime}}\sqrt{\frac{N-\mathcal{N}}{N}}+\text{h.c.}\right],$
$\displaystyle\mathcal{L}_{N,t}^{(4)}$ $\displaystyle=\frac{1}{2N}\int
dxdydx^{\prime}dy^{\prime}(Q_{N,t}\otimes Q_{N,t}w_{N}Q_{N,t}\otimes
Q_{N,t})(x,y;x^{\prime},y^{\prime})a^{*}_{x}a^{*}_{y}a_{x^{\prime}}a_{y^{\prime}}$

with

$\displaystyle\mu_{N}(t)$
$\displaystyle:=\left\langle\varphi_{N,t},[(w_{N}(1-f_{N})\ast|\varphi_{N,t}|^{2}]\varphi_{N,t}\right\rangle$

and

$\displaystyle K_{1,N,t}$ $\displaystyle=Q_{N,t}\widetilde{K}_{1,N,t}Q_{N,t}$
$\displaystyle K_{2,N,t}$ $\displaystyle=Q_{N,t}\otimes
Q_{N,t}\widetilde{K}_{2,N,t}$

where $\widetilde{K}_{1,N,t}$ is the operator on $L^{2}(\mathbb{R}^{3})$ with
integral kernel

$\widetilde{K}_{1,N,t}(x,y)=\varphi_{N,t}(x)w_{N}(x-y)\overline{\varphi_{N,t}(y)}$

and $\widetilde{K}_{2,N,t}$ is a function in
$L^{2}(\mathbb{R}^{3}\times\mathbb{R}^{3})$:

$\widetilde{K}_{2,N,t}(x,y)=\varphi_{N,t}(x)w_{N}(x-y)\varphi_{N,t}(y).$

Next, we have to remove the singular correlation structure from $\Phi_{N,t}$.
Since $\Psi_{N,t}=U^{*}_{\varphi_{N,t}}\Phi_{N,t}$ and since
$U^{*}_{\varphi_{N,t}}$ just adds products of solutions of the nonlinear
equation (53), it is clear that all correlations developed by $\Psi_{N,t}$
must be contained in $\Phi_{N,t}$. To remove correlations from $\Phi_{N,t}$ we
apply the Bogoliubov transformation $T_{N,t}$ defined in (54). Unfortunately,
$T_{N,t}$ does not preserve the number of particles, and therefore it does not
leave the truncated Fock space $\mathcal{F}_{\perp\varphi_{N,t}}^{\leq N}$
invariant. Since $T_{N,t}$ only creates few particles, this should not be a
serious obstacle. To circumvent it, it seems natural to give up the
restriction on the number of particles and consider $\Phi_{N,t}$ as a vector
in the untruncated Fock space $\mathcal{F}_{\perp\varphi_{N,t}}$. The drawback
of this approach is the fact that the generator $\mathcal{L}_{N,t}$ computed
in (62) is defined only on sectors with at most $N$ particles. So, we proceed
as follows; first we approximate $\Phi_{N,t}$ by a new, modified, fluctuation
vector $\widetilde{\Phi}_{N,t}$, whose dynamics is governed by a modified
generator $\widetilde{\mathcal{L}}_{N,t}$ which, on the one hand, is close to
$\mathcal{L}_{N,t}$ when acting on vectors with a small number of particles
and, on the other hand, is well-defined on the full untruncated Fock space
$\mathcal{F}_{\perp\varphi_{N,t}}$. To define $\widetilde{\mathcal{L}}_{N,t}$
we proceed as follows. Starting from the expression on the r.h.s. of (62), we
replace first of all the factor $\sqrt{(N-\mathcal{N})(N-\mathcal{N}-1)}$ by
$N-\mathcal{N}$ and then we replace $\sqrt{N-\mathcal{N}}$ by
$\sqrt{N}G_{b}(\mathcal{N}/N)$ where $G_{b}(t)$ is the Taylor series for
$\sqrt{1-x}$ around $x=0$.

Finally, we add a term of the form
$C_{b}e^{C_{b}|t|}\mathcal{N}(\mathcal{N}/N)^{2b}$ with a sufficiently large
constant $C_{b}$. Since the generators $\mathcal{L}_{N}$ and
$\widetilde{\mathcal{L}}_{N}$ will act on states with small number of
particles, one expects this term to have a negligible effect on the dynamics
(on the other hand, it allows for better control the energy). With these
changes, one obtains the modified generator

