A Second Order Upper Bound for the Ground State Energy of a Hard-Sphere Gas in the Gross–Pitaevskii Regime

Abstract

We prove an upper bound for the ground state energy of a Bose gas consisting of N hard spheres with radius $\mathfrak {a}/N$, moving in the three-dimensional unit torus $\Lambda$. Our estimate captures the correct asymptotics of the ground state energy, up to errors that vanish in the limit $N \rightarrow \infty$. The proof is based on the construction of an appropriate trial state, given by the product of a Jastrow factor (describing two-particle correlations on short scales) and of a wave function constructed through a (generalized) Bogoliubov transformation, generating orthogonal excitations of the Bose–Einstein condensate and describing correlations on large scales.

1 Introduction and Main Result

In [24], Lee-Huang-Yang predicted that the ground state energy per particle of a system of N bosons moving in a box with volume $N/\rho$ and interacting through a potential with scattering length $\mathfrak {a}$ is given, as $N \rightarrow \infty$, by

$$\begin{aligned} e(\rho ) = 4\pi \mathfrak {a} \rho \Big [ 1 + \frac{128}{15\sqrt{\pi }} (\rho \mathfrak {a}^3)^{1/2} + \cdots \Big ] \end{aligned}$$

(1.1)

up to corrections that are small, in the low density limit $\rho \mathfrak {a}^3 \ll 1$ (see [28, 32] for the heuristics behind this formula and its relation with the expected occurrence of Bose–Einstein condensation in dilute Bose gases). At leading order, the validity of (1.1) follows from the upper bound obtained in [16] and from the matching lower bound established in [29]. Recently, also the second order term on the r.h.s. of (1.1) has been rigorously justified. The upper bound has been shown in [35] (through a clever modification of a quasi-free trial state proposed in [17]) and (for a larger class of interactions and using a simpler trial state) in [3]. As for the lower bound, it has been first obtained in [20] for integrable potentials and then in [21], for particles interacting through general potentials, including hard-spheres. The upper bound for the case of hard-sphere potential is still an open question. An alternative approach to the study of the ground state energy of the zero temperature Bose gas, still not justified rigorously but possibly valid beyond the dilute regime, has been proposed in [26] and recently revived in [12,13,14].

Trapped Bose gases can be described as systems of N bosons, confined by external fields in a volume of order one and interacting through a radial, repulsive potential V with scattering length of the order $N^{-1}$; this scaling limit is known as the Gross–Pitaevskii regime (see [28, Chapter 6] for an introduction, and [33, 34] for reviews of more recent results). Focussing for simplicity on systems trapped in the unit torus $\Lambda$, the Hamilton operator takes the form

$$\begin{aligned} H_N = \sum _{j=1}^N -\Delta _{x_j} + \sum _{i<j}^N N^2 V (N (x_i - x_j)) \end{aligned}$$

(1.2)

and acts on $L^2_s (\Lambda ^N)$, the subspace of $L^2 (\Lambda ^N)$ consisting of functions that are symmetric w.r.t. permutations of the N particles. Note that $x_i-x_j$ is here the difference between the position vectors of particles i and j on the torus. Equivalently, we can think of $x_i - x_j$ as the difference in $\mathbb {R}^3$; however, in this case, V has to be replaced by its periodisation. As proven in [27, 29, 30], the ground state energy $E_N$ of (1.2) is given, to leading order, by

$$\begin{aligned} E_N = 4\pi \mathfrak {a} N + o (N) \end{aligned}$$

(1.3)

in the limit $N \rightarrow \infty$. For $V \in L^3 (\mathbb {R}^3)$, more precise information on the low-energy spectrum of (1.2) has been determined in [8]. Here, the ground state energy was proven to satisfy

$$\begin{aligned} \begin{aligned} E_N = \,&4 \pi \mathfrak {a}(N -1) \, +\, e_\Lambda \mathfrak {a}^2 \\&-\frac{1}{2} \sum _{p \in \Lambda ^*_+} \bigg [ p^2 + 8 \pi \mathfrak {a}- \sqrt{|p|^4 + 16 \pi \mathfrak {a}p^2} - \frac{(8 \pi \mathfrak {a})^2}{2p^2}\bigg ] + \mathcal {O}(N^{-1/4}) \end{aligned} \end{aligned}$$

(1.4)

where $\Lambda ^*_+ = 2 \pi \mathbb {Z}^3 {\setminus } \{0\}$ and

$$\begin{aligned} e_\Lambda = 2 - \lim _{M \rightarrow \infty } \sum _{\begin{array}{c} p \in {\mathbb Z}^3 {\setminus } \{0\}:\\ |p_1|, |p_2|, |p_3| \le M \end{array}} \frac{\cos (|p|)}{p^2}\,.\end{aligned}$$

(1.5)

Additionally, the spectrum of $H_N - E_N$ below a threshold $\zeta > 0$ was shown to consist of eigenvalues having the form

$$\begin{aligned} \sum _{p \in 2\pi \mathbb {Z}^3 \backslash \{ 0 \}} n_p \sqrt{|p|^4 + 16 \pi \mathfrak {a} p^2} + \mathcal {O}(N^{-1/4} \zeta ^3)\,. \end{aligned}$$

(1.6)

A new and simpler proof of (1.4), (1.6) was recently obtained in [22], for $V \in L^2 (\Lambda )$. Moreover, these results have been also extended to the non-homogeneous case of Bose gases trapped by external fields in [11, 31].

While the approach of [31] applies to $V \in L^1 (\mathbb {R}^3)$, the validity of (1.4), (1.6) for bosons interacting through non-integrable potentials is still an open question. The goal of this paper is to prove that (1.4) remains valid, as an upper bound, for particles interacting through a hard-sphere potential.

We consider N bosons in $\Lambda =[-\tfrac{1}{2}, \tfrac{1}{2}]^3 \subset \mathbb {R}^3$, with periodic boundary conditions. We assume particles to interact through a hard-sphere potential, with radius $\mathfrak {a}/N$, for some $\mathfrak {a} > 0$. We are interested in the ground state energy of the system, defined by

$$\begin{aligned} E^\text {hs}_N = \inf \, \Big \langle \Psi , \sum _{j=1}^N -\Delta _{x_j} \Psi \Big \rangle \end{aligned}$$

(1.7)

where the infimum is taken over all normalized wave functions $\Psi \in L_s^2 (\Lambda ^N)$ satisfying the hard-core condition

$$\begin{aligned} \Psi (x_1, \dots , x_N ) = 0 \end{aligned}$$

(1.8)

almost everywhere on the set

$$\begin{aligned} \bigcup _{i<j}^N \big \{ (x_1, \dots , x_N) \in \mathbb {R}^{3N}: |x_ i - x_j| \le \mathfrak {a}/N \big \}\,. \end{aligned}$$

Theorem 1.1

Let $E_N^{\text {hs}}$ be defined as in (1.7). There exist $C, \varepsilon > 0$ such that

$$\begin{aligned} \begin{aligned} E^{\text {hs}}_N \le \,&4 \pi \mathfrak {a}(N -1) \, +\, e_\Lambda \mathfrak {a}^2 \\&-\frac{1}{2} \sum _{p \in \Lambda ^*_+} \bigg [ p^2 + 8 \pi \mathfrak {a}- \sqrt{|p|^4 + 16 \pi \mathfrak {a}p^2} - \frac{(8 \pi \mathfrak {a})^2}{2p^2}\bigg ] + C N^{-\varepsilon } \end{aligned} \end{aligned}$$

(1.9)

for all N large enough, with $e_\Lambda$ defined as in (1.5).

