Large Deviation Estimates for Weakly Interacting Bosons

Abstract

We study the many-body dynamics of an initially factorized bosonic wave function in the mean-field regime. We prove large deviation estimates for the fluctuations around the condensate. We derive an upper bound extending a recent result to more general interactions. Furthermore, we derive a new lower bound which agrees with the upper bound in leading order.

1 Introduction and Main Results

1.1 Introduction

We consider the dynamics of N bosons in the mean-field regime described through the bosonic wave function $\psi _{N,t} \in L_\mathrm{s}^2 ( \mathbb {R}^{3N})$, the symmetric subspace of $L^2 ( \mathbb {R}^{3N} )$. The bosons evolve according to the Schrödinger equation

$$\begin{aligned} i \partial _t \psi _{N,t} = H_N \psi _{N,t} \; \end{aligned}$$

(1.1)

where $H_N$ denotes the Hamiltonian

$$\begin{aligned} H_N = \sum _{j=1}^N - \Delta _{x_j} + \frac{1}{N} \sum _{i<j}^N v( x_i - x_j) \,. \end{aligned}$$

(1.2)

The coupling constant 1/N in front of the interaction term corresponds to weak and long-range interactions of mean-field type. In the following we assume the two-particle interaction potential v to satisfy

$$\begin{aligned} v^2 \le C \left( 1 -\Delta \right) \end{aligned}$$

(1.3)

for a positive constant $C>0$. We consider factorized initial data $\psi _{N,0} = \varphi ^{\otimes N}$ exhibiting complete Bose–Einstein condensation (BEC), i.e. their reduced one-particle density $\gamma _N$ satisfies

$$\begin{aligned} \gamma _N = \vert \varphi \rangle \langle \varphi \vert \quad \text {for every { N},} \end{aligned}$$

(1.4)

for a one-particle orbital $\varphi \in H^4 (\mathbb {R}^3)$. Although the factorization is not preserved along the time evolution, the property of BEC is known to be preserved, i.e. the reduced one-particle density $\gamma _{N,t}$ associated to the solution $\psi _{N,t}$ of the Schrödinger equation (1.1) satisfies

$$\begin{aligned} \gamma _{N,t} \rightarrow \vert \varphi _t \rangle \langle \varphi _t \vert \quad \text {as} \quad N \rightarrow \infty \end{aligned}$$

(1.5)

where the time evolution of the condensate wave function $\varphi _t$ is governed by the Hartree equation

$$\begin{aligned} i \partial _t \varphi _t = h_\mathrm{H}(t) \; \varphi _t, \quad \text {with} \quad h_\mathrm{H} (t) = - \Delta + v*\vert \varphi _t \vert ^2 \end{aligned}$$

(1.6)

with initial data $\varphi _{0} = \varphi$. (For more details see e.g. [1,2,3, 10,11,12,13,14,15, 19, 25, 26].)

1.2 Main Results

From a probabilistic point of view, BEC implies a law of large numbers for bounded one-particle observables. To be more precise, for a bounded, self-adjoint one-particle operator O on $L^2 ( \mathbb {R}^3)$ we define the N-particle operator

$$\begin{aligned} O^{(j)} = \mathbb {1} \otimes \cdots \otimes \mathbb {1} \otimes O \otimes \mathbb {1} \otimes \cdots \otimes \mathbb {1} \end{aligned}$$

(1.7)

as the operator acting as O on the j-th particle and as identity elsewhere. We consider $O^{(j)}$ as a random variable with probability distribution determined by $\psi _N$ and given through

$$\begin{aligned} \mathbb {P}_{\psi _{N}} \left[ O^{(j)} \in A \right] = \langle \psi _{N}, \chi _A \left( O^{(j)} \right) \psi _N \rangle \end{aligned}$$

(1.8)

where $\chi _A$ denotes the characteristic function of the set $A \subset \mathbb {R}$. Since the expectation value with respect to factorized states $\psi _N = \varphi ^{\otimes N}$ is

$$\begin{aligned} \mathbb {E}_{\varphi ^{\otimes N}} \left[ O^{(j)} \right] = \langle \varphi , \; O \varphi \rangle \quad \text {for all} \quad j = 1, \dots ,N , \end{aligned}$$

(1.9)

the random variables are i.i.d. and thus, in this case, they satisfy a law of large numbers, i.e. for the averaged sum $O_N = N^{-1} \sum _{j=1}^N \left( O^{(j)} - \langle \varphi , O \varphi \rangle \right)$, we have for any $\delta >0$

$$\begin{aligned} \mathbb {P}_{\varphi ^{\otimes N}} \left[ \; \vert O_N \vert > \delta \right] \rightarrow 0 \quad \text {as} \quad N \rightarrow \infty \; . \end{aligned}$$

(1.10)

The large deviation principle goes one step further and investigates the rate of convergence through the rate function given by

$$\begin{aligned} \Lambda ^*_{\psi _{N}} (x) := - \lim _{N \rightarrow \infty } N^{-1} \log \mathbb {P}_{\psi _N} \left[ O_N > x \right] \; , \end{aligned}$$

(1.11)

assuming the limit exists. For i.i.d. random variables, i.e. $\psi _N=\varphi ^{\otimes N}$, Cramér’s Theorem [9] shows that the rate function is given by

$$\begin{aligned} \Lambda _{\varphi ^{\otimes N}}^* ( x) = \inf _{\lambda \in \mathbb {R}} \left[ - \lambda x + \Lambda _{\varphi ^{\otimes N}} (\lambda ) \right] \end{aligned}$$

(1.12)

where the rate function’s Legendre–Fenchel transform $\Lambda _{\varphi ^{\otimes N}}$ is the logarithmic moment generating function

$$\begin{aligned} \Lambda _{\varphi ^{\otimes N}} ( \lambda ) = \log \langle \varphi , \; e^{\lambda \left( O^{(1)} - \langle \varphi , O \varphi \rangle \right) }\varphi \rangle . \end{aligned}$$

(1.13)

Recall that we consider the time evolution of factorized initial data with respect to (1.1). Thus, initially the random variables are i.i.d. and therefore a law of large numbers and a large deviation principle with rate function (1.13) hold true. Although for times $t >0$ the random variables are not i.i.d. anymore (as the factorization is not preserved), the condensation property (1.5) ensures the validity of a law of large numbers [4], i.e. for any $\delta >0$

$$\begin{aligned} \mathbb {P}_{\psi _{N,t}} \left[ \vert O_N \vert > \delta \right] \rightarrow 0 \quad \text {as} \quad N \rightarrow \infty \; . \end{aligned}$$

(1.14)

In the following theorem, we show that for $t >0$ large deviation estimates hold true as well.

Before stating our main theorem, let us introduce some notation. For O a bounded self-adjoint operator on $L^2(\mathbb {R}^3)$, we define the norm

$$\begin{aligned} {\left| \left| \left| O \right| \right| \right| } = \Vert \left( - \Delta + 1 \right) O \left( - \Delta + 1 \right) ^{-1} \Vert \end{aligned}$$

(1.15)

where $\Vert \cdot \Vert$ denotes the usual operator norm. Moreover, for $0\le s \le t$, let $f_{s;t} \in L^2 ( \mathbb {R}^3 )$ denote the solution to

$$\begin{aligned} i\partial _s f_{s;t} = \left( h_\mathrm{{H}}(s) + \widetilde{K}_{1,s} - \widetilde{K}_{2,s} J \right) f_{s;t} \end{aligned}$$

(1.16)

with initial datum $f_{t;t} = q_t O \varphi _t = O \varphi _t -\langle \varphi _t, \; O \varphi _t \rangle \varphi _t$, where $q_s = 1- \vert \varphi _s \rangle \langle \varphi _s \vert$, J denotes the anti-linear operator $Jf = \overline{f}$, the Hartree Hamiltonian $h_\mathrm{H}(s)$ is defined in (1.6), and

$$\begin{aligned} \widetilde{K}_{1,s}= q_s K_{1,s} \; q_s, \quad \widetilde{K}_{2,s} = q_s K_{2,s} \; q_s \end{aligned}$$

(1.17)

with $K_{j,s}$ the operators defined by the integral kernels

$$\begin{aligned} K_{1,s} (x,y) = v(x-y) \varphi _s (x) \overline{\varphi _s (y)}, \quad K_{2,s} (x,y) = v(x-y)\varphi _s (x) \varphi _s (y)\; . \end{aligned}$$

(1.18)

Theorem 1.1

Assume that the interaction potential v satisfies (1.3) and $\varphi \in H^4 ( \mathbb {R}^3 )$ with $\Vert \varphi \Vert _{2} =1$. For $t >0$, let $\psi _{N,t}$ denote the solution of the Schrödinger equation (1.1) with initial datum $\psi _{N,0} = \varphi ^{\otimes N}$ and $\varphi _t$ the solution to the Hartree equation (1.6) with $\varphi _0 = \varphi$.

Let O be a self-adjoint operator on $L^2 \left( \mathbb {R}^3 \right)$ such that ${\left| \left| \left| O \right| \right| \right| } < \infty$, and let $f_{s;t}$ be as defined above. With $O^{(j)}$ from (1.7), we define $O_{N,t} = N^{-1} \sum _{j=1}^N \left( O^{(j)} - \langle \varphi _t, O \varphi _t \rangle \right)$. There exist $C_1,C_2>0$ (independent of O) such that

• (i) for all $t \ge 0$ and $0\le x \le e^{- e^{C_1 t}} \Vert f_{0;t}\Vert _2^2/ {\left| \left| \left| O \right| \right| \right| }$

  $$\begin{aligned} \limsup _{N \rightarrow \infty } N^{-1}\log \mathbb {P}_{\psi _{N,t}} \left[ O_{N,t} > x\right] \le - \frac{x^2}{2 \Vert f_{0;t}\Vert _2^2} + x^3 \frac{C_1 e^{e^{C_1 t}}{\left| \left| \left| O \right| \right| \right| }^3}{\Vert f_{0;t}\Vert _2^6} . \end{aligned}$$

  (1.19)

• (ii) for all $t \ge 0$ and $0\le x \le e^{-e^{ C_2 t} } \Vert f_{0;t}\Vert _{2}^4 / ( C_2 {\left| \left| \left| O \right| \right| \right| }^3)$

  $$\begin{aligned} \liminf _{N\rightarrow \infty } N^{-1} \log \mathbb {P}_{\psi _N} \left[ O_{N,t} > x \right] \ge - \frac{x^2}{2 \Vert f_{0;t}\Vert _2^2} - x^{5/2} \frac{C_2 e^{e^{C_2 t}} {\left| \left| \left| O \right| \right| \right| }^{3/2}}{\Vert f_{0;t}\Vert _{2}^4} . \end{aligned}$$

  (1.20)

We remark that the function $f_{s;t}$ is determined through Bogoliubov’s quasi-free approximation of the fluctuations around the condensate (see (3.27) below). For a detailed explanation see [4, Theorem 2.2 and subsequent Remark]. In fact, with the notation of [4], $f_{s;t} = q_s (U(t;s) + J V(t;s)) O \varphi _t$.

The bounds (1.19) and (1.20) show that the rate function of the system is, if it exists, for sufficiently small $x>0$ given by

$$\begin{aligned} \Lambda ^*_{\psi _{N,t}}(x) = - \frac{x^2}{2 \Vert f_{0;t}\Vert _2^2} + O(x^{5/2}) . \end{aligned}$$

(1.21)

In particular, Theorem 1.1 determines the rate function $\Lambda ^*_{\psi _{N,t}}$ up to quadratic order. Note that for time $t=0$ the quadratic term in (1.21) agrees with the one of Cramér’s theorem (1.13) as

$$\begin{aligned} \Vert f_{0;0} \Vert _2^2 = \Vert q_0 O \varphi \Vert _2^2 = \langle \varphi , \, O^2 \varphi \rangle - \vert \langle \varphi , \, O \varphi \rangle \vert ^2 . \end{aligned}$$

(1.22)

In the regime $x= O(1/\sqrt{N})$, our findings agree with the central limit theorems previously obtained in [4, 7] proving that

$$\begin{aligned} \lim _{N \rightarrow \infty }\mathbb {P}_{\psi _{N,t}} \left[ \sqrt{N} O_{N,t} < x \right] = \frac{1}{\sqrt{2\pi } \Vert f_{0;t}\Vert _2} \int _{- \infty }^x e^{-\frac{r^2}{2 \Vert f_{0;t}\Vert _2^2}} dr \; . \end{aligned}$$

(1.23)

We remark that a central limit theorem still holds true when replacing the weak mean field potential (given by $N^{3\beta } v(N^\beta x )$ for $\beta =0$) with more singular interactions in the intermediate regime (corresponding to $0<\beta <1$) [23]. In the physically most relevant Gross–Pitaevski regime ($\beta =1$), a central limit theorem holds for the ground state [24] showing, in particular, that the fluctuations around the condensate are approximately quasi-free. The validity of large deviation estimates for fluctuations around the condensate for Bose–Einstein condensates in the ground state is still an open question, however.