$\begin{split}&\widetilde{\mathcal{L}}_{N,t}=\frac{N+1}{2}\left\langle\varphi_{N,t},[w_{N}(1-2f_{N})\ast|\varphi_{N,t}|^{2}]\varphi_{N,t}\right\rangle-\mu_{N}(t)\\\
&+\frac{1}{2}\left\langle\varphi_{N,t},[w_{N}\ast|\varphi_{N,t}|^{2}]\varphi_{N,t}\right\rangle\frac{\mathcal{N}(\mathcal{N}+1)}{N}\\\
&+\Big{[}\sqrt{N}a^{*}(Q_{N,t}[(w_{N}(1-f_{N}))\ast|\varphi_{N,t}|^{2}]\varphi_{N,t})G_{b}(\mathcal{N}/N)+{\rm
h.c.}\Big{]}\\\
&-\Big{[}a^{*}(Q_{N,t}[w_{N}\ast|\varphi_{N,t}|^{2}]\varphi_{N,t})\frac{\mathcal{N}}{\sqrt{N}}G_{b}(\mathcal{N}/N)+{\rm
h.c.}\Big{]}\\\ &+{\rm
d\Gamma}\Big{(}-\Delta+(w_{N}f_{N})*|\varphi_{N,t}|^{2}+K_{1,N,t}-\mu_{N,t}\Big{)}\\\
&+{\rm
d\Gamma}\Big{(}Q_{N,t}(w_{N}(1-f_{N})*|\varphi_{N,t}|^{2})Q_{N,t}\Big{)}\\\
&-{\rm
d\Gamma}\Big{(}Q_{N,t}(w_{N}*|\varphi_{N,t}|^{2})Q_{N,t}+K_{1,N,t}\Big{)}\frac{\mathcal{N}}{N}\\\
&+\bigg{[}\frac{1}{2}\int{\rm d}x{\rm
d}y\,K_{2,N,t}(x,y)a^{*}_{x}a^{*}_{y}\frac{N-\mathcal{N}}{N}+{\rm
h.c.}\bigg{]}\\\ &+\bigg{[}\frac{1}{\sqrt{N}}\int
dxdydx^{\prime}dy^{\prime}(Q_{N,t}\otimes Q_{N,t}w_{N}Q_{N,t}\otimes
1)(x,y;x^{\prime},y^{\prime})\times\\\
&\qquad\qquad\times\varphi_{N,t}(y^{\prime})a_{x}^{*}a_{y}^{*}a_{x^{\prime}}G_{b}(\mathcal{N}/N)+\text{h.c.}\bigg{]}\\\
&+\frac{1}{2N}\int dxdydx^{\prime}dy^{\prime}\ \,(Q_{N,t}\otimes
Q_{N,t}w_{N}Q_{N,t}\otimes
Q_{N,t})(x,y;x^{\prime},y^{\prime})a^{*}_{x}a^{*}_{y}a_{x^{\prime}}a_{y^{\prime}}\\\
&+C_{b}e^{C_{b}|t|}\,\mathcal{N}(\mathcal{N}/N)^{2b}.\end{split}$ (63)

Using this modified generator, we define the modified fluctuation dynamics
$\widetilde{\Phi}_{N,t}$ as the solution of the Schrödinger equation

$\displaystyle
i\partial_{t}\widetilde{\Phi}_{N,t}=\widetilde{\mathcal{L}}_{N,t}\widetilde{\Phi}_{N,t},$
(64)

with the appropriately transformed initial data. One can then prove that for
all $\alpha<(1-\beta)/2$, there exists a constant $C>0$ such that

$\|\Phi_{N,t}-\widetilde{\Phi}_{N,t}\|^{2}\leq CN^{-\alpha}\exp(C\exp(C|t|))$

for all $t\in\mathbb{R}$.

Finally, one applies the Bogoliubov transformation (54) to the modified
fluctuation evolution $\widetilde{\Phi}_{N,t}$ defined in (64). Let

$\xi_{N,t}=T_{N,t}\widetilde{\Phi}_{N,t}$ (65)

Then $\xi_{N,t}\in\mathcal{F}_{\perp\varphi_{N,t}}$ (with no restriction on
the number of particles) and it solves the Schrödinger equation

$\displaystyle i\partial_{t}\xi_{N,t}=\mathcal{G}_{N,t}\xi_{N,t},$ (66)

with the generator

$\displaystyle\mathcal{G}_{N,t}=(i\partial_{t}T_{N,t})T_{N,t}^{*}+T_{N,t}\widetilde{\mathcal{L}}_{N,t}T_{N,t}^{*}.$
(67)

As explained above, the application of the Bogoliubov transformation $T_{N,t}$
takes care of correlations and makes it possible for us to approximate the
evolution (66) with the unitary evolution $\mathcal{U}_{2,N}$, having as
generator the quadratic part of (67). The generator $\mathcal{U}_{2}$ that
appears in the statement of Theorem 4.1 is what one obtains from
$\mathcal{U}_{2,N}$ in limit $N\to\infty$.