Remarks

1) Theorem 1.1 establishes an upper bound for the ground state energy (1.7). With minor modifications, it would also be possible to obtain upper bounds for low-energy excited eigenvalues, agreeing with (1.6). To conclude the proof of the estimates (1.4), (1.6) for particles interacting through hard-sphere potentials, we would need to establish matching lower bounds. A possible approach to achieve this goal (at least for the ground state energy) consists in taking the lower bound established in [21], for particles in the thermodynamic limit, and to translate it to the Gross–Pitaevskii regime.

2) We believe that the statement of Theorem 1.1 and its proof can also be extended to bosons in the Gross–Pitaevskii regime interacting through a larger class of potentials, combining a hard-sphere potential at short distances and an integrable potential at larger distances. This would require the extension of Lemma 2.1 to more general interactions. To keep our analysis as simple as possible, we focus here on hard-sphere bosons.

3) Theorem 1.1 and its proof could also be extended to systems of N bosons interacting through a hard-sphere potential with radius of the order $N^{-1+\kappa }$ for sufficiently small $\kappa > 0$ (results for integrable potentials with scattering length of the order $N^{-1+\kappa }$ have been recently discussed in [1, 2, 9, 19]).

The proof of (1.4), (1.6) obtained in [8] is based on a rigorous version of Bogoliubov theory, developed in [5,6,7]. The starting point of Bogoliubov theory is the observation that, at low energies, the Bose gas exhibits complete condensation; all particles, up to a fraction vanishing in the limit $N \rightarrow \infty$, can be described by the same zero-momentum orbital $\varphi _0$ defined by $\varphi _0 (x) = 1$, for all $x \in \Lambda$. This, however, does not mean that the factorized wave function $\varphi _0^{\otimes N}$ is a good approximation for the ground state of (1.2); in fact, its energy does not even approximate the ground state energy to leading order. To decrease the energy and approach (1.3), correlations are crucial. The strategy developed in [5,6,7,8] is based on the idea that most correlations can be inserted through the action of (generalized) Bogoliubov transformations, having the form

$$\begin{aligned} T = \exp \left[ \frac{1}{2} \sum _{p \in \Lambda ^*_+} \eta _p \big ( b_p^* b_{-p}^* - b_p b_{-p} \big ) \right] \end{aligned}$$

(1.10)

where the (modified) creation and annihilation operators $b_p^*, b_p$ act on the Fock space of orthogonal excitations of the Bose–Einstein condensate; the precise definitions are given below, in Sect. 5 (to be more precise, the action of (1.10) has to be corrected through an additional unitary operator, given by the exponential of a cubic, rather than quadratic, expression in creation and annihilation operators; see [8] for details). An important feature of (generalized) Bogoliubov transformations of the form (1.10), which plays a major role in the derivation of (1.4), (1.6), is the fact that their action on creation and annihilation operators is (almost) explicit. This makes computations relatively easy and it gives the possibility of including correlations also at very large length scales.

Unfortunately, Bogoliubov transformations of the form (1.10) do not seem compatible with the hard-core condition (1.8). As a consequence, they do not seem appropriate to construct trial states approximating the ground state energy of a system of particles interacting through a hard-sphere potential. A different class of trial states, for which (1.8) can be easily verified, consists of products having the form

$$\begin{aligned} \Psi _N (x_1, \dots , x_N) = \prod _{j=1}^N f (x_i - x_j) \, \end{aligned}$$

(1.11)

for a function f satisfying $f (x) = 0$, for all $|x| < \mathfrak {a}/N$ (as mentioned after (1.2), also here $x_i -x_j$ is interpreted as difference on the torus). Such an ansatz was first used in the physics literature in [4, 15, 23]; it is often known as Jastrow factor. In order for (1.11) to provide a good approximation for the ground state energy, f must describe two-particle correlations. Probably the simplest possible choice of f is given by the solution

$$\begin{aligned} f(x) = \left\{ \begin{array}{ll} 0 &{}\quad \text {if } |x| < \mathfrak {a}/N \\ 1 - \frac{\mathfrak {a}}{N|x|} &{}\quad \text {if } |x| \ge \mathfrak {a}/N \end{array} \right. \end{aligned}$$

of the zero-energy scattering equation $-\Delta f = 0$, with the hard-core requirement $f(x) = 0$ for $|x| < \mathfrak {a}/N$ and the boundary condition $f(x) \rightarrow 1$, as $|x| \rightarrow \infty$. The problem with this choice is the fact that f has long tails; as a consequence, it is extremely difficult to control the product (1.11). To make computations possible, we need to cutoff f at some intermediate length scale $\mathfrak {a}/N \ll \ell \ll 1$, requiring that $f (x) = 1$ for $|x| \ge \ell$ (the cutoff can be implemented in different ways; below, we will choose f as the solution of a Neumann problem on the ball $|x| \le \ell$ and we will keep it constant outside the ball). Choosing $\ell$ small enough (in particular, smaller than the typical distance among particles, which is of the order $N^{-1/3}$), the Jastrow factor becomes more manageable and it is not too difficult to show that its energy matches, to leading order, the ground state energy (1.3). In the thermodynamic limit, this was first verified in [16], using a modification of (1.11), considering only correlations among neighbouring particles.

While Jastrow factors can lead to the correct leading order term in the ground state energy, it seems much more difficult to use (1.11) to obtain an upper bound matching also the second order term on the r.h.s. of (1.9). The point is that the second order corrections are generated by correlations at much larger length scales; to produce the term on the second line of (1.9) we would need to take $\ell$ of order one, making computations very difficult.

In order to prove Theorem 1.1, we will therefore consider a trial state given by the product of a Jastrow factor (1.11), describing correlations up to a sufficiently small length scale $1/N \ll \ell \ll 1$, and of a wave function $\Phi _N$, constructed through a Bogoliubov transformation, describing correlations on length scales larger than $\ell$. This allows us to combine the nice features of the Jastrow factor (in particular, the fact that it automatically takes care of the hard core condition (1.8)) and of the Bogoliubov transformation (in particular, their (almost) explicit action on creation and annihilation operators, which enables us to insert correlations at large length scales).