The proof of Theorem 1.1 (given in Sect. 2) is based on a lower and an upper bound on the logarithmic moment generating function stated in the following (and proven in Sect. 3).

Theorem 1.2

Under the same assumptions as in Theorem 1.1,

• (i’) there exists a constant $C_1>0$ such that for all $0\le \lambda \le e^{-e^{C_1 t}} / {\left| \left| \left| O \right| \right| \right| }$ we have

  $$\begin{aligned} \limsup _{N\rightarrow \infty } N^{-1} \log \mathbb {E}_{\psi _{N,t}} e^{\lambda N O_{N,t} } \le \frac{\lambda ^2}{2} \Vert f_{0;t}\Vert _2^2 + C_1 e^{e^{C_1 t}}\lambda ^3 {\left| \left| \left| O \right| \right| \right| }^3 . \end{aligned}$$

  (1.24)

• (ii’) there exists a constant $C_2 >0$ such that for all $0 \le \lambda \le e^{-e^{C_2 t}} / {\left| \left| \left| O \right| \right| \right| }$ we have

  $$\begin{aligned} \liminf _{N\rightarrow \infty } N^{-1} \log \mathbb {E}_{\psi _{N,t}} e^{\lambda N O_{N,t} } \ge \frac{\lambda ^2}{2} \Vert f_{0;t}\Vert _2^2 - C_2e^{e^{C_2 t}} \lambda ^3 {\left| \left| \left| O \right| \right| \right| }^3 . \end{aligned}$$

  (1.25)

The upper bound (i’) on the logarithmic moment generating function (as well as the resulting upper bound on the rate function in Theorem 1.1(i)) is an extension of the large deviation estimate obtained in [18] to more general interaction potentials. In particular, the assumptions on the potential in Theorem 1.1 involve the physical interesting Coulomb potential which was excluded by the assumptions $v \in L^1( \mathbb {R}^3) \cap L^\infty ( \mathbb {R}^3)$ in [18]. Note that the term cubic in $\lambda$ in (1.24) depends on time through a double exponential, compared to a term exponential in time in [18], which is a consequence of allowing less regular interaction potentials here (entering the proof through Lemmas 3.1 and 3.2). The quadratic term of (1.24) agrees with the findings from [[18, Theorem 1.1]]. In particular, the definition of $f_{0;t}$ in (1.16) here is the same as in [18, Eq. (1.1)].¹

In contrast to [18], we prove here also a matching lower bound (ii’) on the logarithmic moment generating function (resulting, together with the upper bound, in the lower bound on the rate function in Theorem 1.1(ii)). This allows to determine the rate function $\Lambda _{\psi _{N,t}}^*$ up to quadratic order. In particular, we show that $\Lambda _{\psi _{N,t}}^*$ coincides up to quadratic order with the rate function of Bogoliubov’s quasi-free approximation of the fluctuat ions around the condensate. Whether this holds true for higher order terms remains an open question. In fact, we don’t expect that the rate function of Bogoliubov’s approximation of the fluctuations agrees with $\Lambda ^*_{\psi _{N,t}}$ to all orders.

2 Proof of Theorem 1.1

The proof of Theorem 1.1 uses ideas of the proof of Cramer’s theorem involving estimates on the logarithmic moment generating function in Theorem 1.2. The upper bound (i) follows by Chebychev’s inequality from (i’), while the proof of the lower bound is more involved and uses both (i’) and (ii’).

Proof

Upper bound (i): For $\lambda >0$, we have

$$\begin{aligned} \mathbb {P}_{\psi _{N,t}} \left[ O_{N,t}> x \right] = \mathbb {P}_{\psi _{N,t}} \left[ e^{-\lambda N x}e^{ \lambda N O_{N,t} } > 1 \right] . \end{aligned}$$

(2.1)

Chebychev’s inequality implies that

$$\begin{aligned} \mathbb {P}_{\psi _{N,t}} \left[ O_{N,t}> x \right] \le e^{-\lambda N x}\; \mathbb {E}_{\psi _{N,t}} \left[ e^{ \lambda N O_{N,t} } \right] , \end{aligned}$$

(2.2)

and we find with Theorem 1.1 for $\lambda < e^{-e^{C_1 t}} / {\left| \left| \left| O \right| \right| \right| }$

$$\begin{aligned} \limsup _{N \rightarrow \infty } N^{-1} \log \mathbb {P}_{\psi _{N,t}} \left[ O_{N,t}> x \right] \le -\lambda x + \frac{\lambda ^2}{2} \Vert f_{0;t}\Vert _2^2 + C_1 e^{e^{C_1 t}}\lambda ^3 {\left| \left| \left| O \right| \right| \right| }^3 \; . \end{aligned}$$

(2.3)

For

$$\begin{aligned} x < e^{-e^{C_1 t}} \Vert f_{0;t}\Vert _2^2 / {\left| \left| \left| O \right| \right| \right| } \end{aligned}$$

(2.4)

let $\lambda = x/ \Vert f_{0;t}\Vert _2^2$. Then

$$\begin{aligned} \limsup _{N \rightarrow \infty } N^{-1} \log \mathbb {P}_{\psi _{N,t}} \left[ O_{N,t}> x \right] \le - \frac{x^2}{2 \Vert f_{0;t}\Vert _2^2} +\frac{x^3 C_1 e^{e^{C_1 t}}{\left| \left| \left| O \right| \right| \right| }^3}{\Vert f_{0;t}\Vert _2^6} \; . \end{aligned}$$

(2.5)

Lower Bound (ii): For arbitrary $\varepsilon >0$, we have

$$\begin{aligned} N^{-1} \log \mathbb {P}_{\psi _{N,t}} \left[ O_{N,t} > x\right] \ge N^{-1}\log \mathbb {P}_{\psi _{N,t}} \left[ O_{N,t} \in ( x , x+\varepsilon )\right] \end{aligned}$$

(2.6)

and it suffices to consider in the following

$$\begin{aligned} \mathbb {P}_{\psi _{N,t}} \left[ O_{N,t} \in ( x, x+\varepsilon )\right]&= \langle \psi _{N,t}, \chi _{(x, x+ \varepsilon )} \left( O_{N,t} \right) \psi _{N,t} \rangle \nonumber \\&= \langle \psi _{N,t}, \chi _{(x, x+\varepsilon )} \left( O_{N,t} \right) e^{- \lambda N O_{N,t} } e^{\lambda N O_{N,t}} \psi _{N,t} \rangle . \end{aligned}$$

(2.7)

On the support of $\chi _{(x, x+ \varepsilon )} (O_{N,t})$ we have $e^{-\lambda N O_{N,t}}\ge e^{- \left( x + \varepsilon \right) \lambda N }$ for $\lambda >0$ and we find

$$\begin{aligned} \mathbb {P}_{\psi _{N,t}} \left[ O_{N,t} \in (x, x+ \varepsilon )\right] \ge e^{- \left( x + \varepsilon \right) \lambda N }\langle \psi _{N,t}, \chi _{(x, x+\varepsilon )} \left( O_{N,t} \right) e^{\lambda N O_{N,t}} \psi _{N,t} \rangle . \end{aligned}$$

(2.8)

It is easy to check that

$$\begin{aligned} \widetilde{\mathbb {P}}_{\psi _{N,t}} \left[ O_{N,t} \in A \right] = e^{-N \Lambda _{N,t} ( \lambda ) } \langle \psi _{N,t}, \chi _{A} \left( O_{N,t} \right) e^{\lambda N O_{N,t} } \psi _{N,t} \rangle \end{aligned}$$

(2.9)

for $A\subset \mathbb {R}$ and

$$\begin{aligned} \Lambda _{N,t} ( \lambda ) = N^{-1}\log \langle \psi _{N,t}, e^{\lambda N O_{N,t} } \psi _{N,t} \rangle \end{aligned}$$

(2.10)

defines a probability distribution. We use (2.9) to rewrite the expression (2.8) as

$$\begin{aligned} \mathbb {P}_{\psi _{N,t}} \left[ O_{N,t} \in (x, x+ \varepsilon )\right]&\ge e^{ \lambda N \left( - x - \varepsilon \right) + N\Lambda _{N,t}(\lambda ) } \widetilde{\mathbb {P}}_{\psi _{N,t}} \left[ O_{N,t}\in (x, x+ \varepsilon ) \right] \nonumber \\&= e^{ \lambda N \left( - x - \varepsilon \right) +N \Lambda _{N,t}(\lambda ) } \left( 1 -\widetilde{\mathbb {P}}_{\psi _{N,t}} \left[ O_{N,t} \le x \right] \right. \nonumber \\&\quad \left. - \widetilde{\mathbb {P}}_{\psi _{N,t}} \left[ O_{N,t} \ge x + \varepsilon \right] \right) . \end{aligned}$$

(2.11)

Similarly to the upper bound’s proof, we use Chebychev’s inequality for the last two terms on the r.h.s. and obtain for arbitrary $\lambda , \mu , \widetilde{\mu } \ge 0$

$$\begin{aligned}&\mathbb {P}_{\psi _{N,t}} \left[ O_{N,t} \in ((x, x+ \varepsilon )\right] \nonumber \\&\quad \ge e^{ \lambda N \left( - x - \varepsilon \right) +N \Lambda _{N,t}(\lambda ) } \nonumber \\&\quad \quad \times \left( 1 -e^{N \left( - \Lambda _{N,t} (\lambda ) + \widetilde{\mu } x + \Lambda _{N,t} (\lambda - \widetilde{\mu }) \right) }-e^{N \left( - \Lambda _{N,t} (\lambda ) - \mu (x + \varepsilon ) +\Lambda _{N,t} (\lambda + \mu )\right) } \right) \,. \end{aligned}$$

(2.12)

For given $x \in \mathbb {R}$, we need to choose $\lambda ,\mu , \widetilde{\mu }$ and $\varepsilon$ such that both

$$\begin{aligned} \limsup _{N\rightarrow \infty } \left( - \Lambda _{N,t} (\lambda ) + \widetilde{\mu } x + \Lambda _{N,t} (\lambda - \widetilde{\mu }) \right) <0 \end{aligned}$$

(2.13)

and

$$\begin{aligned} \limsup _{N\rightarrow \infty } \left( - \Lambda _{N,t} (\lambda ) - \mu (x + \varepsilon ) +\Lambda _{N,t} (\lambda + \mu ) \right) <0\,. \end{aligned}$$

(2.14)

In fact, for $0< \delta < \widetilde{\delta }$, let $\lambda = x ( 1 + \delta ) / \Vert f_{0;t} \Vert _2^2, \varepsilon = x \widetilde{\delta }, \widetilde{\mu } = \delta x / \Vert f_{0;t}\Vert ^2_{2}$ and $\mu = ( \widetilde{\delta } - \delta ) x / \Vert f_{0;t}\Vert _{2}^2$. Then, as long as,

$$\begin{aligned} 0\le x \le \frac{\min \lbrace e^{-e^{C_1 t}}, \; e^{-e^{C_2 t}}/(1 + \delta )\rbrace \Vert f_{0;t}\Vert _{2}^2}{ {\left| \left| \left| O \right| \right| \right| } } \end{aligned}$$

(2.15)

we have with Theorem 1.2

$$\begin{aligned}&\limsup _{N\rightarrow \infty } \left( -\Lambda _{N,t}(\lambda ) + \widetilde{\mu }x + \Lambda _{N,t}(\lambda -\widetilde{\mu }) \right) \nonumber \\&\quad \le - \frac{\lambda ^2}{2} \Vert f_{0;t}\Vert _{2}^2 + C_2 e^{e^{C_2 t}} \lambda ^3 {\left| \left| \left| O \right| \right| \right| }^3 + \widetilde{\mu }x + \frac{(\lambda -\widetilde{\mu })^2}{2}\Vert f_{0;t}\Vert _{2}^2 + C_1 e^{e^{C_1 t}} (\lambda -\widetilde{\mu })^3 {\left| \left| \left| O \right| \right| \right| }^3\nonumber \\&\quad = - \frac{ x^2 \delta ^2}{2 \Vert f_{0;t}\Vert _{2}^2} + \frac{x^3 {\left| \left| \left| O \right| \right| \right| }^3}{\Vert f_{0;t}\Vert ^6} \left( C_2 e^{e^{C_2 t}}(1+\delta )^3 + C_1 e^{e^{C_1 t}}\right) <0 \end{aligned}$$