###  5 Fock space approximation

The methods used in the proof of Theorem 4.1 were inspired strongly by the
result of Boccato, Cenatiempo and Schlein who proved in [24] an analogous
(i.e. for $\beta<1$) result in the Fock space setting (that is in the spirit
of (14)). In that work the authors also used the fluctuation dynamics approach
with the correlations described by the Bogoliubov transformation (54) (as in
[16]).

Earlier, the program of deriving effective dynamics in the Fock space setting
for singular interactions was initiated by Grillakis, Machedon and Margetis in
[83]. In this work they considered the mean-field regime and proved a result
of the type (14) for Coulomb potentials (see [84] for an extension). In [79],
Grillakis and Machedon considered the NLS regime with $\beta<1/3$.

The appraoch of Grillakis, Machedon and co-authors is in spirit very similar
to the one of [24]. However, there is one crucial difference that we would
like to point out. To this end let us briefly explain the approach in [79].

As mentioned before, in the NLS regime correlations play an important role and
to include them in the analysis, similarly to (54), Grillakis and Machedon
introduce a Bogoliubov transformation (in fact, they did not use this
terminology in [79])

$T(k_{t})=\exp\left(\frac{1}{2}\int
dxdy\,\left[\overline{k}_{t}(x,y)a_{x}a_{y}-\text{h.c.}\right]\right)$

for some function $k_{t}(x,y)$. They considered initial Fock space states of
the form

$\displaystyle\Phi(0)=W^{*}(\sqrt{N}\varphi_{0})T^{*}(k_{0})\Omega$ (68)

which are, a priori, more general than coherent states. In particular, by
choosing $k_{0}=0$ one obtains a coherent state as an initial state. Their
idea was to approximate the Fock space many-body evolution

$\Phi(t)=e^{-it\mathcal{H}_{N}}\Phi(0)$

by an effective quadratic evolution that would capture the creation and
evolution of correlations. To this end, Grillakis and Machedon postulate that

$\Phi(t)\approx e^{iN\xi(t)}W^{*}(\sqrt{N}\varphi_{t})T^{*}(k_{t})\Omega$

for some phase $\xi(t)$. Next, they introduce the so-called reduced dynamics

$\Phi_{\rm red}(t)=T(k_{t})W(\sqrt{N}\varphi_{t})\Phi(t).$

Note that $\Phi_{\rm red}(0)=c\Omega$ (for some $c$ such that $|c|=1$) and the
goal is to find such a $k_{t}$ so that also the evolved reduced state
satisfies

$\displaystyle\Phi_{\rm red}(t)\approx\Omega.$ (69)

Thus, as we can see, so far the general idea - the analysis of the fluctuation
dynamics - is the same as in the work Boccato, Cenatiempo and Schlein. Here
comes the main difference, however. While Boccato, Cenatiempo and Schlein
postulated the kernel of $k_{t}$ to be of the form (55) straight away,
Grillakis and Machedon derived an equation for $k_{t}$ so that (69) can be
satisfied. More precisely, using the properties of coherent states and
Bogoliubov transformations one can compute (similarly to (67)) the time
evolution

$i\partial_{t}\Phi_{\rm red}(t)=\mathcal{H}_{\rm red}(t)\Phi_{\rm red}(t)$

and determine $\mathcal{H}_{\rm red}(t)$. The goal is to choose such
$\varphi_{t}$ and $k_{t}$ so that

$\displaystyle\mathcal{H}_{\rm red}(t)=N\mu(t)+\int
L_{t}(x,y)a_{x}^{*}a_{y}dxdy+N^{-1/2}\mathcal{E}(t)$ (70)

where $\mathcal{E}(t)$ is an error term containing polynomials in $a$ and
$a^{*}$ up to degree four, $L_{t}(x,y)$ is the kernel of some (self-adjoint)
operator and $\mu(t)$ is an appropriate phase. This leads to the following set
of equations

$\displaystyle\begin{aligned}
i\partial_{t}\varphi_{t}&=\big{(}-\Delta+w_{N}*|\varphi_{t}|^{2}\big{)}\varphi_{t},\\\
i\partial_{t}\mathrm{sh}(2k_{t})&=-g_{N}^{T}\circ\mathrm{sh}(2k_{t})-\mathrm{sh}(2k_{t})\circ
g_{N}+m_{N}\circ\mathrm{ch}(2k_{t})+\mathrm{ch}^{T}(2k_{t})\circ
m_{N}\end{aligned}$ (71)

where $\circ$ denotes the composition of operators and the operators
$g_{N},m_{N}$ are given by