The paper is organised as follows. In Sect. 2, we define our trial state $\Psi _N$ as the product of a Jastrow factor and an N-particle wave function $\Phi _N$, to be specified later on, and we compute its energy. One of the contributions to the energy of $\Psi _N$ is a three-body term; under certain conditions on $\Phi _N$ (see (3.1)), we show that this term is negligible in Sect. 3. In Sect. 4 we then prove that the remaining contributions to the energy can be reduced (again under suitable assumptions on $\Phi _N$; see (4.4)) to the expectation of an effective Hamiltonian $H_N^{\textrm{eff}}$, defined in (4.3). Sections 5 and 6 are devoted to the study of $H_N^{\textrm{eff}}$; the goal is to find $\Phi _N$ so that the expectation of $H_N^{\textrm{eff}}$ produces the energy on the r.h.s of (1.9), up to negligible errors. Here, we use the approach developed in [5,6,7]. In Sect. 7, we show that the chosen wave function $\Phi _N$ satisfies the bounds that were used in Sects. 3 and 4. Finally, in Sect. 8, we put all ingredients together to conclude the proof of Theorem 1.1. The proof of important properties concerning the solution of the scattering equations is deferred to Appendix A.

2 The Jastrow Factor and Its Energy

As explained in the introduction, our trial state involves a Jastrow factor, to describe short-distance correlations. To define the Jastrow factor, we choose $1/N \ll \ell \ll 1$ and we consider the ground state solution of the Neumann problem

$$\begin{aligned} \left\{ \begin{array}{ll} - \Delta f_\ell (x) =\lambda _\ell f_{\ell }(x) &{}{}\quad {\text{ for } }\; \mathfrak {a}/N \le |x| \le \ell \\ \partial _r f_\ell (x) = 0 &{}{}\quad {\text{ if } }\; |x| = \ell \end{array} \right. \end{aligned}$$

(2.1)

on the ball $B_\ell = \{ x \in \mathbb {R}^3: |x| \le \ell \}$, with the hard-core condition $f_\ell (x) = 0$ for $|x| \le \mathfrak {a}/N$ and the normalization $f_\ell (x) = 1$ for $|x| = \ell$ (we denote here by $\partial _r$ the radial derivative). We extend $f_\ell$ to $\Lambda$ setting $f_\ell (x)=1$ for $|x|\in \Lambda \backslash B_\ell$. We have

$$\begin{aligned} - \Delta f_\ell (x) = \lambda _\ell \chi _\ell (x) f_{\ell }(x) \end{aligned}$$

(2.2)

where $\chi _\ell$ denotes the characteristic function of $B_\ell$. The following lemma establishes properties of $\lambda _\ell$, $f_\ell$, of the difference $\omega _\ell (x)= 1- f_\ell (x)$ and of its Fourier coefficients

$$\begin{aligned} {\widehat{\omega }}_\ell (p) = \int e^{ip \cdot x} \omega _\ell (x) dx \end{aligned}$$

defined for $p \in \Lambda ^* = 2\pi \mathbb {Z}^3$ (since $\omega _\ell$ has compact support inside $[-1/2; 1/2]^3$, we can think of the integral as being over $\mathbb {R}^3$).

Lemma 2.1

Let $\lambda _\ell$ denote the ground state eigenvalue appearing in (2.1). Then

$$\begin{aligned} \tan \big ( \sqrt{\lambda _\ell }\, (\ell - \mathfrak {a}/N) \big ) = \sqrt{\lambda _\ell } \,\ell \,. \end{aligned}$$

(2.3)

For $N \ell \rightarrow \infty$, we find

$$\begin{aligned} \lambda _\ell =\frac{3 {\mathfrak {a}}}{N \ell ^3} \left[ 1+\frac{9}{5}\frac{{\mathfrak {a}}}{N \ell } + {\mathcal {O}}\Big (\frac{{\mathfrak {a}}^2}{N^2 \ell ^2}\Big ) \right] . \end{aligned}$$

(2.4)

The corresponding eigenvector $f_\ell$ is given by

$$\begin{aligned} f_\ell (x) = \frac{\ell }{|x|} \frac{\sin (\sqrt{\lambda }_\ell (|x|-\mathfrak {a}/N))}{\sin (\sqrt{\lambda }_\ell (\ell -\mathfrak {a}/N))} \end{aligned}$$

(2.5)

for all $\mathfrak {a}/N \le |x| \le \ell$ ($f_\ell (x) = 0$ for $|x| \le \text{\AA} /N$ and $f_\ell (x) = 1$ for $|x| > \ell$). We find

$$\begin{aligned} N \lambda _{\ell } \int \chi _{\ell } f^2_{\ell }\, dx = 4 \pi \mathfrak {a}+ \frac{24}{5}\pi \frac{\mathfrak {a}^2}{\ell N} + \mathcal {O}\Big ( \frac{\mathfrak {a}^2}{N^{2}\ell ^2} \Big ). \end{aligned}$$

(2.6)

With the notation $\omega _\ell (x) = 1 - f_\ell (x)$, we have $\omega _\ell (x) = 0$ for $|x| \ge \ell$ and, for $|x| \le \ell$, the pointwise bounds

$$\begin{aligned} 0 \le \omega _\ell (x) \le \frac{C \mathfrak {a}}{N|x|}, \qquad |\nabla \omega _\ell (x)| \le \frac{C \mathfrak {a}}{N|x|^2} \end{aligned}$$

(2.7)

for a constant $C>0$. Furthermore, there exists a constant $C>0$ so that

$$\begin{aligned} \Big | \Vert \omega _\ell \Vert _1 - \frac{2}{5} \,\pi {\mathfrak {a}} \, \frac{\ell ^2}{N} \Big | \le C \frac{\mathfrak {a}^2 \ell }{N^2} \end{aligned}$$

(2.8)

and, for all $p \in [1, 3)$ and $q \in [1, 3/2)$,

$$\begin{aligned} \Vert \omega _\ell \Vert _{p} \le C \ell ^{\frac{3}{p}-1}N^{-1}, \qquad \Vert \nabla \omega _\ell \Vert _{q} \le C \ell ^{\frac{3}{q}-2}N^{-1}. \end{aligned}$$

(2.9)

Finally, for $p \in \Lambda ^*$, let $\widehat{\omega }_p$ denote the Fourier coefficients of $\omega _\ell$. Then

$$\begin{aligned} |\widehat{\omega }_\ell (p)| \le C \min \left\{ \frac{\ell ^2 }{N} ;\, \frac{1}{N |p|^2};\, \frac{1}{|p|^3}\right\} . \end{aligned}$$

(2.10)

We defer the proof of Lemma 2.1 to Appendix A.

With the solution $f_\ell$ of the Neumann problem (2.1), we consider trial states of the form

$$\begin{aligned} \Psi _N (x_1, \dots , x_N) = \Phi _N (x_1, \dots , x_N) \prod _{i<j}^N f_\ell (x_i - x_j) \end{aligned}$$

(2.11)

for $\Phi _N \in L^2_s (\Lambda ^N)$ to be specified later on. Again, $x_i -x_j$ should be interpreted as difference on the torus (or $f_\ell$ should be replaced with its periodic extension). Note that a similar trial state has been used in [27]. However, for us the wave function $\Phi _N$ serves a completely different purpose (in our analysis, $\Phi _N$ carries correlations on length scales larger than $\ell$; in [27], on the other hand, it was a product state, describing the condensate trapped in an external potential).