(2.16)

if

$$\begin{aligned} x {\left| \left| \left| O \right| \right| \right| }^3 \left( C_2 e^{e^{C_2 t}} (1+\delta )^3 + C_1 e^{e^{C_1 t}}\right) < \frac{ \delta ^2}{2} \Vert f_{0;t}\Vert _{2}^4\,. \end{aligned}$$

(2.17)

Similarly, as long as

$$\begin{aligned} 0\le x \le \frac{ \min \lbrace e^{-e^{C_1 t}}/(1+\widetilde{\delta }),e^{-e^{C_2 t}}/(1+\delta ) \rbrace \Vert f_{0;t}\Vert _{2}^2 }{{\left| \left| \left| O \right| \right| \right| }} \end{aligned}$$

(2.18)

we have from Theorem 1.2

$$\begin{aligned}&\limsup _{N\rightarrow \infty } \left( -\Lambda _{N,t}(\lambda ) - \mu (x + \varepsilon ) + \Lambda _{N,t}(\lambda + \mu ) \right) \nonumber \\&\quad \le - \frac{\lambda ^2}{2}\Vert f_{0;t}\Vert _{2}^2 + C_2 e^{e^{C_2 t}}\lambda ^3 {\left| \left| \left| O \right| \right| \right| }^3- \mu (x + \varepsilon ) + \frac{(\lambda + \mu )^2}{2} \Vert f_{0;t}\Vert _{2}^2 \nonumber \\&\quad + C_1 e^{e^{C_1 t}} (\lambda + \mu )^3 {\left| \left| \left| O \right| \right| \right| }^3 = - \frac{ x^2 (\widetilde{\delta }-\delta )^2}{2 \Vert f_{0;t}\Vert _{2}^2 } + \frac{x^3 {\left| \left| \left| O \right| \right| \right| }^3}{\Vert f_{0;t}\Vert _{2}^6} \left( C_2 e^{e^{C_2 t}}(1+\delta )^3 + C_1 e^{e^{C_1 t}} (1+\widetilde{\delta })^3\right) <0 \end{aligned}$$

(2.19)

if

$$\begin{aligned} x {\left| \left| \left| O \right| \right| \right| }^3 \left( C_2 e^{e^{C_2 t}} (1+\delta )^3 + C_1 e^{e^{C_1 t}}(1+\widetilde{\delta })^3\right) < \frac{ (\widetilde{\delta }- \delta )^2}{2} \Vert f_{0;t}\Vert _{2}^4\,. \end{aligned}$$

(2.20)

In particular, under conditions (2.15), (2.17), (2.18) and (2.20), we have from (2.12)

$$\begin{aligned} \liminf _{N \rightarrow \infty } N^{-1} \log \mathbb {P}\left[ O_{N,t} >x\right]&\ge \liminf _{N \rightarrow \infty } \Lambda _{N,t}(\lambda ) -\lambda (x+\varepsilon ) \nonumber \\&\ge - \frac{x^2 }{2 \Vert f_{0;t}\Vert _{2}^2}(1+ 2 \widetilde{\delta }(1+\delta ) ) - C_2 e^{e^{C_2 t}} x^3 (1+\delta )^3 \frac{ {\left| \left| \left| O \right| \right| \right| }^3}{ \Vert f_{0;t}\Vert _{2}^6}\,.\nonumber \\ \end{aligned}$$

(2.21)

With $C_3=\max \{C_1,C_2\}$ we can take

$$\begin{aligned} \delta ^2 = \left( \frac{\tilde{\delta }}{2}\right) ^2 = 70\, C_3 e^{e^{C_3 t} } \frac{ x {\left| \left| \left| O \right| \right| \right| }^3}{ \Vert f_{0;t}\Vert _{2}^4} \end{aligned}$$

(2.22)

and (2.17) as well as (2.20) are satisfied as long as $\delta <1$. For

$$\begin{aligned} x \le \min \left\{ \tfrac{1}{3} e^{-e^{C_3 t}} \Vert f_{0;t}\Vert _{2}^2 / {\left| \left| \left| O \right| \right| \right| }, \; \tfrac{1}{70\, C_3} e^{- e^{ C_3 t} } \Vert f_{0;t}\Vert _{2}^4 / {\left| \left| \left| O \right| \right| \right| }^3 \right\} \end{aligned}$$

(2.23)

we can thus conclude that

$$\begin{aligned} \liminf _{N \rightarrow \infty } N^{-1} \log \mathbb {P}\left[ O_{N,t}>x\right] \ge - \frac{x^2 }{2 \Vert f_{0;t}\Vert _{2}^2} - C_4 e^{e^{C_4 t}} x^{5/2} \frac{{\left| \left| \left| O \right| \right| \right| }^{3/2}}{\Vert f_{0;t}\Vert _{2}^4} \end{aligned}$$

(2.24)

for $C_4 >0$ large enough. We shall show in Lemma 3.2 below that $\Vert f_{0;t}\Vert _2 \le {\left| \left| \left| O \right| \right| \right| } e^{C|t|}$ for suitable $C>0$, which allows for the simpler condition on x as stated in Theorem 1.1(ii). $\square$

3 Proof of Theorem 1.2

3.1 Properties of $K_{j,s}$ and $f_{s;t}$

In this section, we show in Lemma 3.2 useful estimates on the function $f_{s;t}$ defined in (1.16). To this end, we first collect in Lemma 3.1 properties of the kernels $K_{j,s}$ defined in (1.18). These Lemmas are the crucial ingredient to generalize the result of [18] to more singular interaction potentials. The main difference is that in [18] estimates of the form (3.6) rely on the propagation of the $H^1$-norm of $\varphi _s$ using, in particular, that by conservation of energy and (1.3) we have

$$\begin{aligned} \Vert \varphi _s \Vert _{H^1(\mathbb {R}^3)} \le C \Vert \varphi \Vert _{H^1(\mathbb {R}^3)} \end{aligned}$$

(3.1)

for a constant $C>0$. In contrast, here, we need the propagation of higher Sobolev norms of $\varphi _s$ in (3.11), i.e. bounds of the form

$$\begin{aligned} \Vert \varphi _s \Vert _{H^k (\mathbb {R}^3)} \le C e^{C s}\Vert \varphi _0 \Vert _{H^k (\mathbb {R}^3)} \end{aligned}$$

(3.2)

for $k \ge 2$ which are well-known (see e.g. [8]). These lead to bounds exponential in time in (3.6) and, thus, to bounds double exponential in time in (3.13) because of the use of a Gronwall type estimate. These bounds effect Lemma 3.4 and, consequently, the error terms in Theorems 1.1 and  1.2.

Lemma 3.1

For $s \in \mathbb {R}$ and v satisfying (1.3), let $\varphi _s$ denote the solution to the Hartree equation (1.6) with initial data $\varphi \in H^4( \mathbb {R}^3)$. There exists a constant $C>0$ such that

$$\begin{aligned} \Vert v * \vert \varphi _s \vert ^2 \Vert _{\infty }&\le C \; , \end{aligned}$$

(3.3)

$$\begin{aligned} \Vert \nabla v * \vert \varphi _s \vert ^2 \Vert _{\infty }&\le C\, e^{C \vert s \vert } \ , \quad \Vert \Delta v * \vert \varphi _s \vert ^2 \Vert _{\infty } \le C e^{C \vert s \vert } \; , \end{aligned}$$

(3.4)

and, furthermore for $j=1,2$ and $f \in H^2 ( \mathbb {R}^3)$

$$\begin{aligned} \Vert K_{j,s}&\Vert _{L^2 \left( \mathbb {R}^3 \times \mathbb {R}^3 \right) } \le C \end{aligned}$$

(3.5)

$$\begin{aligned} \Vert \nabla K_{j,s} f \Vert _2 \le C e^{C \vert s \vert } \Vert f \Vert _{H^1 (\mathbb {R}^3)},&\quad \Vert \Delta K_{j,s} f \Vert _2 \le C e^{C \vert s \vert } \Vert f \Vert _{H^2 (\mathbb {R}^3)} . \end{aligned}$$

(3.6)

Proof

From (1.3), we have

$$\begin{aligned} \vert \left( v* \vert \varphi _s \vert ^2 \right) (x) \vert \le \int \vert v(x-y) \vert ^2 \; \vert \varphi _s (y) \vert ^2 dy+ C \Vert \varphi _s \Vert _{2}^2 \le C \Vert \varphi _s \Vert _{H^1( \mathbb {R}^3)}^2 \end{aligned}$$

(3.7)

and (3.3) follows from (3.1). Similarly, since

$$\begin{aligned} \nabla v* \vert \varphi _s \vert ^2 = 2 v* \mathrm {Re}\, \overline{\varphi _s} \nabla \varphi _s \end{aligned}$$

(3.8)

we have with (3.2)

$$\begin{aligned} \Vert \nabla v * \vert \varphi _s \vert ^2 \Vert _{\infty } \le C \Vert \varphi _s \Vert _{H^2 (\mathbb {R}^3)}^2 \le C e^{C \vert s \vert } \, . \end{aligned}$$

(3.9)

The second bound in (3.4) follows in the same way.

Moreover, with (1.18), we have for $j=1,2$

$$\begin{aligned} \Vert K_{j,s} \Vert _{L^2 \left( \mathbb {R}^3 \times \mathbb {R}^3 \right) }^2 = \int \vert \varphi _s (x) \vert ^2 v^2(x-y)\vert \varphi _s (y)\vert ^2 \; dxdy = \langle \varphi _s, \;\left( v^2 * \vert \varphi _s \vert ^2 \right) \varphi _s \rangle \end{aligned}$$

(3.10)

and thus (3.5) follows by arguing as in (3.7) above. In order to show (3.6), we integrate by parts

$$\begin{aligned} \int \left( \nabla _x K_{1,s} \right) (x,y) f (y ) dy =&\int v(x-y) \left( \nabla \varphi _s\right) (x ) \overline{\varphi _s (y)} f(y) dy \nonumber \\&+ \int v(x-y) \varphi _s (x) \left( \overline{\nabla \varphi _s}\right) (y) f (y) dy \nonumber \\&+ \int v(x-y) \varphi _s (x) \overline{ \varphi _s (y)} \nabla f (y) dy \end{aligned}$$

(3.11)

and estimate with (1.3) similarly as above

$$\begin{aligned} \Vert \nabla K_{1,s} f \Vert _2 \le C \Vert \varphi _s \Vert _{H^3( \mathbb {R}^3)}^2 \Vert f \Vert _{2}+ C \Vert \varphi _s \Vert _{H^2 ( \mathbb {R}^3)}^2 \Vert f \Vert _{H^1 (\mathbb {R}^3)} \le C e^{C \vert s \vert } \Vert f \Vert _{H^1 (\mathbb {R}^3)} \; . \end{aligned}$$

(3.12)

The second estimate in (3.6) follows in the same way. $\square$

Because of (3.2), one readily checks that the same bounds hold with $\widetilde{K}_{j,s}$ in place of $K_{j,s}$. Those bounds are in fact the ones we need below.

Lemma 3.2

Under the same assumptions as in Theorem 1.2, let $f_{s;t}$ be defined as in (1.16). Then, there exists a constant $C>0$ such that for all $0\le s \le t$ we have

$$\begin{aligned} \Vert f_{s;t} \Vert _2 \le \Vert O \Vert e^{C \vert t -s \vert } , \quad \Vert f_{s;t} \Vert _{H^2 \left( \mathbb {R}^3 \right) } \le Ce ^{e^{C \vert t \vert }}{\left| \left| \left| O \right| \right| \right| } \end{aligned}$$

(3.13)

where ${\left| \left| \left| O \right| \right| \right| }$ is defined in (1.15).