$\displaystyle g_{N}$ $\displaystyle=-\Delta+|\varphi_{t}|^{2}\ast
w_{N}+\widetilde{K}_{1}(t),$ $\displaystyle m_{N}$
$\displaystyle=\widetilde{K}_{2}(t).$

Here $\widetilde{K}_{1}(t)$ is the operator with kernel
$\widetilde{K}_{1}(t,x,y)=\overline{\varphi}_{t}(x)w_{N}(x-y)\varphi_{t}(y)$,
and $\widetilde{K}_{2}(t,x,y)=-\varphi_{t}(x)w_{N}(x-y)\varphi_{t}(y)$.

Thus, as one could expect from earlier results, the dynamics of $\varphi_{t}$
is governed by the Hartree equation in the NLS regime. What is important, the
Hartree equation is uncoupled from the equation for $k_{t}$. The latter
equation is in fact, under appropriate assumptions, equivalent to the
Bogoliubov equations (51) (see [130] for more details). In [120] this equation
has been derived in a different manner. In fact, the derivation of Grillakis
and Machedon is closely related to the diagonalization problem of quadratic
Hamiltonians (see [126] for more details).

In the Grillakis–Machedon approach one has to derive the properties of $k_{t}$
using the equation. In particular, in order to prove that $\mathcal{E}(t)$ in
(70) can be treated as an error term, various norms of $k_{t}$ need to be
estimated uniformly in $N$. Obtaining such estimates is more difficult in the
case of attractive interactions and this has been done in [53]. In [99] Kuz
extended the analysis to cover the case when $\beta<1/2$. In that work the
equations (71) remained unchanged.

To cover the NLS regime with $\beta>1/2$ it turns out that the uncoupled
equations (71) are not sufficient. In that case, in [80] Grillakis and
Machedon suggested a new set of coupled equations that would allow to treat
correlations for larger $\beta$. Briefly, the equations have been derived from
the condition that

$X_{1}=0,\,\,\,\text{and}\,\,\,X_{2}=0$

where $X_{1},X_{2}$ are one and two-body states in the Fock space given by

$\mathcal{H}_{\rm red}\Omega=(X_{0},X_{1},X_{2},X_{3},X_{4},0,\ldots).$

In fact, this condition can be obtained by minimizing the term $X_{0}$ over
$\varphi_{t}$ and $k_{t}$ (this can be seen as a time-dependent version of the
Beliaev theorem introduced in [59] and used in [57]). The resulting equations
are then quite similar to (71), but now the operator $m_{N}$ has to be
replaced by the operator $\Theta$ with the kernel

$\Theta(x,y)=-w_{N}(x-y)(\varphi_{t}(x)\varphi_{t}(y)+\frac{1}{2N}\mathrm{sh}(2k_{t})(x,y))$

and a similar $O\big{(}\frac{1}{N}\big{)}$ correction appears in the Hartree
equation. Under certain smoothness assumptions on $\varphi_{0}$ and $k_{0}$,
in [81] Grillakis and Machedon were able to show for
$\beta\in(\frac{1}{3},\frac{2}{3})$ that if $\varphi_{t}$ and $k_{t}$ are
solution of the coupled equations, then

$\|e^{it\mathcal{H}_{N}}W^{*}(\sqrt{N}\varphi_{0})T^{*}(k_{0})\Omega-e^{i\chi_{t}}W^{*}(\sqrt{N}\varphi_{t})T^{*}(k_{t})\Omega\|_{\mathcal{F}}\leq\frac{C}{N^{1/6}}$

locally in time (i.e. for some $t\in(0,T_{0})$) and for an appropriate phase
factor $\chi_{t}$.

The main difficulty in the proof of the result above lies in the analysis of
the coupled equations, in particular in establishing uniform in $N$ estimates
on the solutions $\varphi_{t}$ and $k_{t}$ in certain function spaces. This
result has been further extended for all $\beta<1$ (still locally in time) in
[82] and then for all $\beta<1$ but globally in time in [54] (see also [52]
for results in one dimension). Finally, let us mention that equations similar
to those coupled equations used by Grillakis and Machedon are often called
Hartree–Fock–Bogoliubov equations and have been analyzed also in [18, 11].

Acknowledgments. The author’s work was supported by the National Science
Centre (NCN project Nr. 2016/21/D/S). He would like to thank Phan Thành Nam
for useful discussions and remarks on the initial version of this manuscript
and the Institute of Mathematical Sciences at the National University of
Singapore for support, which allowed him to attend a section of the Workshop
on ‘Density Functionals for Many Particle Systems’ in September 2019.

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