We compute

$$\begin{aligned} \begin{aligned} \frac{-\Delta _{x_j} \Psi _N (x_1, \dots , x_N) }{\prod _{i<j}^N f_\ell (x_i - x_j) } = \;&\Big [ -\Delta _{x_j} - 2 \sum _{i \not = j}^N \frac{\nabla f_\ell (x_j - x_i)}{f_\ell (x_j - x_i)} \cdot \nabla _{x_j} \Big ] \Phi _N (x_1, \dots , x_N) \\ {}&+ \sum _{i \not = j}^N \frac{-\Delta f_\ell (x_j - x_i)}{f_\ell (x_j - x_i)} \Phi _N (x_1, \dots , x_N) \\&- \sum ^N_{i,m,j} \frac{\nabla f_\ell (x_j - x_i)}{f_\ell (x_j. -x_i)} \cdot \frac{\nabla f_\ell (x_j - x_m)}{f_\ell (x_j - x_m)} \Phi _N (x_1, \dots , x_N) \end{aligned} \end{aligned}$$

where the sum in the last term runs over $i,j, m \in \{1, \dots , N \}$ all different. Noticing that the operator on the first line is the Laplacian with respect to the measure defined by (the square of) the Jastrow factor, and using (2.2) in the second line, we conclude that

$$\begin{aligned} \begin{aligned} \langle \Psi _N, \sum _{j=1}^N -\Delta _{x_j} \Psi _N \rangle&= \, \sum _{j=1}^N \int |\nabla _{x_j} \Phi _N ({\textbf {x}})|^2 \prod _{n<m}^N f^2_\ell (x_n - x_m) d{\textbf {x}} \\ {}&\quad + \sum _{i<j}^N 2\lambda _\ell \int \chi _\ell (x_i - x_j) |\Phi _N ({\textbf {x}})|^2 \prod _{n<m}^N f^2_\ell (x_n - x_m) d{\textbf {x}} \\ {}&\quad - \sum _{i,j,k} \int \frac{\nabla f_\ell (x_j - x_i)}{f_\ell (x_j - x_i)} \cdot \frac{\nabla f_\ell (x_j - x_k)}{f_\ell (x_j - x_k)} |\Phi _N ({\textbf {x}})|^2 \prod _{m<n}^N f^2_\ell (x_m - x_n) d{\textbf {x}} \end{aligned}\nonumber \\ \end{aligned}$$

(2.12)

where we introduced the notation $\textbf{x} = (x_1, \dots ,x_N) \in \Lambda ^N$.

3 Estimating the Three-Body Term

In the next proposition, we control the last term on the r.h.s. of (2.12). To this end, we need to assume some regularity on the N-particle wave function $\Phi _N$, appearing in (2.11) (we will later make sure that our choice of $\Phi _N$ satisfies these estimates).

Proposition 3.1

Let $N^{-1+\nu } \le \ell \le N^{-1/2-\nu }$, for some $\nu > 0$. Suppose $\Phi _N \in L^2_s (\Lambda ^N)$ is such that

$$\begin{aligned} \langle \Phi _N, (1-\Delta _{x_1}) (1- \Delta _{x_2})(1-\Delta _{x_3}) \Phi _N \rangle \le C \left( 1 + \frac{1}{N^2 \ell ^3} \right) \end{aligned}$$

(3.1)

and define $\Psi _N$ as in (2.11). Then, for every $\delta > 0$, there exists $C > 0$ such that

$$\begin{aligned} \Big | \frac{1}{\Vert \Psi _N \Vert ^2} \sum _{i,j,k} \int \frac{\nabla f_\ell (x_j - x_i)}{f_\ell (x_j - x_i)} \cdot&\frac{\nabla f_\ell (x_j - x_k)}{f_\ell (x_j - x_k)} |\Phi _N ({\textbf {x}})|^2 \prod _{n<m}^N f^2_\ell (x_n - x_m) d{\textbf {x}} \Big | \nonumber \\ {}&\le C N \ell ^{2-\delta } \left( 1 + \frac{1}{N^2 \ell ^3} \right) . \end{aligned}$$

(3.2)

To prove this proposition, we will use the following lemma.

Lemma 3.2

Let $W: \mathbb {R}^3 \rightarrow \mathbb {R}$, with $\text {supp } W \subset [-1/2; 1/2]^3$. Then W can be extended to a periodic function (i.e. a function on the torus $\Lambda$) satisfying, on $L^2 (\Lambda ) \otimes L^2 (\Lambda )$, the operator inequalities

$$\begin{aligned} \begin{aligned} \pm W(x-y) \le \,&C \Vert W\Vert _{3/2} \left( 1-\Delta _x \right) \\ \pm W(x-y) \le \,&C \Vert W\Vert _{2} \left( 1-\Delta _x \right) ^{3/4} \end{aligned}\end{aligned}$$

for a constant $C>0$, independent on W. Moreover, for every $\delta \in [0,1/2)$ there exists $C > 0$ such that

$$\begin{aligned} \pm W(x-y) \le \; C \Vert W\Vert _{1} \Big \{ 1 + \left( -\Delta _x\right) ^{3/4+\delta /2} \left( -\Delta _y\right) ^{3/4+\delta /2} \Big \}\,. \end{aligned}$$

(3.3)

Additionally, for any $r > 1$, there exists $C > 0$ such that

$$\begin{aligned} \begin{aligned} \pm W (x-y) W (x-z) \le \,&C \Vert W\Vert ^2_r \, \left( 1-\Delta _x \right) \left( 1-\Delta _y\right) \left( 1-\Delta _z \right) . \end{aligned} \end{aligned}$$

(3.4)

Proof

The proof is an adaptation to the torus of arguments that are, by now, standard on $\mathbb {R}^3$. For example, (3.3) follows by writing, in momentum space

$$\begin{aligned} \begin{aligned} \big | \langle \varphi , W(x-y) \varphi \rangle \big |&= \Big | \sum _{p_1, p_2, q_1, q_2 \in \Lambda ^*} {\widehat{W}} (p_1 - q_1) \widehat{\varphi } (p_1, p_2) \overline{\widehat{\varphi } (q_1, q_2)} \, \delta _{p_1 + p_2, q_1 + q_2} \Big | \\ {}&\le C \Vert {\widehat{W}} \Vert _\infty \sup _{p \in \Lambda ^*} \sum _{q \in \Lambda ^*_+} \frac{1}{1+|q|^{3/2+ \delta } |p-q|^{3/2+\delta }} \\ {}&\quad \qquad \times \big \langle \varphi , \big [ 1 + (-\Delta _x)^{3/4+\delta /2} (-\Delta _y)^{3/4+\delta /2}\big ] \varphi \big \rangle \\ {}&\le C \Vert W \Vert _1 \big \langle \varphi , \big [ 1 + (-\Delta _x)^{3/4+\delta /2} (-\Delta _y)^{3/4+\delta /2} \big ] \varphi \big \rangle \,. \end{aligned} \end{aligned}$$