Proof

Since

$$\begin{aligned} \Vert f_{s;t} \Vert _2^2 =&\Vert f_{t;t} \Vert _2^2 - \int _s^t \partial _\tau \Vert f_{\tau ;t} \Vert _2^2 \; d\tau = \Vert f_{t;t} \Vert _2^2 - 2 \int _s^t \mathrm {Im}\langle f_{\tau ;t}, \widetilde{K}_{2,\tau } J f_{\tau ;t} \rangle \; d\tau \end{aligned}$$

(3.14)

we have with (3.5)

$$\begin{aligned} \Vert f_{s;t} \Vert _2^2 \le \Vert f_{t;t} \Vert _2^2 + C \int _s^t \Vert f_{\tau ;t} \Vert _2^2 \; d\tau \; . \end{aligned}$$

(3.15)

Since $\Vert f_{t;t} \Vert _2 = \Vert q_t O \varphi _t \Vert _2 \le \Vert O \Vert$, the first bound in (3.13) is a consequence of Gronwall’s inequality.

In order to show the second, we compute

$$\begin{aligned} \partial _s \Vert f_{s;t} \Vert _{H^2 \left( \mathbb {R}^3 \right) }^2&= 2\, \mathrm {Im}\, \langle \left( \Delta ^2 - 2 \Delta \right) f_{s;t}, \; \left( v* \vert \varphi _s \vert ^2 + \widetilde{K}_{1,s} \right) f_{s;t}\rangle \nonumber \\&\quad - 2\, \mathrm {Im}\, \langle \left( - \Delta + 1 \right) ^2 f_{s;t}, \; \widetilde{K}_{2,s} J f_{s;t}\rangle \nonumber \\&= 4\, \mathrm {Im}\, \langle \Delta f_{s;t}, \; \left( \nabla v* \vert \varphi _s \vert ^2 \right) \nabla f_{s,t} \rangle + 2\, \mathrm {Im}\, \langle \Delta f_{s;t}, \left( \Delta v* \vert \varphi _s \vert ^2 \right) f_{s,t} \rangle \nonumber \\&\quad + 2\, \mathrm {Im}\, \langle \Delta f_{s;t}, \; \Delta \widetilde{K}_{1,s} f_{s,t}\rangle - 4\,\mathrm {Im}\, \langle \Delta f_{s;t}, \; \left( v* \vert \varphi _s \vert ^2 + \widetilde{K}_{1,s} \right) f_{s,t} \rangle \nonumber \\&\quad - 2\, \mathrm {Im}\, \langle \left( - \Delta + 1 \right) f_{s;t}, \; \left( - \Delta +1 \right) \widetilde{K}_{2,s} J f_{s,t} \rangle \,. \end{aligned}$$

(3.16)

It follows from Lemma 3.1 that all the terms on the r.h.s. can be bounded by $C\Vert f_{s;t}\Vert _{H^2(\mathbb {R}^3)}^2 e^{Cs}$. The second bound in (3.13) thus also follows from Gronwall’s inequality, together with

$$\begin{aligned} \Vert f_{t;t} \Vert _{H^2 \left( \mathbb {R}^3 \right) }&= \Vert q_t O \varphi _t \Vert _{H^2 \left( \mathbb {R}^3 \right) } \le \Vert \varphi _t\Vert _{H^2(\mathbb {R}^3)} \Vert O\Vert + \Vert O \varphi _t \Vert _{H^2 \left( \mathbb {R}^3 \right) } \nonumber \\&\quad \le \left( \Vert O \Vert + {\left| \left| \left| O \right| \right| \right| } \right) \Vert \varphi _t \Vert _{H^2 \left( \mathbb {R}^3 \right) } \end{aligned}$$

(3.17)

and (3.2). $\square$

Note that the generalization of the interaction potential comes into play when using the estimates (3.55) and (3.72) from Lemma 3.1 and Lemma 3.2. These estimates lead to the bounds double exponential in time.

3.2 Fluctuations Around the Condensate

For the proof of Theorem 1.2, we need to study the fluctuations around the condensate in the truncated Fock space of excitations. This description is based on the observation of [21] that any N-particle bosonic wave function $\psi _N \in L_\mathrm{s}^2( \mathbb {R}^{3N})$ can be decomposed as

$$\begin{aligned} \psi _N = \eta _0 \; \varphi _t^{\otimes N} + \eta _1 \otimes _\mathrm{s} \varphi _t^{\otimes (N-1)} + \cdots + \eta _N \end{aligned}$$

(3.18)

with $\eta _j \in L^2_{\perp \varphi _t} \left( \mathbb {R}^3 \right) ^{\otimes _\mathrm{s} j}$, where $L^2_{\perp \varphi _t} ( \mathbb {R}^3 )$ denotes the orthogonal complement in $L^2 ( \mathbb {R}^3)$ of the condensate wave function $\varphi _t$ and $\otimes _\mathrm{s}$ the symmetric tensor product. In particular, this observation allows to define the unitary operator

$$\begin{aligned} \mathcal {U}_t: L_\mathrm{s}^2 \left( \mathbb {R}^{3N} \right) \rightarrow \mathcal {F}_{\varphi _t}^{\le N} = \bigoplus _{j=0}^N L^2_{\perp \varphi _t} \left( \mathbb {R}^3 \right) ^{\otimes _\mathrm{s} j } \end{aligned}$$

(3.19)

mapping an N-particle bosonic wave function $\psi _N$ onto an element of the truncated Fock space, with $\mathcal {U}_t \psi _N = \lbrace \eta _0, \dots , \eta _N \rbrace$ describing the excitations orthogonal to the condensate. On the full bosonic Fock space (built over $L^2( \mathbb {R}^3)$) we have the usual creation and annihilation operators, given for $f \in L ^2( \mathbb {R}^3 )$ by

$$\begin{aligned} a^*(f) = \int f(x) \, a^*_x \, dx, \quad a(f) = \int \overline{f(x)} \, a_x \, dx \end{aligned}$$

(3.20)

and the number of particles operator $\mathcal {N}= \int a_x^* a_x dx$. Moreover, we have the modified creation and annihilation operators $b^*(f), b(f)$ which (in contrast to $a^*(f), a(f)$) leave the truncated Fock space $\mathcal {F}_{\varphi _t}^{\le N}$ invariant and are given for $f \in L^2_{\perp \varphi _t} ( \mathbb {R}^3 )$ by

$$\begin{aligned} b^*(f)&= \mathcal {U}_t \; a^*(f) \tfrac{a(\varphi _t)}{\sqrt{N}}\; \mathcal {U}_t^* = a^*(f) \sqrt{ 1- \tfrac{\mathcal {N}_+(t)}{N}} \nonumber \\ b(f)&= \mathcal {U}_t \; \tfrac{a^*(\varphi _t)}{\sqrt{N}} a(f) \; \mathcal {U}_t^* =\sqrt{ 1- \tfrac{\mathcal {N}_+(t)}{N}} a(f) \end{aligned}$$

(3.21)

where $\mathcal {N}_+(t) = \mathcal {N}- a^*(\varphi _t) a(\varphi _t)$ is the number of excitations. Note that the operators $b^*(f), b(f)$ are time dependent, yet we omit the time dependence in their notation for simplicity. Their commutators given for $f_1,f_2 \in L^2_{\perp \varphi _t} ( \mathbb {R}^3 )$ by

$$\begin{aligned} \left[ b(f_1), b^*(f_2) \right]&= \left( 1 - \frac{\mathcal {N}_+(t)}{N} \right) \langle f_1, \; f_2 \rangle - \frac{1}{N} a^*(f_2) a(f_1) , \nonumber \\ \left[ b(f_1), b(f_2) \right]&= \left[ b^*(f_1), b^*(f_2) \right] = 0 \end{aligned}$$

(3.22)

behave in the limit $N \rightarrow \infty$ similarly as the standard commutation relations of $a^*(f_1), a(f_2)$; however, the correction terms of order $N^{-1}$ lead to technical difficulties in the proofs below. With (3.21) and the following further properties of $\mathcal {U}_t$

$$\begin{aligned} \mathcal {U}_t a^*( \varphi _t) a( \varphi _t) \mathcal {U}_t^*&= N - \mathcal {N}_+ (t) \quad \nonumber \\ \mathcal {U}_t a^*(f) a(g) \mathcal {U}_t^*&= a^*(f) a(g) \end{aligned}$$

(3.23)

for $f,g \in L_{\perp \varphi _t }^2 ( \mathbb {R}^3 )$, we can compute the generator $\mathcal {L}_N(t)$ of the fluctuation dynamics

$$\begin{aligned} \mathcal {W}_N(t_2;t_1) = \mathcal {U}_{t_2} e^{-i H_N (t_2 -t_1)}\mathcal {U}_{t_1}^*: \mathcal {F}_{\varphi _{t_1}}^{\le N} \rightarrow \mathcal {F}_{\varphi _{t_2}}^{\le N} \; \end{aligned}$$

(3.24)

defined by

$$\begin{aligned} i \partial _{t_2} \mathcal {W}_N (t_2; t_1) = \mathcal {L}_N (t_2) \mathcal {W}_N(t_2;t_1) \,. \end{aligned}$$

(3.25)

For $\xi _1, \xi _2 \in \mathcal {F}_{\perp \varphi _t}^{\le N}$ it is given by

$$\begin{aligned} \langle \xi _1, \mathcal {L}_N (t) \xi _2 \rangle&= \langle \xi _1, \left[ i \partial _t \mathcal {U}_t \right] \mathcal {U}_t^* + \mathcal {U}_t H_N \mathcal {U}_t^* \xi _2 \rangle \nonumber \\&= \langle \xi _1, d\Gamma (h_\mathrm{H} (t) + K_{1,t}) \xi _2 \rangle + \mathrm {Re}\int \; K_{2,t} (x,y) \, \langle \xi _1, b_x^* b_y^* \xi _2 \rangle \; dxdy \nonumber \\&\quad - \frac{1}{2N} \langle \xi _1 , d\Gamma (v *|\varphi _{t}|^2 + K_{1,t} - \mu _{t}) (\mathcal {N}_+ (t) - 1) \xi _2 \rangle \nonumber \\&\quad + \frac{2}{\sqrt{N}} \mathrm {Re}\, \langle \xi _1, \mathcal {N}_+(t) b ((v*|\varphi _{t}|^2) \varphi _{t}) \xi _2 \rangle \nonumber \\&\quad +\frac{2}{\sqrt{N}} \int \; v (x-y) \mathrm {Re}\, \varphi _{t} (x) \langle \xi _1, a_y^*a_{x}b_{y} \xi _2 \rangle \; dx dy \nonumber \\&\quad + \frac{1}{2N} \int \,v (x-y) \langle \xi _1 , a_x^* a_y^* a_{x} a_{y} \xi _2 \rangle \, dx dy \end{aligned}$$

(3.26)

where we used the notation introduced in (1.6), (1.18) and $2\mu _{t} = \int dx dy \; v(x-y) \vert \varphi _{t} (x) \vert ^2 \vert \varphi _{t} (y) \vert ^2$.

In the limit of large N, the fluctuation dynamics $\mathcal {W}_N (t_2; t_1)$ can be approximated by a limiting dynamics $\mathcal {W}_\infty (t_2 ;t_1) : \mathcal {F}_{\perp \varphi _{t_1}} \rightarrow \mathcal {F}_{\perp \varphi _{t_2}}$ which is obtained by taking a formal limit $N\rightarrow \infty$ in (3.26). It satisfies the equation

$$\begin{aligned} i\partial _t \mathcal {W}_\infty (t_2 ; t_1 ) = \mathcal {L}_\infty (t_2) \mathcal {W}_\infty (t_2;t_1) \end{aligned}$$

(3.27)

with the generator $\mathcal {L}_\infty (t)$ whose matrix elements are given for $\xi _1, \xi _2 \in \mathcal {F}_{\perp \varphi _{t}}$ by

$$\begin{aligned} \langle \xi _1 , \mathcal {L}_\infty (t) \xi _2 \rangle = \langle \xi _1, d\Gamma (h_\mathrm{H} (t) + K_{1,t}) \xi _2 \rangle + \mathrm {Re}\int K_{2,t} (x;y) \langle \xi _1, a_x^* a_y^* \xi _2 \rangle \; dx dy \end{aligned}$$

(3.28)

For more details see [20] resp. [11, 16, 17, 22]. The generator $\mathcal {L}_\infty (t)$ of the limiting fluctuation dynamics is quadratic in creation and annihilation operators and thus gives rise to a Bogoliubov transformation [4, 5, 23] related to the function $f_{0;t}$ defined in (1.16) (see [[4, Theorem 1.2 et seq.]]).

3.3 Proof of Theorem 1.2

The proof follows closely the ideas of [18] and is based on Baker–Campbell–Hausdorff formulas proved therein (see [18, Propositions 2.2–2.5]. The main difference compared to [18] is, on the one hand, our weaker assumptions on the interaction potential (entering in the estimates (3.55) and (3.72) through Lemmas 3.1 and 3.2). On the other hand, we prove lower bounds in Lemmas 3.3–3.5 as well, based on similar ideas as for the upper bounds (see also [18, subsequent discussion of Theorem 1.1]).

With the map $\mathcal {U}_0$ defined in (3.19), we observe that $\varphi ^{\otimes N} = \mathcal {U}_0^* \Omega$ and thus by definition of the fluctuation dynamics in (3.24), we have

$$\begin{aligned} \psi _{N,t} = e^{-iH_N t} \varphi ^{\otimes N} = e^{-iH_N t} \mathcal {U}_0^* \Omega = \mathcal {U}_t^* \mathcal {W}_N(t;0 ) \Omega . \end{aligned}$$

(3.29)

Hence we can write the moment generating function as

$$\begin{aligned} \mathbb {E}_{\psi _{N,t}} \left[ e^{\lambda N O_{N,t} } \right] =&\left\langle \psi _{N,t}, \;e^{\lambda N O_{N,t}} \psi _{N,t} \right\rangle \nonumber \\ =&\left\langle \Omega , \; \mathcal {W}_N^*(t;0) \mathcal {U}_t e^{\lambda d \Gamma ( \widetilde{O}_t ) } \mathcal {U}_t^* \mathcal {W}_N (t;0) \Omega \right\rangle \end{aligned}$$

(3.30)

with $\widetilde{O}_t = O - \langle \varphi _t, O \varphi _t \rangle$. The properties (3.21), (3.23) of $\mathcal {U}_t$ allow to compute

$$\begin{aligned} \mathcal {U}_t d \Gamma (\widetilde{O}_t ) \mathcal {U}_t^* = d\Gamma ( q_t \widetilde{O}_t q_t ) + \sqrt{N} \phi _+ ( q_t O \varphi _t ) \end{aligned}$$

(3.31)

where we used that $\langle \varphi _t, \widetilde{O}_t \varphi _t \rangle =0$ and we introduced the notation

$\phi _+(h) = b(h) + b^*(h)$ for $h \in L^2_{\perp \varphi _t} ( \mathbb {R}^3 )$. Thus, we arrive at

$$\begin{aligned} \mathbb {E}_{\psi _{N,t}} \left[ e^{\lambda N O_{N,t} } \right] = \left\langle \Omega , \; \mathcal {W}_N^*(t;0) e^{\lambda d \Gamma ( q_t\widetilde{O}_t q_t) + \lambda \sqrt{N} \phi _+( q_t O \varphi _t )} \mathcal {W}_N (t;0) \Omega \right\rangle \; . \end{aligned}$$

(3.32)

As in [18], we split the proof into three steps. The first step, Lemma 3.3, can be proved as [[18, Lemma 3.1]].

Lemma 3.3

There exists a constant $C>0$ such that for all $t \in \mathbb {R}$ and $\lambda \le \Vert O\Vert ^{-1}$

$$\begin{aligned}&e^{-C N \Vert O \Vert ^3 \lambda ^3} \left\langle \Omega , \; \mathcal {W}_N^*(t;0) e^{\lambda \sqrt{N} \phi _+ (q_t O \varphi _t )/2 } e^{-2 \lambda \Vert O \Vert \mathcal {N}_+(t) } e^{\lambda \sqrt{N} \phi _+ (q_t O \varphi _t )/2 } \mathcal {W}_N(t;0) \Omega \right\rangle \nonumber \\&\quad \le \left\langle \mathcal {W}_N^* (t;0) \Omega , \; e^{\lambda d \Gamma ( q_t \widetilde{O}_t q_t ) + \lambda \sqrt{N} \phi _+ (q_t O \varphi _t)} \mathcal {W}_N(t;0) \Omega \right\rangle \nonumber \\&\quad \le e^{C N \Vert O \Vert ^3 \lambda ^3} \left\langle \Omega , \; \mathcal {W}_N^*(t;0) e^{\lambda \sqrt{N} \phi _+ (q_t O \varphi _t )/2 } e^{2 \lambda \Vert O \Vert \mathcal {N}_+(t) } e^{\lambda \sqrt{N} \phi _+ (q_t O \varphi _t )/2 } \mathcal {W}_N(t;0) \Omega \right\rangle \,. \end{aligned}$$

(3.33)

Proof

The proof of the upper bound in (3.33) is the same as in [[18, Lemma 3.1]]. The lower bound can be proved in essentially the same way. For completeness we carry it out in the following. As in [18] (but replacing $\kappa$ with $- \kappa$), we define for $s \in [0,1]$ and $\kappa > 0$ the vector

$$\begin{aligned} \xi _s = e^{-(1-s)\lambda \kappa \mathcal {N}_+ (t) /2} e^{(1-s) \lambda \sqrt{N} \phi _+ (q_t O \varphi _t)/2} e^{s \lambda \left[ d \Gamma (q_t \widetilde{O}_t q_t ) + \sqrt{N} \phi _+(q_t O \varphi _t) \right] /2} \mathcal {W}_N (t;0) \Omega . \end{aligned}$$

(3.34)

We have

$$\begin{aligned} \Vert \xi _0 \Vert ^2 =\left\langle \Omega , \; \mathcal {W}_N^*(t;0) e^{\lambda \sqrt{N} \phi _+ (q_t O \varphi _t )/2 } e^{-\kappa \lambda \mathcal {N}_+(t) } e^{\lambda \sqrt{N} \phi _+ (q_t O \varphi _t )/2 } \mathcal {W}_N(t;0) \Omega \right\rangle \end{aligned}$$

(3.35)

and

$$\begin{aligned} \Vert \xi _1 \Vert ^2 = \left\langle \Omega , \; \mathcal {W}_N^* (t;0) e^{\lambda d \Gamma ( q_t \widetilde{O}_t q_t ) + \lambda \sqrt{N} \phi _+ (q_t O \varphi _t)} \mathcal {W}_N(t;0) \Omega \right\rangle . \end{aligned}$$

(3.36)

To control the difference of (3.35) and (3.36), we compute the derivative

$$\begin{aligned} \partial _s \Vert \xi _s \Vert ^2 = 2 \mathrm {Re}\langle \xi _s, \, \partial _s \xi _s \rangle = 2 \mathrm {Re}\langle \xi _s, \, \mathcal {M}_s \xi _s \rangle \end{aligned}$$

(3.37)

where the operator $\mathcal {M}_s$ is given by

$$\begin{aligned} \mathcal {M}_s =&\frac{\lambda }{2} e^{-(1-s)\lambda \kappa \mathcal {N}_+ (t)/2} e^{(1-s)\lambda \sqrt{N} \phi _+ (q_t O \varphi _t )/2} d \Gamma (q_t \widetilde{O}_t q_t ) e^{-(1-s)\lambda \sqrt{N} \phi _+ (q_t O \varphi _t )/2} e^{(1-s)\lambda \kappa \mathcal {N}_+ (t)/2} \nonumber \\&+ \frac{\lambda \kappa }{2} \mathcal {N}_+ (t) . \end{aligned}$$

(3.38)

With [18, Propositions 2.2–2.4], we can compute $\mathcal {M}_s$ explicitly. Note that only the hermitian part of $\mathcal {M}_s$ enters in (3.37). Using the notation $h_t = (1-s) \lambda q_t O \varphi _t$ and $\gamma _s = \cosh s, \sigma _s = \sinh s$, we find

$$\begin{aligned} \frac{\mathcal {M}_s + \mathcal {M}_s^*}{\lambda }&= d \Gamma (q_t \widetilde{O}_t q_t) - \frac{\sigma _{\Vert h_t \Vert }^2}{\Vert h_t \Vert ^2} \langle h_t , \widetilde{O}_t h_t \rangle \left( N - {\mathcal {N}_+ (t)} \right) \nonumber \\&\quad + \left( \frac{\gamma _{\Vert h_t \Vert } - 1}{\Vert h_t \Vert ^2} \right) ^2 \langle h_t , \widetilde{O}_t h_t \rangle a^* (h_t) a (h_t)\nonumber \\&\quad + \frac{\gamma _{\Vert h_t \Vert } -1}{\Vert h_t \Vert ^2} (a^* (h_t) a (q_t \widetilde{O}_t h_t) + a^* (q_t \widetilde{O}_t h_t) a (h_t) ) + \kappa \mathcal {N}_+ (t)\nonumber \\&\quad + \sqrt{N} \, \frac{\sigma _{\Vert h_t \Vert }}{\Vert h_t \Vert } \sinh ((s-1) \lambda \kappa /2) \left[ \frac{\gamma _{\Vert h_t \Vert } - 1}{\Vert h_t \Vert ^2} \langle h_t , \widetilde{O}_t h_t \rangle \phi _+ (h_t) + \phi _+ (q_t \widetilde{O}_t h_t) \right] \,. \end{aligned}$$

(3.39)

For any $h \in L^2_{\perp \varphi } ( \mathbb {R}^3 )$ and any bounded operator H on $L^2_{\perp \varphi } ( \mathbb {R}^3 )$, we have the bounds

$$\begin{aligned} \Vert b(h) \xi \Vert \le \Vert h \Vert _2 \Vert \mathcal {N}_+(t)^{1/2} \xi \Vert ,&\quad \Vert b^*(h) \xi \Vert \le \Vert h\Vert _2 \Vert \left( \mathcal {N}_+(t) +1 \right) ^{1/2} \xi \Vert , \pm d\Gamma (H) \le \Vert H \Vert \mathcal {N}_+ (t) \,. \end{aligned}$$

(3.40)

Consequently, all terms on the r.h.s. of (3.39) can be bounded by a constant of order N. Furthermore, since

$$\begin{aligned} \Vert \widetilde{O}_t \Vert \le \Vert O \Vert ( 1 + \Vert \varphi _t \Vert _2^2 ) =2 \Vert O \Vert \end{aligned}$$

(3.41)

we can bound $d \Gamma (q_t \widetilde{O}_t q_t) \ge - 2 \Vert O\Vert \mathcal {N}_+(t)$ and hence the choice $\kappa = 2 \Vert O\Vert$ gives $d \Gamma (q_t \widetilde{O}_t q_t) + \kappa \mathcal {N}_+(t) \ge 0$. Moreover, since

$$\begin{aligned} \Vert h_t \Vert _2 \le \lambda \Vert q_t O \varphi _t\Vert _2 \le \lambda \Vert O \Vert \, \Vert \varphi _t\Vert _2\le 1 \end{aligned}$$

(3.42)

for all $\lambda \le \Vert O \Vert ^{-1}$, all the other terms on the r.h.s. of (3.39) are at least of order $\lambda ^2$. Thus, using (3.40) and $\kappa = 2 \Vert O\Vert$ we obtain the lower bound

$$\begin{aligned} \frac{2}{\lambda } \text {Re } \langle \xi _{s} , \mathcal {M}_{s} \xi _{s} \rangle \ge - C \lambda ^2 N \Vert O \Vert ^3 \Vert \xi _{s} \Vert ^2 \,. \end{aligned}$$

(3.43)

In combination with (3.37) the lower bound in (3.33) now follows from Gronwall’s inequality.

The proof of the upper bound in [18] works in the same way, simply replacing $\kappa$ by $-\kappa$ and estimating the terms in (3.39) from above instead of from below. $\square$

The second step, Lemma 3.4, is a generalization of Lemma [[18, Lemma 3.2]] to more singular interaction potentials. The proof involves Lemmas 3.1 and 3.2 (see in particular (3.55) and (3.72)) for the estimates yielding to an double exponential in time of the term cubic in $\lambda$ and in the definition of $\kappa _s$(compared to an exponential in time in [18]). We remark that for Lemma 3.4 it is a crucial observation that $f_{s;t} \in L^2_{\perp \varphi _s} ( \mathbb {R}^3)$ for all $0\le s \le t$. This follows from the fact that $\langle \varphi _t, f_{t;t} \rangle = \langle \varphi _t , q_t O \varphi _t\rangle =0$ by construction, as well as

$$\begin{aligned} \partial _s \langle \varphi _s, f_{s;t} \rangle = - i \, \mathrm {Im}\, \langle \varphi _s, \left[ \widetilde{K}_{1,s} - \widetilde{ K}_{2,s} J \right] f_{s;t }\rangle =0 \end{aligned}$$

(3.44)

using the definitions (1.16) and (1.17).