To show (3.4), we proceed similarly, writing

$$\begin{aligned} \begin{aligned}&\big | \langle \varphi , W(x-y) W(x-z) \varphi \rangle \big | \\ {}&\qquad = \, \Big |\sum {\widehat{W}} (p_2 - q_2) {\widehat{W}} (p_3 - q_3) \widehat{\varphi } (p_1, p_2,p_3) \overline{\widehat{\varphi } (q_1, q_2,q_3)} \, \delta _{p_1 + p_2+p_3, q_1 + q_2+q_3} \Big | \\ {}&\qquad \le \, C \sup _p \sum _{q_2, q_3 \in \Lambda ^*} \frac{|{\widehat{W}} (p_2 - q_2)| | {\widehat{W}} (p_3 - q_3)|}{(1+|p-q_2 - q_3|^2)(1+|q_2|^2) (1+|q_3|^2)} \\ {}&\qquad \quad \times \langle \varphi , (1-\Delta _x) ( 1-\Delta _y) ( 1- \Delta _z) \varphi \rangle \\ {}&\qquad \le \, C \Vert {\widehat{W}} \Vert _{r'}^2 \langle \varphi , (1-\Delta _x) ( 1-\Delta _y) ( 1- \Delta _z) \varphi \rangle \end{aligned} \end{aligned}$$

where $1/r+ 1/r' = 1$ and where we used the bound

$$\begin{aligned} \sum _{q_2, q_3 \in \Lambda ^*} \frac{1}{(1+|p-q_2 - q_3|^2)^r (1+|q_2|^2)^r (1+|q_3|^2)^r} \le C \end{aligned}$$

uniformly in p, for any $r > 1$. $\square$

We are now ready to show Proposition 3.1.

Proof of Prop. 3.1

Using the permutation symmetry, $0 \le f_\ell \le 1$ and then Lemma 3.2 (in particular, (3.4)), the bound (2.9) and the assumption (3.1), we can estimate the numerator in (3.2) by

$$\begin{aligned}{} & {} {} \left| \sum _{i,j,k} \int \frac{\nabla f_\ell (x_j - x_i)}{f_\ell (x_j - x_i)}\cdot \frac{\nabla f_\ell (x_j - x_k)}{f_\ell (x_j - x_k)} |\Phi _N ({\textbf {x}})|^2 \prod _{m<n}^N f^2_\ell (x_m - x_n) d{\textbf {x}} \right| \nonumber \\{}{} & {} {} \quad \le C N^3 \int |\nabla f_\ell (x_1 - x_2)||\nabla f_\ell (x_1 - x_3)| |\Phi _N ({\textbf {x}})|^2 d{\textbf {x}} \nonumber \\{}{} & {} {} \quad \le C N^3 \Vert \nabla f_\ell \Vert _r^2 \langle \Phi _N, (1- \Delta _{x_1}) (1-\Delta _{x_2)} (1-\Delta _{x_3}) \Phi _N \rangle \nonumber \\{}{} & {} {} \quad \le C N \ell ^{\frac{6}{r} -4} \Big ( 1 + \frac{1}{N^2 \ell ^3} \Big ) \end{aligned}$$

(3.5)

for any $r > 1$. As for the denominator in (3.2), we write $u_\ell = 1- f_\ell ^2 = 2\omega _\ell - \omega _\ell ^2$, with $\omega _\ell$ defined after (2.2), and we bound (see (4.6) below for a justification of this inequality)

$$\begin{aligned} \prod _{n<m}^N f_\ell ^2 (x_n - x_m) \ge 1 - \sum _{n<m}^N u_\ell (x_n - x_m). \end{aligned}$$

Using $\Vert \Phi _N \Vert = 1$, Lemma 3.2 (in particular, (3.3)), the bound (2.9) and again the assumption (3.1), we arrive at

$$\begin{aligned} \begin{aligned} \int |\Phi _N ({\textbf {x}})|^2 \prod _{n<m}^N f_\ell ^2 (x_i - x_j) d{\textbf {x}}&\ge 1 - \sum _{n<m}^N \int |\Phi _N ({\textbf {x}})|^2 u_\ell (x_n - x_m) d{\textbf {x}} \\ {}&\ge 1 - C N^2 \Vert u_\ell \Vert _1 \langle \Phi _N, (1- \Delta _{x_1}) (1-\Delta _{x_2}) \Phi _N \rangle \\ {}&\ge 1 - C N \ell ^2 \Big ( 1 + \frac{C}{N^2 \ell ^3} \Big ) \ge 1 - C N \ell ^2 - \frac{C}{N\ell } \ge 1/2 \end{aligned} \end{aligned}$$

for $N^{-1} \ll \ell \ll N^{-1/2}$. Combining this estimate with (3.5) and choosing $r > 1$ so that $6/r - 4 > 2 - \delta$, we obtain the desired bound. $\square$

4 Reduction to an Effective Hamiltonian

Let us introduce the notation

$$\begin{aligned} E_\text{ kin } (\Phi _N)= & {} {} \sum _{j=1}^N \int |\nabla _{x_j} \Phi _N (x_1, \dots , x_N)|^2 \prod _{n<m}^N f_\ell ^2 (x_n - x_m) dx_1 \dots dx_N \nonumber \\ E_\text{ pot } (\Phi _N)= & {} {} \sum _{i<j}^N 2\lambda _\ell \int \chi _\ell (x_i - x_j) |\Phi _N (x_1, \dots , x_N)|^2 \prod _{n<m}^N f_\ell ^2 (x_n -x_m) dx_1 \dots dx_N. \nonumber \\ \end{aligned}$$

(4.1)

It follows from (2.12) and Prop. 3.1 that

$$\begin{aligned} \frac{1}{\Vert \Psi _N \Vert ^2} \langle \Psi _N, \sum _{j=1}^N -\Delta _{x_j} \Psi _N \rangle = \frac{1}{\Vert \Psi _N \Vert ^2} \big [ E_\text {kin} (\Phi _N) + E_\text {pot} (\Phi _N) \big ] + \mathcal {E}\end{aligned}$$

(4.2)

where $\pm \mathcal {E}\le C N \ell ^{2-\delta } (1+ 1/ (N^2\ell ^3))$, provided $\Phi _N$ satisfies (3.1).