Lemma 3.4

For $0\le s \le t$, let $f_{s;t} \in L^2_{\perp \varphi _s} \left( \mathbb {R}^3 \right)$ be defined by (1.16). Let O be a self-adjoint operator on $L^2 \left( \mathbb {R}^3 \right)$ such that ${\left| \left| \left| O \right| \right| \right| } < \infty$ as defined in (1.15). There exists a constant $C>0$ such that for $\kappa$ defined as

$$\begin{aligned} \kappa = C {\left| \left| \left| O \right| \right| \right| } e^{e^{C t}} \end{aligned}$$

(3.45)

we have for all $0\le \lambda \kappa \le 1$

$$\begin{aligned}&\left\langle \Omega , \mathcal {W}_N (t;0) e^{\lambda \sqrt{N} \phi _+ (q_t O \varphi _t)/2} e^{2 \lambda \Vert {O} \Vert \mathcal {N}_+ (t)} e^{\lambda \sqrt{N} \phi _+ (q_t O \varphi _t)/2} \mathcal {W}_N (t;0) \Omega \right\rangle \nonumber \\&\quad \le e^{ \kappa \left( N \lambda ^3 {\left| \left| \left| O \right| \right| \right| }^2+\lambda \right) } \left\langle \Omega , e^{\lambda \sqrt{N} \phi _+ (f_{0;t})/2} e^{\lambda \kappa \mathcal {N}_+ (0)} e^{\lambda \sqrt{N} \phi _+ (f_{0;t})/2} \Omega \right\rangle \; \end{aligned}$$

(3.46)

and

$$\begin{aligned}&\left\langle \Omega , \mathcal {W}_N (t;0) e^{\lambda \sqrt{N} \phi _+ (q_t O \varphi _t)/2} e^{-2 \lambda \Vert O \Vert \mathcal {N}_+ (t)} e^{\lambda \sqrt{N} \phi _+ (q_t O \varphi _t)/2} \mathcal {W}_N (t;0) \Omega \right\rangle \nonumber \\&\quad \ge e^{ -\kappa \left( N \lambda ^3{\left| \left| \left| O \right| \right| \right| }^2 + \lambda \right) } \left\langle \Omega , e^{\lambda \sqrt{N} \phi _+ (f_{0;t})/2} e^{- \lambda \kappa \mathcal {N}_+ (0)} e^{\lambda \sqrt{N} \phi _+ (f_{0;t})/2} \Omega \right\rangle \; . \end{aligned}$$

(3.47)

Proof

The lower bound (3.47) follows with ideas from [[18, Lemma 3.2]] and from Lemmas 3.1 and 3.2. For $0\le s \le t$ and some (differentiable) $\kappa _s \ge 0$ with $\kappa _t = 2 \Vert O\Vert$, define the vector

$$\begin{aligned} \xi _t (s) = e^{-\lambda \kappa _{s} \mathcal {N}_+ (s) /2} e^{\lambda \sqrt{N} \phi _+ (f_{s;t}) /2} \mathcal {W}_N (s;0) \Omega \in \mathcal {F}_{\perp \varphi _s}^{\le N} \,. \end{aligned}$$

(3.48)

It satisfies

$$\begin{aligned} \Vert \xi _t (0) \Vert ^2 = \langle \Omega , e^{\lambda \sqrt{N} \phi _+ (f_{0;t}) /2} e^{ - \lambda \kappa _0 \mathcal {N}_+ (0)} e^{\lambda \sqrt{N} \phi _+ (f_{0;t}) /2} \Omega \rangle \end{aligned}$$

(3.49)

and

$$\begin{aligned} \Vert \xi _t (t) \Vert ^2 = \left\langle \Omega , \mathcal {W}_N (t;0)^* e^{\lambda \sqrt{N} \phi _+ (q_t O \varphi _t) /2} e^{-2 \lambda \Vert O\Vert \mathcal {N}_+ (t)} e^{\lambda \sqrt{N} \phi _+ (q_t O \varphi _t) /2} \mathcal {W}_N (t;0) \Omega \right\rangle \, . \end{aligned}$$

(3.50)

Note that the definition (3.48) is similar to the vector defined at the beginning of the proof of [18, Lemma 3.2]. The crucial difference is that, here, in (3.48), for the lower bound, the exponential of the number of particles operator comes with a negative constant in front (in contrast to a positive one in [18] for the upper bound).

As in the proof of Lemma 3.3, we want to control the difference of (3.49) and (3.50) through the derivative

$$\begin{aligned} \partial _s \Vert \xi _t (s) \Vert ^2 = -2i \, \mathrm {Im}\left\langle \xi _t (s) , \mathcal {J}_{N,t} (s) \xi _t (s) \right\rangle \end{aligned}$$

(3.51)

where $\mathcal {J}_N(s)$ is (in the sense of a quadratic form on $\mathcal {F}_{\perp \varphi _s}^{\le N}$ as in (3.26)) given by

$$\begin{aligned} \mathcal {J}_{N,t} (s) = \;&e^{-\lambda \kappa _{s} \mathcal {N}_+ (s)/ 2} e^{\lambda \sqrt{N} \phi _+ (f_{s;t}) /2} \mathcal {L}_N (s) e^{-\lambda \sqrt{N} \phi _+ (f_{s;t}) /2} e^{\lambda \kappa _{s} \mathcal {N}_+ (s) / 2} \nonumber \\&+ e^{-\lambda \kappa _{s} \, \mathcal {N}_+ (s) / 2} \left[ i \partial _s e^{\lambda \sqrt{N} \phi _+ (f_{s;t}) /2} \right] e^{-\lambda \sqrt{N} \phi _+ (f_{s;t}) /2} e^{\lambda \kappa _{s} \, \mathcal {N}_+ (s) /2} -\frac{i\lambda }{2} {\dot{\kappa }_s} \, \mathcal {N}_+ (s) \,, \end{aligned}$$

(3.52)

where we denote $\dot{\kappa }_s = d\kappa _s/ds$. For this computation it is convenient to embed $\mathcal {F}_{\perp \varphi _s}^{\le N}$ into the full Fock space $\mathcal {F}$ in which case $\mathcal {N}_+(s)$ can be replaced by the s-independent $\mathcal {N}$ (for more details see the discussion before [18, Eq. (3.3)]). We proceed as in [18] and compute the anti-symmetric part of $\mathcal {J}_{N,t }(s)$ explicitly with the help of [18, Propositions 2.2–2.4], and show that its norm is bounded by terms of order $N \lambda ^3$ and $\lambda$.

To this end, recalling the definition of $\mathcal {L}_N(s)$ in (3.26) and analogous calculations as in [18, (3.4)–(3.5)], we have

$$\begin{aligned}&e^{-\lambda \kappa _{s} \mathcal {N}_+ (s)/ 2} e^{\lambda \sqrt{N} \phi _+ (f_{s;t}) /2} d\Gamma (h_\mathrm{H} (s) + K_{1,s}) e^{-\lambda \sqrt{N} \phi _+ (f_{s;t}) /2} e^{\lambda \kappa _{s} \mathcal {N}_+ (s) / 2}\nonumber \\&\quad = \frac{i \lambda \sqrt{N}}{2} \phi _- ((h_\mathrm{H} (s) + K_{1,s}) f_{s;t}) + T_1 + S_1 \end{aligned}$$

(3.53)

where we introduced the notation $\phi _-(f) = -i (b(f)-b^*(f))$, $S_1 = S_1^*$ is symmetric and

$$\begin{aligned} T_1&= i \sqrt{N} \left( \frac{\sigma _{\Vert h_{s;t} \Vert }}{\Vert h_{s;t} \Vert } \cosh (\lambda \kappa _{s} /2) -1 \right) \phi _- ((h_\mathrm{H} (s) + K_{1,s}) h_{s;t})\nonumber \\&\quad + i \sqrt{N} \frac{\sigma _{\Vert h_{s;t} \Vert }}{\Vert h_{s;t} \Vert } \frac{\gamma _{\Vert h_{s;t} \Vert } - 1}{\Vert h_{s;t} \Vert ^2} \langle h_{s;t} , (h_\mathrm{H} (s) + K_{1,s}) h_{s;t} \rangle \cosh (\lambda \kappa _{s} /2) \phi _- (h_{s;t}) \end{aligned}$$

(3.54)

with $h_{s;t} = \lambda f_{s;t}/2$, as well as $\gamma _s = \cosh s$ and $\sigma _s = \sinh s$ as in the proof of Lemma 3.3. We use the bounds (3.40) and

$$\begin{aligned} \Vert h_\mathrm{H} (s) h_{s;t} \Vert _2 \le C \Vert h_{s;t} \Vert _{H^2 \left( \mathbb {R}^3 \right) } \le C \lambda {\left| \left| \left| O \right| \right| \right| } e^{ e^{C t}} , \quad \Vert K_{1,s} h_{s;t} \Vert _2 \le C \lambda \Vert O \Vert e^{ C t} \end{aligned}$$

(3.55)

for all $0\le s\le t$ by Lemmas 3.1 and 3.2, and conclude that for all $0 \le \lambda \kappa _s \le 1$ and $s \in [0,t]$, we have $\Vert T_1 \Vert \le C e^{e^{C t}} N {\left| \left| \left| O \right| \right| \right| }^3 \lambda ^3$.

We proceed similarly with the remaining terms of (3.52). For the second term of $\mathcal {L}_{N,t}(s)$ in (3.26), we find with analogous calculations as the ones leading to [18, Eq. (3.6)], using that

$$\begin{aligned} \Vert K_{2,s} \Vert \le \Vert K_{2,s} \Vert _{L^2( \mathbb {R}^3 \times \mathbb {R}^3)} \le C, \quad \Vert f_{s;t} \Vert _2 \le e^{C t } \Vert O \Vert \end{aligned}$$

(3.56)

for $0\le s\le t$ by Lemmas 3.1 and 3.2,

$$\begin{aligned}&e^{-\lambda \kappa _{s} \mathcal {N}_+ (s) / 2} e^{\sqrt{N} \phi _+ (h_{s;t}) } \left( \frac{1}{2} \int \left[ \overline{K_{2,s} (x,y)} b_x b_y + K_{2,s} (x,y) b_x^* b_y^* \right] dx dy \right) e^{- \sqrt{N} \phi _+ (h_{s;t}) } e^{\lambda \kappa _{s} \mathcal {N}_+ (s) / 2}\nonumber \\&\quad = - \frac{i \lambda \sqrt{N}}{2} \phi _- (K_{2,s}\overline{ f_{s;t}}) + S_2 + T_2 + i R_2 \end{aligned}$$

(3.57)

where $S_2=S_2^*$ is symmetric, $T_2$ is bounded as $T_1$ above by $\Vert T_2 \Vert \le C e^{e^{C t}} N {\left| \left| \left| O \right| \right| \right| }^3 \lambda ^3$, and $R_2$ contains all the remaining terms of order $\lambda$, which are given by

$$\begin{aligned} R_2&= -i \frac{\lambda \kappa _{s}}{2} \int \left[ \overline{K_{2,s} (x,y)} b_x b_y - K_{2,s} (x,y) b_x^* b_y^* \right] dx dy \nonumber \\&\quad + i \frac{\lambda \sqrt{N}}{2} \left[ \left( 1 - \frac{\mathcal {N}_+(s)+1/2}{N}\right) b (K_{2,s} \overline{f_{s;t}}) -b^* (K_{2,s} \overline{f_{s;t}})\left( 1 - \frac{\mathcal {N}_+(s)+1/2}{N}\right) \right] \nonumber \\&\quad -i \frac{\lambda }{4\sqrt{N}} \int \left[ \overline{K_{2,s} (x,y)} b^*(f_{s;t}) a_x a_y - K_{2,s} (x,y) a_y^* a_x^* b(f_{s;t}) \right] dx dy \nonumber \\&\quad -i \frac{\lambda }{4\sqrt{N}} \int \left[ \overline{K_{2,s} (x,y)} a^*(f_{s;t}) a_x b_y - K_{2,s} (x,y) b_y^* a_x^* a(f_{s;t}) \right] dx dy \,. \end{aligned}$$

(3.58)

The bounds in Lemmas 3.1 and 3.2 imply that²

$$\begin{aligned} R_2 \ge - C \lambda \left( \kappa _s + \Vert O\Vert e^{Ct} \right) \left( \mathcal {N}_+ (s) + 1 \right) \end{aligned}$$

(3.59)

for $0\le s \le t$.

For the third term of (3.26), we proceed as in [18, Eq. (3.7)] and use

$$\begin{aligned} \Vert K_{1,s}\Vert \le C, \quad \Vert v* \vert \varphi _s \vert ^2 \Vert _\infty \le C , \quad \Vert f_{s;t}\Vert _2 \le e^{ C t } \Vert O \Vert \end{aligned}$$

(3.60)

from Lemmas 3.1 and 3.2 to conclude that

$$\begin{aligned}&e^{-\lambda \kappa _{s} \mathcal {N}_+ (s) / 2} e^{ \sqrt{N} \phi _+ (h_{s;t}) } d\Gamma (v *|\varphi _{s}|^2 + K_{1,s} - \mu _{s}) \frac{ \mathcal {N}_+ (s) - 1}{2N} e^{- \sqrt{N} \phi _+ (h_{s;t}) } e^{\lambda \kappa _{s} \mathcal {N}_+ (s) / 2}\nonumber \\&\quad = S_3 + T_3 + i R_3 \end{aligned}$$

(3.61)

where $S_3 = S_3^*$ is symmetric, $\Vert T_3 \Vert \le C e^{e^{C t}} N {\left| \left| \left| O \right| \right| \right| }^3 \lambda ^3$ and

$$\begin{aligned} R_3 \ge - C e^{C t } \Vert O \Vert \lambda (\mathcal {N}_+ (s)+1) \,. \end{aligned}$$

(3.62)

For the forth term on the r.h.s. of (3.26), we have with ${d}_s = \left( v * \vert \varphi _s \vert ^2 \right) \varphi _s$

$$\begin{aligned}&\frac{1}{\sqrt{N}} e^{-\lambda \kappa _{s} \mathcal {N}_+ (s) / 2} e^{ \sqrt{N} \phi _+ (h_{s;t})} \left( \mathcal {N}_+ (s)\, b ( {d}_s )+ b^* ( {d}_s )\, \mathcal {N}_+ (s) \right) e^{- \sqrt{N} \phi _+ (h_{s;t}) } e^{\lambda \kappa _{s} \mathcal {N}_+ (s) / 2}\nonumber \\&\qquad = S_4 + T_4 + i R_4 \end{aligned}$$

(3.63)

where $S_4$ and $R_4$ are symmetric and $\Vert T_4 \Vert \le C N ( e^{C t} \Vert O \Vert + \kappa _{s})^3 \lambda ^3$ for all $0\le s \le t$ and $\lambda \kappa _s \le 1$. The term $R_4$ equals

$$\begin{aligned} R_4&= \frac{ \lambda \kappa _{s}}{2 i \sqrt{N}} \left( \mathcal {N}_+(s) b({d}_s) - b^*({d}_s) \mathcal {N}_+(s) \right) + i \mathcal {N}_+(s) ( \mathrm {Im}\langle {d}_s, h_{s;t} \rangle \left( 1 - \mathcal {N}_+(s)/N \right) \nonumber \\&\quad - i \frac{\mathcal {N}_+(s) }{N} \left( a^*(h_{s;t}) a ( {d}_s) - a^* ( {d}_s) a(h_{s;t})\right) + \phi _-(h_{s;t} ) b({d}_s) +b^* ( {d}_s) \phi _-(h_{s;t} ). \end{aligned}$$

(3.64)

Since

$$\begin{aligned} \Vert {d}_s \Vert _2 \le \Vert v* \vert \varphi _s \vert ^2 \Vert _\infty \Vert \varphi _s \Vert _2 \le C , \quad \Vert f_{s;t} \Vert _2 \le e^{C t} \Vert O \Vert \end{aligned}$$

(3.65)

by Lemmas 3.1 and 3.2, we have

$$\begin{aligned} R_4 \ge - C \lambda \left( \kappa _s + \Vert O\Vert e^{Ct} \right) \left( \mathcal {N}_+ (s) + 1 \right) \end{aligned}$$

(3.66)

for all $0\le s \le t$.

Next, we consider the fifth term on the r.h.s. of (3.26) and follow the same strategy as the one leading to [18, Eq. (3.8)]. With

$$\begin{aligned} \Vert v* \left( f_{s;t} \varphi _s \right) \Vert _\infty \le C \Vert f_{s;t}\Vert _{2} \Vert \varphi _s \Vert _{H^1(\mathbb {R}^3)} \le C e^{C t} \Vert O \Vert \end{aligned}$$

(3.67)

for $0\le s \le t$ from (3.1) and (3.13) we find that

$$\begin{aligned}&e^{-\lambda \kappa _{s} \mathcal {N}_+ (s) / 2} e^{\sqrt{N} \phi _+ (h_{s;t})} \int v(x-y) \left[ \varphi _s (x) a^*_y a_x b_y + \overline{\varphi _s(x)} b_y^* a_x^* a_y \right] dxdy e^{-\sqrt{N} \phi _+ (h_{s;t})} e^{\lambda \kappa _{s} \mathcal {N}_+ (s) / 2} \nonumber \\&\quad = S_5 + T_5 + i R_5 \end{aligned}$$

(3.68)

where $S_5^* = S_5$ is symmetric, $\Vert {T}_5 \Vert \le C e^{Ct} N \Vert O \Vert ^3 \lambda ^3$ and

$$\begin{aligned} i R_5&= - \frac{\lambda \kappa _{s}}{2\sqrt{N}} \int v(x-y) \, \left[ \varphi _s (y) b_x^* a_y^* a_x - \overline{\varphi _s (y)} a_x^* a_y b_x \right] \; dx dy \nonumber \\&\quad - \int v(x-y) \left[ h_{s;t} (y) \varphi _s (y) b_x^* b_x - \overline{h_{s;t} (y)} \overline{\varphi _s (y)} b_x^* b_x \right] \; dx dy \nonumber \\&\quad - \int v(x-y) \left[ h_{s;t} (x) \varphi _s (y) b_x^* b_y^* - \overline{h_{s;t} (x)} \overline{\varphi _s (y)} b_y b_x \right] \; dx dy \, \nonumber \\&\quad + \int v(x-y) \left[ \overline{h_{s;t} (x)} \varphi _s (y) (1-\mathcal {N}_+ (s)/ N) a_y^* a_x \right. \nonumber \\&\quad \left. - h_{s;t} (x) \overline{\varphi _s(y)} a_x ^* a_y (1-\mathcal {N}_+(s) /N) \right] \; dx dy \nonumber \\&\quad - \frac{1}{N} \int v(x-y) \left[ \varphi _s (y) a_x^* a(h_{s;t} ) a_y^* a_x - \overline{\varphi _s (y)} a_x^* a_y a^* (h_{s;t}) a_x \right] \; dx dy \,. \end{aligned}$$

(3.69)

Using again (3.40) and (3.67) as well as

$$\begin{aligned} \Vert v^2 * \vert \varphi _s \vert ^2 \Vert _\infty \le C \Vert \varphi _s \Vert _{H^1(\mathbb {R}^3)}^2 \le C \end{aligned}$$

(3.70)

by (1.3), we find that

$$\begin{aligned} R_5 \ge - C \lambda \left( \kappa _s + \Vert O\Vert e^{Ct} \right) \left( \mathcal {N}_+ (s) + 1 \right) \end{aligned}$$

(3.71)

for all $0 \le s \le t$.

Finally, we consider the last term on the r.h.s. of (3.26), proceeding as in [18, Eq. (3.9)]. With (3.40) and

$$\begin{aligned} \Vert v^2 * f^2_{s;t}\Vert _\infty \le C \Vert f_{s;t}\Vert ^2_{H^1(\mathbb {R}^3)} \le C e^{e^{Ct}} {\left| \left| \left| O \right| \right| \right| } \end{aligned}$$

(3.72)

we find that

$$\begin{aligned}&\frac{1}{2N} e^{-\lambda \kappa _{s} \mathcal {N}_+ (s) / 2} e^{ \sqrt{N} \phi _+ (h_{s;t}) } \int dxdy \, v(x-y) a^*_x a^*_y a_y a_x e^{- \sqrt{N} \phi _+ (h_{s;t})} e^{\lambda \kappa _{s} \mathcal {N}_+ (s) / 2} \nonumber \\&\quad = S_6 + T_6 + i R_6 \end{aligned}$$

(3.73)

with

$$\begin{aligned} i R_6 =\frac{\lambda }{2\sqrt{N}} \int v(x-y) \, \left[ \overline{f_{s;t} (y)} a_x^* a_x b_y - f_{s;t} (y) b^*_y a_x^* a_x \right] \; dxdy \,. \end{aligned}$$

(3.74)

Again $S_6 = S_6^*$ is symmetric and $\Vert T_6 \Vert \le Ce^{e^{C t }} N {\left| \left| \left| O \right| \right| \right| }^3 \lambda ^3$. Furthermore, with (3.72) a Cauchy–Schwarz inequality yields

$$\begin{aligned} R_6 \ge - Ce^{e^{Ct} } {\left| \left| \left| O \right| \right| \right| } \lambda \mathcal {N}_+ (s) \,. \end{aligned}$$

(3.75)

If we combine (3.53), (3.57), (3.61), (3.63), (3.68) and (3.73), we conclude that the first term on the r.h.s. of (3.52) is given by

$$\begin{aligned}&e^{-\lambda \kappa _{s} \mathcal {N}_+ (s) / 2} e^{\sqrt{N} \phi _+ (h_{s;t})} \mathcal {L}_N (s) e^{-\sqrt{N} \phi _+ (h_{s;t})} e^{ \lambda \kappa _{s} \mathcal {N}_+ (s) / 2} \nonumber \\&\quad = \frac{i \lambda \sqrt{N}}{2} \phi _- ((h_\mathrm{H} (s) + K_{1,s} + K_{2,s} J ) f_{s;t}) + S +T + i R \end{aligned}$$

(3.76)

where $S^* = S$ is symmetric, $\Vert T \Vert \le C e^{e^{C t} } N {\left| \left| \left| O \right| \right| \right| }^3 \lambda ^3$ and

$$\begin{aligned} R \ge -C \lambda ({\left| \left| \left| O \right| \right| \right| } e^{e^{C t} } + \kappa _{s}) \left( \mathcal {N}_+ (s) +1 \right) \end{aligned}$$

(3.77)

for all $0\le s\le t$ and $0\le \lambda \kappa _s \le 1$. For the second term of the r.h.s. of (3.52) we find as in [18, p. 2613] using the definition of $f_{s;t}$ in (1.16) that

$$\begin{aligned}&e^{-\lambda \kappa _{t-s} \mathcal {N}_+ (s) / 2} \left[ i \partial _s e^{ \sqrt{N} \phi _+ (h_{s;t}) } \right] e^{- \sqrt{N} \phi _+ (h_{s;t}) } e^{\lambda \kappa _{t-s} \mathcal {N}_+ (s) /2} \nonumber \\&\quad = - \frac{i \lambda \sqrt{N}}{2} \, \phi _- ( i \partial _s f_{s;t}) + \widetilde{S} + \widetilde{T}\nonumber \\&\quad = - \frac{i \lambda \sqrt{N}}{2} \, \phi _- \left( \left( h_\mathrm{H} (s) + \widetilde{K}_{1,s} - \widetilde{K}_{2,s} J \right) f_{s;t}\right) + \widetilde{S} + \widetilde{T} \nonumber \\&\quad = - \frac{i \lambda \sqrt{N}}{2} \, \phi _- \left( \left( h_\mathrm{H} (s) + K_{1,s} - K_{2,s} J \right) f_{s;t}\right) + \widetilde{S} + \widetilde{T} \end{aligned}$$

(3.78)

where $\widetilde{S} = \widetilde{S}^*$ is symmetric and $\Vert \widetilde{T} \Vert \le C e^{e^{Ct}} N {\left| \left| \left| O \right| \right| \right| }^3 \lambda ^3$. We remark that the last equality holds as an identity in the sense of a quadratic form on $\mathcal {F}_{\perp \varphi _s}^{\le N}$ where the projection $q_s$ acts as the identity.