The goal of this subsection is to rewrite the main term on the r.h.s. of (4.2) as the expectation, in the state $\Phi _N \in L^2_s (\Lambda ^N)$, of an effective N-particle Hamiltonian having the form

$$\begin{aligned} H_N^\text {eff} = \sum _{j=1}^N - \Delta _{x_j} + 2 \sum _{i<j}^N \nabla _{x_j} \cdot u_\ell (x_i - x_j) \nabla _{x_j} + 2 \sum _{i<j}^N \lambda _\ell \chi _\ell (x_i - x_j) f_\ell ^2 (x_i - x_j)\nonumber \\ \end{aligned}$$

(4.3)

where $u_\ell = 1 - f_\ell ^2$. To achieve this goal, we will make use of the following regularity bounds on the wave function $\Phi _N$ (when we will define $\Phi _N$ in the next sections, we will prove that it satisfies these estimates):

$$\begin{aligned} \langle \Phi _N, (-\Delta _{x_1}) \Phi _N \rangle \le&{} \frac{C}{N\ell } \nonumber \\ \langle \Phi _N, (-\Delta _{x_1})(-\Delta _{x_2}) \Phi _N \rangle \le&{} \frac{C}{N^2 \ell ^3} \nonumber \\ \langle \Phi _N, (-\Delta _{x_1})(- \Delta _{x_2})(- \Delta _{x_3}) \Phi _N \rangle \le&{} \frac{C}{N^3 \ell ^4} \nonumber \\ \langle \Phi _N, (-\Delta _{x_1})(- \Delta _{x_2})(- \Delta _{x_3}) (-\Delta _{x_4}) \Phi _N \rangle \le&{} \frac{C}{N^4 \ell ^6} \nonumber \\ \langle \Phi _N, (-\Delta _{x_1})^{3/4+\delta } (-\Delta _{x_2})^{3/4+\delta } \dots (-\Delta _{x_n})^{3/4+ \delta } \Phi _N \rangle \le&{} \frac{C}{N^n \ell ^{\alpha _n}} \nonumber \\ \langle \Phi _N, (-\Delta _{x_1}) (- \Delta _{x_2})^{3/4+\delta } (-\Delta _{x_3})^{3/4+\delta } \dots (-\Delta _{x_n})^{3/4+\delta } \Phi _N \rangle \le&{} \frac{C}{N^n \ell ^{\beta _n}} \end{aligned}$$

(4.4)

for all $n \le 6$ and $\delta > 0$ small enough and for sequences $\alpha _n, \beta _n$ defined by $\alpha _n = (7/6+\delta ) n -(4/9)(1- (-1/2)^n)$ and $\beta _n = \alpha _n + 1/2 -\delta$. In applications (in particular, in Prop. 4.1 below) we will only need the last two bounds in (4.4) for $n = 2,4,6$ and, respectively, for $n = 3,4,5$. The relevant values of $\alpha _n, \beta _n$ are given by: $\alpha _2 = 2+2\delta$, $\alpha _4 = 17/4+4\delta$, $\alpha _6 = 105/16+6\delta$, $\beta _3 = 7/2+2\delta$, $\beta _4= 19/4+3\delta$, $\beta _5 =47/8+4\delta$.

Proposition 4.1

Consider a sequence $\Phi _N \in L^2_s (\Lambda ^N)$ of normalized wave functions, satisfying the bounds (4.4) and such that $\langle \Phi _N, H_N^\text {eff} \Phi _N \rangle \le 4 \pi \mathfrak {a} N + C$, for a constant $C > 0$ (independent of N), and for all N large enough. Suppose $N^{-1 + \nu } \le \ell \le N^{-3/4-\nu }$, for some $\nu > 0$. Then, there exist $C, \varepsilon > 0$ such that

$$\begin{aligned}&{} \frac{1}{\Vert \Psi _N \Vert ^2} \big [ E_{\mathrm {{kin}}} (\Phi _N) + E_{\mathrm {{pot}}} (\Phi _N) \big ] \le \langle \Phi _N, H_N^\text{ eff } \Phi _N \rangle - \frac{N(N-1)}{2}\nonumber \\{}&{} \qquad \times \Big \langle \Phi _N, \Big \{ \big [ H_{N-2}^\text{ eff } - 4 \pi \mathfrak {a} N \big ] \otimes u_{\ell } (x_{N-1} - x_N) \Big \} \Phi _N \Big \rangle + C N^{-\varepsilon }.\end{aligned}$$

(4.5)

Remark

We will later prove a lower bound for $H_{N-2}^\text {eff} - 4 \pi \mathfrak {a} N$ which will allow us to show that the second term on the r.h.s. of (4.5) is negligible, in the limit $N \rightarrow \infty$.

Proof

Writing again $u_\ell = 1- f_\ell ^2$, we can estimate

$$\begin{aligned} \prod _{i<j}^N f_\ell ^2 (x_i - x_j)\ge & {} {} 1- \sum _{i<j} u_\ell (x_i - x_j) \nonumber \\ \prod _{i<j}^N f_\ell ^2 (x_i - x_j)\le & {} {} 1 - \sum _{i<j} u_\ell (x_i - x_j) + \frac{1}{2} \sum _{\begin{array}{c} i<j; \, m<n:\\ (i,j) \not = (m,n) \end{array}} u_\ell (x_i - x_j) u_\ell (x_m - x_n).\nonumber \\ \end{aligned}$$

(4.6)

These bounds follow by setting $h(s) = \prod _{i<j}^N (1 - s u_\ell (x_i -x_j))$, for $s \in [0;1]$, and by proving that

$$\begin{aligned} h' (s) \ge - \sum _{i<j}^N u_\ell (x_i -x_j), \qquad h'' (s) \le \sum _{\begin{array}{c} i<j; m<n:\\ (i,j) \not = (m,n) \end{array}} u_\ell (x_i - x_j) u_\ell (x_m - x_n) \end{aligned}$$

for all $s \in (0;1)$. Thus, we obtain the upper bound

$$\begin{aligned} E_\text {kin}{} & {} E_\text{ kin }{} {} (\Phi _N) \nonumber \\ {}{}{} & {} {} \quad \le \, N \int |\nabla _{x_1} \Phi _N ({{{\textbf {x}}}})|^2 d{\textbf {x}} - N \int |\nabla _{x_1} \Phi _N ({\textbf {x}})|^2 \sum _{i<j} u_\ell (x_i - x_j) \, d {\textbf {x}} \nonumber \\ {}{}{} & {} {} \qquad + \frac{N}{2} \int |\nabla _{x_1} \Phi _N ({\textbf {x}})|^2 \sum _{\begin{array}{c} i<j; m<n:\\ (i,j) \not = (m,n) \end{array}} u_\ell (x_i - x_j) u_\ell (x_m - x_n) \, d{\textbf {x}} \nonumber \\ {}{}{} & {} {} = \, N \int |\nabla _{x_1} \Phi _N ({\textbf {x}}) |^2 (1- (N-1) u_\ell (x_1 - x_2)) d{\textbf {x}} \nonumber \\ {}{}{} & {} {} \quad - \frac{N (N-1)(N-2)}{2} \int |\nabla _{x_1} \Phi _N ({\textbf {x}})|^2 (1 - (N-3) u_\ell (x_1 - x_2)) u_\ell (x_3 - x_4) \, d{\textbf {x}} \nonumber \\ {}{}{} & {} {} \quad + \mathcal {E}_\text{ kin } \end{aligned}$$