With (3.76) and (3.78), we conclude that

$$\begin{aligned} \frac{1}{i} \left[ \mathcal {J}_{N,t} (s) - \mathcal {J}_{N,t}^* (s) \right]&\ge -C e^{e^{C t} } N {\left| \left| \left| O \right| \right| \right| }^3 \lambda ^3 - \lambda \left[ C ({\left| \left| \left| O \right| \right| \right| } e^{e^{Ct}} + \kappa _s) + \dot{\kappa }_s \right] \mathcal {N}_+ (s) \nonumber \\&\quad - C \lambda ({\left| \left| \left| O \right| \right| \right| } e^{e^{Ct}} + \kappa _s) \end{aligned}$$

(3.79)

for all $0\le s \le t$ and $0\le \lambda \kappa _s \le 1$. We shall choose

$$\begin{aligned} \kappa _s = 2 \Vert O\Vert e^{C (t-s) } + {\left| \left| \left| O \right| \right| \right| } e^{e^{C t}} \left( e^{C(t-s)} - 1 \right) \end{aligned}$$

in which case the second term on the r.h.s. of (3.79) vanishes. With this choice of $\kappa$, we thus have from (3.51)

$$\begin{aligned} \partial _s \Vert \xi _t (s) \Vert ^2 \ge - C e^{e^{C t}} \left[ N \lambda ^3 {\left| \left| \left| O \right| \right| \right| }^3 + \lambda {\left| \left| \left| O \right| \right| \right| }\right] \Vert \xi _t (s) \Vert ^2 \end{aligned}$$

(3.80)

for suitable $C>0$. With Gronwall’s inequality, we arrive at

$$\begin{aligned} \Vert \xi _t (t) \Vert ^2 \ge e^{-C ( N \lambda ^3 {\left| \left| \left| O \right| \right| \right| }^3 + \lambda {\left| \left| \left| O \right| \right| \right| }) e^{e^{Ct}} t } \, \Vert \xi _t (0) \Vert ^2 \,. \end{aligned}$$

(3.81)

This concludes the proof of the lower bound.

As already mentioned at the beginning of the proof, the upper bound follows along the same lines. One simply replaces $\kappa _s$ by $-\kappa _s$ and estimates the various error terms $R_j$ for $2\le j\le 6$ from above instead of from below. $\square$

The third step, Lemma 3.5, is proven similarly to [[18, Lemma 3.3]].

Lemma 3.5

There exists a constant $C_1 > 0$ such that for all $t>0$, $0 \le \kappa \le C_1 {\left| \left| \left| O \right| \right| \right| } e^{e^{C_1} t}$ and $0\le \lambda \le e^{-e^{C_1 t}} / (C_1 {\left| \left| \left| O \right| \right| \right| } )$

$$\begin{aligned} \ln \left\langle \Omega , e^{\lambda \sqrt{N} \phi _+ (f_{0;t})/2} e^{\lambda \kappa \mathcal {N}_+ (0)} e^{\lambda \sqrt{N} \phi _+ (f_{0;t})/2} \Omega \right\rangle \le \frac{\lambda ^2 N}{2} \Vert f_{0;t} \Vert ^2 + C_1 N \lambda ^3 {\left| \left| \left| O \right| \right| \right| }^3 e^{e^{C_1 t}} \end{aligned}$$

(3.82)

and

$$\begin{aligned} \ln \left\langle \Omega , e^{\lambda \sqrt{N} \phi _+ (f_{0;t})/2} e^{-\lambda \kappa \mathcal {N}_+ (0)} e^{\lambda \sqrt{N} \phi _+ (f_{0;t})/2} \Omega \right\rangle \ge \frac{\lambda ^2 N}{2} \Vert f_{0;t} \Vert ^2 -C_1 N \lambda ^3 {\left| \left| \left| O \right| \right| \right| }^3 e^{e^{C_1 t}} \,. \end{aligned}$$

(3.83)

Proof

We start with the lower bound (3.83), we proceed similarly as in the proof of the previous Lemmas. Following [[18, Lemma 3.3]], we define for $s \in [0,1]$ the vector

$$\begin{aligned} \xi _s = e^{(1-s)^2 (N - 2 \mathcal {N}_+(0)) \Vert h_t \Vert ^2 /2} e^{-\lambda \kappa \mathcal {N}_+ (0) /2} e^{s \sqrt{N} \phi _+ (h_t)} e^{(1-s) \sqrt{N} b^* (h_t)} e^{(1-s) \sqrt{N} b(h_t)}\Omega \end{aligned}$$

(3.84)

where we introduced the notation $h_t = \lambda f_{0;t}/2 \in L^2_{\perp \varphi } (\mathbb {R}^3)$. Note that the last exponential factor in (3.84) could be omitted since $b(h_t) \Omega = 0$, but it is actually convenient to keep it for the calculation of the derivative of $\partial _s \Vert \xi _s\Vert ^2$. Compared to the upper bound in [18], we need the additional term $e^{-(1-s)^2\mathcal {N}_+(0)\Vert h_t\Vert ^2}$ in (3.84), as will be seen below. We have

$$\begin{aligned} \Vert \xi _1 \Vert ^2 = \left\langle \Omega , e^{\lambda \sqrt{N} \phi _+ (f_{0;t})/2} e^{-\lambda \kappa \mathcal {N}_+ (0)} e^{\lambda \sqrt{N} \phi _+ (f_{0;t})/2} \Omega \right\rangle \end{aligned}$$

(3.85)

and

$$\begin{aligned} \Vert \xi _0 \Vert ^2 = e^{N \Vert h_t \Vert ^2} \langle e^{\sqrt{N} b^* (h_t)} \Omega , e^{- (\lambda \kappa + 2 \Vert h_t\Vert ^2) \mathcal {N}_+ (0)} e^{\sqrt{N} b^* (h_t)} \Omega \rangle . \end{aligned}$$

(3.86)

The latter quantity will lead to the desired bound on the r.h.s. of (3.83). In order to compare (3.85) and (3.86), we compute the derivative of $\xi _s$ as

$$\begin{aligned} \partial _s \Vert \xi _s \Vert ^2 = 2\, \mathrm {Re}\, \langle \xi _s , \mathcal {G}_s \xi _s \rangle \end{aligned}$$

(3.87)

where, following [[18, Eq. 3.12 et seq.]],

$$\begin{aligned} \mathcal {G}_s&= 2 (1-s) \mathcal {N}_+(0) \Vert h_t\Vert ^2 \nonumber \\&\quad - e^{-\lambda \kappa \mathcal {N}_+ (0) /2} e^{s \sqrt{N} \phi _+ (h_t)} \left[ (1-s) \Vert h_t \Vert ^2 \mathcal {N}_+ (0) + (1-s) a^* (h_t) a(h_t) \right. \nonumber \\&\quad \left. - \sqrt{N} \Vert h_t \Vert ^2 (1-s)^2 b^* (h_t) \right] e^{-s \sqrt{N} \phi _+ (h_t)} e^{\lambda \kappa _t \mathcal {N}_+ (0) /2} \; . \end{aligned}$$

(3.88)

Using that $\Vert h_t \Vert _2 \le \lambda \Vert O\Vert e^{C t }/2$ by Lemma 3.2, it follows from the calculation [[18, Eq. 3.12 et seq.]] that

$$\begin{aligned} \mathcal {G}_s = (1-s) \mathcal {N}_+(0) \Vert h_t\Vert ^2 - (1-s) a^*(h_s) a(h_s) + T \end{aligned}$$

(3.89)

with $\Vert T\Vert \le C N \lambda ^3 \Vert O\Vert ^3 e^{C t}$ as long as $\lambda \kappa \le 1$. Since $\mathcal {N}_+(0) \Vert h_t\Vert ^2 \ge a^*(h_t) a(h_t)$, the remaining terms are positive, hence

$$\begin{aligned} \partial _s \Vert \xi _s \Vert ^2 \ge - CN \lambda ^3 \Vert O\Vert ^3 e^{C t} \Vert \xi _s\Vert ^2 \,. \end{aligned}$$

(3.90)

With Gronwall’s inequality we arrive at

$$\begin{aligned} \Vert \xi _1 \Vert ^2 \ge e^{-CN\lambda ^3 \Vert O\Vert ^3 e^{Ct}} \Vert \xi _0 \Vert ^2\,. \end{aligned}$$

(3.91)

It remains to compute (3.86). To this end, let us introduce $\kappa ' = \kappa + 2 \Vert h_t\Vert ^2 /\lambda = \kappa + \lambda \Vert f_{0;t}\Vert ^2/2$. As in [[18, Lemma 3.3]] we compute

$$\begin{aligned}&e^{N \lambda ^2 \Vert f_{0;t} \Vert ^2/4} \langle e^{\sqrt{N} \lambda b^* (f_{0;t})/2} \Omega , e^{-\lambda \kappa ' \mathcal {N}_+ (0) }e^{\sqrt{N} \lambda b^* (f_{0;t})/2} \Omega \rangle \nonumber \\&\quad = e^{N \lambda ^2 \Vert f_{0;t} \Vert ^2/4} \sum _{n=0}^N \frac{N^n \lambda ^{2n}}{4^{n} (n!)^2} e^{-\lambda \kappa ' n} \Vert b^* (f_{0;t})^n \Omega \Vert ^2 \; \end{aligned}$$

(3.92)

and furthermore

$$\begin{aligned}&\Vert b^* (f_{0;t})^n \Omega \Vert ^2 \nonumber \\&\quad = \left\| a^* (f_{0;t}) (1- \mathcal {N}_+ (0) /N)^{1/2} a^* (f_{0;t}) (1-\mathcal {N}_+ (0)/N)^{1/2} \cdots a^* (f_{0;t}) (1-\mathcal {N}_+ (0) /N)^{1/2} \Omega \right\| ^2 \nonumber \\&\quad = \frac{(N - (n-1)) \cdots (N-1)}{N^{(n-1)}} \Vert a^* (f_{0;t})^n \Omega \Vert ^2 = \frac{(N-1)! n! }{N^{(n-1)} (N-n)!} \Vert f_{0;t} \Vert _2^{2n} \; . \end{aligned}$$

(3.93)

Thus we have

$$\begin{aligned}&e^{N \lambda ^2 \Vert f_{0;t} \Vert ^2/4} \langle e^{\sqrt{N}\lambda b^* (f_{0;t})/2} \Omega , e^{-\lambda \kappa ' \mathcal {N}_+ (0) }e^{\sqrt{N} \lambda b^* (f_{0;t})/2} \Omega \rangle \nonumber \\&\quad = e^{N \lambda ^2 \Vert f_{0;t} \Vert _2^2/4} \sum _{n=0}^N {N \atopwithdelims ()n} \frac{\lambda ^{2n} \Vert f_{0;t} \Vert _2^{n} }{4^{n}} e^{-\lambda \kappa ' n} \nonumber \\&\quad = e^{N \lambda ^2 \Vert f_{0;t} \Vert _2^2/4} \, \left( 1 + \frac{\lambda ^{2} \Vert f_{0;t} \Vert _2^2}{4} e^{-\lambda \kappa '} \right) ^N \nonumber \\&\quad = e^{N \left( \lambda ^2 \Vert f_{0;t} \Vert _2^2/4 + \ln \left( 1 + \lambda ^2 \Vert f_{0;t} \Vert _2^2 e^{-\lambda \kappa '} /4 \right) \right) } \nonumber \\&\quad \ge e^{N \left( \lambda ^2 \Vert f_{0;t} \Vert _2^2 ( 1 + e^{-\lambda \kappa '}) \right) /4 - N \lambda ^4 \Vert f_{0;t} \Vert _2^4 / 32} \end{aligned}$$

(3.94)

where we used that $\ln (1+x) \ge x - x^2/2$ for $x\ge 0$. Using in addition that $e^{-\lambda \kappa '} \ge 1- \lambda \kappa '$ and $\Vert f_{0;t}\Vert _2 \le \Vert O\Vert e^{Ct}$, we arrive at the desired bound (3.83).

The upper bound (3.82) follows in essentially the same way, see [18, Lemma 3.3]. $\square$

Proof of Theorem 1.2

The upper bound (1.24) is an immediate consequence of (3.32), the upper bound in (3.33), (3.46) and (3.82). Similarly, the lower bound (1.25) follows by combining (3.32) with the lower bound in (3.33), (3.47) and (3.83). $\square$