(4.7)

where

$$\begin{aligned} \mathcal {E}_\text {kin}\le & {} C N^3 \int |\nabla _{x_1} \Phi _N (\textbf{x})|^2 u_\ell (x_1 - x_2) u_\ell (x_1-x_3) d\textbf{x}\nonumber \\ {}{} & {} +C N^3 \int |\nabla _{x_1} \Phi _N (\textbf{x})|^2 u_\ell (x_1 - x_2) u_\ell (x_2 - x_3) d\textbf{x} \nonumber \\{} & {} + C N^4 \int |\nabla _{x_1} \Phi _N (\textbf{x})|^2 u_\ell (x_2 - x_3) u_\ell (x_2 - x_4) d\textbf{x} \nonumber \\{} & {} + C N^5 \int |\nabla _{x_1} \Phi _N (\textbf{x})|^2 u_\ell (x_2 - x_3) u_\ell (x_4 - x_5) d\textbf{x} \nonumber \\ {}= & {} \, \mathcal {E}_\text {kin}^{(1)} + \mathcal {E}_\text {kin}^{(2)} + \mathcal {E}_\text {kin}^{(3)}+ \mathcal {E}_\text {kin}^{(4)}. \end{aligned}$$

(4.8)

Consider the first error term on the r.h.s. of (4.8). Writing $\mathfrak {p} = |\varphi _0 \rangle \langle \varphi _0|$ for the orthogonal projection onto the condensate wave function $\varphi _0 (x) \equiv 1$, $\mathfrak {p}_j$ for $\mathfrak {p}$ acting on the j-th particle and $\mathfrak {q}_j = 1 - \mathfrak {p}_j$, we find

$$\begin{aligned} \mathcal {E}_\text {kin}^{(1)}= & {} C N^3 \langle \nabla _{x_1} \Phi _N, u_\ell (x_1 - x_2) u_\ell (x_1 - x_3) \nabla _{x_1} \Phi _N \rangle \nonumber \\\le & {} C N^3 \langle \nabla _{x_1} \mathfrak {q}_3 \Phi _N, u_\ell (x_1 - x_2) u_\ell (x_1 - x_3) \nabla _{x_1} \mathfrak {q}_3 \Phi _N \rangle \nonumber \\{} & {} + C N^3 \Vert u_\ell \Vert _1 \langle \nabla _{x_1} \mathfrak {p}_3 \Phi _N, u_\ell (x_1 - x_2) \nabla _{x_1} \mathfrak {p}_3 \Phi _N \rangle \nonumber \\\le & {} C N^3 \langle \nabla _{x_1} \mathfrak {q}_2 \mathfrak {q}_3 \Phi _N, u_\ell (x_1 - x_2) u_\ell (x_1 - x_3) \nabla _{x_1} \mathfrak {q}_2 \mathfrak {q}_3 \Phi _N \rangle \nonumber \\{} & {} + C N^3 \Vert u_\ell \Vert _1 \langle \nabla _{x_1} \mathfrak {q}_2 \mathfrak {p}_3 \Phi _N, u_\ell (x_1 - x_2) \nabla _{x_1} \mathfrak {q}_2 \mathfrak {p}_3 \Phi _N \rangle \nonumber \\{} & {} + C N^3 \Vert u_\ell \Vert ^2_1 \Vert \nabla _{x_1} \mathfrak {p}_2 \mathfrak {p}_3 \Phi _N \Vert ^2. \end{aligned}$$

(4.9)

With Lemma 3.2, and observing that, on the range of $\mathfrak {q}$, $(1-\Delta ) \le - C \Delta$, we obtain

$$\begin{aligned} \begin{aligned} \mathcal {E}_\text {kin}^{(1)} \le \,&C N^3 \Vert u_\ell \Vert _{3/2}^2 \langle \Phi _N, (-\Delta _{x_1})(-\Delta _{x_2}) ( -\Delta _{x_3}) \Phi _N \rangle \\ {}&+ C N^3 \Vert u_\ell \Vert _1 \Vert u_\ell \Vert _{3/2} \langle \Phi _N, (-\Delta _{x_1})(-\Delta _{x_2}) \Phi _N \rangle + C N^3 \Vert u_\ell \Vert _1^2 \langle \Phi _N, (-\Delta _{x_1}) \Phi _N \rangle . \end{aligned} \end{aligned}$$

The term $\mathcal {E}^{(2)}_\text {kin}$ can be treated like $\mathcal {E}^{(1)}_\text {kin}$. Proceeding analogously, we also find, with (3.4),

$$\begin{aligned} \begin{aligned} \mathcal {E}_\text {kin}^{(3)} \le \,&C N^4 \Vert u_\ell \Vert _r^2 \langle \Phi _N, (-\Delta _{x_1}) (-\Delta _{x_2}) (-\Delta _{x_3}) (-\Delta _{x_4}) \Phi _N \rangle \\ {}&+ C N^4 \Vert u_\ell \Vert _1^2 \langle \Phi _N, (-\Delta _{x_1}) \big [ 1 + (\Delta _{x_2} \Delta _{x_3})^{3/4+\delta } \big ] \Phi _N \rangle \end{aligned} \end{aligned}$$

for any $r > 1$, and

$$\begin{aligned} \begin{aligned} \mathcal {E}_\text {kin}^{(4)} \le \,&CN^5 \Vert u_\ell \Vert _1^2 \big \langle \Phi _N, (-\Delta _{x_1}) \big [ 1 + (\Delta _{x_2} \Delta _{x_3})^{3/4+\delta } + (\Delta _{x_2} \Delta _{x_3} \Delta _{x_4} \Delta _{x_5})^{3/4+\delta } \big ] \Phi _N \big \rangle . \end{aligned} \end{aligned}$$

From $u_\ell = 1-f_\ell ^2 = 2\omega _\ell - \omega _\ell ^2$, we obtain $0 \le u_\ell \le 2 \omega _\ell$ and thus, with (2.9),

$$\begin{aligned} \Vert u_\ell \Vert _r \le C \ell ^{\frac{3}{p}-1} / N \end{aligned}$$

(4.10)

for any $p \ge 1$. From the assumption (4.4), we find

$$\begin{aligned} \begin{aligned}\mathcal {E}_\text{ kin}^{(1)}, \mathcal {E}_\text{ kin}^{(2)} \le \frac{C}{N^2 \ell ^2}, \quad \mathcal {E}_\text{ kin}^{(3)} \le \frac{C}{N^2 \ell ^{2 +6 (1-1/r)}}, \quad \mathcal {E}^{(4)}_\text{ kin } \le C N^2 \ell ^3 + C \ell ^{1/2-2\delta } + \frac{C}{N^2 \ell ^{15/8+4\delta }}. \end{aligned} \end{aligned}$$

Choosing $\delta > 0$ sufficiently small and $r > 1$ sufficiently close to 1, we conclude that there exist $C, \varepsilon > 0$ such that $\mathcal {E}_\text {kin} \le C N^{-\varepsilon }$, if $N^{-1+\nu } \le \ell \le N^{-2/3- \nu }$ for a $\nu > 0$, and $N \in \mathbb {N}$ is large enough.

Let us now consider the potential energy. From (4.1), we can estimate

$$\begin{aligned} E_\text {pot} (\Phi _N) \le N (N-1) \lambda _\ell \int \chi _\ell (x_1 - x_2) f_\ell ^2 (x_1 - x_2) |\Phi _N (\textbf{x})|^2 \prod _{3 \le i<j}^N f_\ell ^2 (x_i - x_j) d \textbf{x}. \end{aligned}$$

With (4.6) (applied now to the product over $3 \le i < j$), we obtain

$$\begin{aligned}{} & {} E_\text{ pot }{} {} (\Phi _N) \nonumber \\ {}{}{} & {} {} \quad \le \, N (N-1) \lambda _\ell \int \chi _\ell (x_1 - x_2) f_\ell ^2 (x_1 - x_2) |\Phi _N ({\textbf {x}})|^2 d{\textbf {x}} \nonumber \\ {}{}{} & {} {} \quad - \frac{N (N-1)(N-2)(N-3)}{2} \lambda _\ell \int \chi _\ell (x_1 - x_2) f_\ell ^2 (x_1- x_2) |\Phi _N ({\textbf {x}})|^2 u_\ell (x_3 - x_4) d {\textbf {x}} \nonumber \\ {}{}{} & {} {} \quad + \mathcal {E}_\text{ pot } \end{aligned}$$

(4.11)

where

$$\begin{aligned} \begin{aligned} \mathcal {E}_\text {pot} \le \,&CN^6 \lambda _\ell \int \chi _\ell (x_1 - x_2) f_\ell ^2 (x_1 - x_2) |\Phi _N (\textbf{x})|^2 u_\ell (x_3 - x_4) u_\ell (x_5 - x_6) d\textbf{x} \\&+CN^5 \lambda _\ell \int \chi _\ell (x_1 - x_2) f_\ell ^2 (x_1 - x_2) |\Phi _N (\textbf{x})|^2 u_\ell (x_3 - x_4) u_\ell (x_4 - x_5) d\textbf{x} \\ = \,&\mathcal {E}_\text {pot}^{(1)} + \mathcal {E}_\text {pot}^{(2)}.\end{aligned} \end{aligned}$$

Proceeding similarly to (4.9) (introducing the projections $\mathfrak {p}_j, \mathfrak {q}_j$), we can bound

$$\begin{aligned} \begin{aligned} \mathcal {E}_\text {pot}^{(1)} \le \,&CN^6 \lambda _\ell \Vert \chi _\ell \Vert _1 \Vert u_\ell \Vert _1^2 \Big [ 1 + \langle \Phi _N, (\Delta _{x_1} \Delta _{x_2})^{3/4+\delta } \Phi _N \rangle + \langle \Phi _N, (\Delta _{x_1} \dots \Delta _{x_4})^{3/4+\delta } \Phi _N \rangle \\ {}&+ \langle \Phi _N, (\Delta _{x_1} \dots \Delta _{x_6})^{3/4+\delta } \Phi _N \rangle \Big ] \\ \mathcal {E}_\text {pot}^{(2)} \le \,&CN^5 \lambda _\ell \Vert \chi _\ell \Vert _1 \Vert u_\ell \Vert _1 \Vert u_\ell \Vert _{3/2} \\&\times \Big [ \langle \Phi _N, (-\Delta _{x_1}) (\Delta _{x_2} \dots \Delta _{x_5})^{3/4+\delta } \Phi _N \rangle + \langle \Phi _N, (-\Delta _{x_1}) (\Delta _{x_2} \Delta _{x_3})^{3/4+\delta } \Phi _N \rangle \Big ] \\ {}&+ CN^5 \lambda _\ell \Vert \chi _\ell \Vert _1 \Vert u_\ell \Vert _1^2 \Big [ 1 + \langle \Phi _N, (\Delta _{x_1} \Delta _{x_2})^{3/4+\delta } \Phi _N \rangle + \langle \Phi _N, (\Delta _{x_1} \dots \Delta _{x_4})^{3/4+\delta } \Phi _N \rangle \Big ]. \end{aligned} \end{aligned}$$

From Lemma 2.1, we have $\lambda _\ell \le C/(N\ell ^3)$. From the assumption (4.4) and from (4.10), we obtain

$$\begin{aligned} \begin{aligned} \mathcal {E}_\text {pot}^{(1)}&\le C \Big [ N^3 \ell ^4 + N \ell ^{2-\delta } + \frac{1}{N \ell ^{1/4 + \delta }} + \frac{1}{N^3 \ell ^{41/16+ 6\delta }} \Big ] \\ \mathcal {E}_\text {pot}^{(2)}&\le C \Big [ N^2 \ell ^4 + \ell ^{2-\delta } + \frac{1}{N^2 \ell ^{1/4 + 4\delta }} + \frac{1}{N \ell ^{1/2 +\delta }} + \frac{1}{N^3 \ell ^{23/8+4\delta }} \Big ]. \end{aligned} \end{aligned}$$

Thus, choosing $\delta > 0$ small enough, we can find $C, \varepsilon >0$ such that $\mathcal {E}_\text {pot} \le C N^{-\varepsilon }$, if $N^{-1+\nu } \le \ell \le N^{-3/4-\nu }$ for a $\nu > 0$, and $N \in \mathbb {N}$ is large enough.

Finally, we consider the denominator on the r.h.s. of (4.2). With the lower bound in (4.6) (and the assumption $\Vert \Phi _N \Vert _2 = 1$), we find

$$\begin{aligned} \int |\Phi _N ({\textbf {x}})|^2 \prod _{i<j}^N f_\ell ^2 (x_i - x_j) d{\textbf {x}} \ge 1 - \frac{N(N-1)}{2} \int u_\ell (x_1 - x_2) |\Phi _N ({\textbf {x}})|^2 d{\textbf {x}}.\end{aligned}$$

Observing that, by (3.3), (4.10) and by the assumption (4.4),

$$\begin{aligned} \begin{aligned} \frac{N(N-1)}{2} \int |\Phi _N&(\textbf{x})|^2 u_\ell (x_1 - x_2) d\textbf{x} \\ {}&\le C N^2 \Vert u_\ell \Vert _1 \big [ 1 + \langle \Phi _N, (\Delta _{x_1} \Delta _{x_2})^{3/4+\delta } \Phi _N \rangle \big ] \le C \Big [ N \ell ^2 + \frac{1}{N \ell ^{2\delta }} \Big ] \end{aligned} \end{aligned}$$

we conclude, choosing $\delta > 0$ sufficiently small and recalling that $\ell \le N^{-3/4-\nu }$, that

$$\begin{aligned} \frac{1}{ \int |\Phi _N (\textbf{x})|^2 \prod _{i<j}^N f_\ell ^2 (x_i - x_j) d\textbf{x}} \le 1 + \frac{N(N-1)}{2} \int u_\ell (x_1 - x_2) |\Phi _N (\textbf{x})|^2 d\textbf{x} + C N^{-1-\varepsilon } \end{aligned}$$

for $\varepsilon > 0$ small enough. Combining the last equation with (4.7), (4.11) we arrive at (recall the assumption $\langle \Phi _N, H_N^\text {eff} \Phi _N \rangle \le 4 \pi \mathfrak {a} N + C$)

$$\begin{aligned} \begin{aligned}&\frac{1}{\Vert \Psi _N \Vert ^2} \big [ E_\text {kin} (\Phi _N) + E_\text {pot} (\Phi _N) \big ] \\&\quad \le \Big [ 1 + \frac{N(N-1)}{2} \int u_\ell (x_1 - x_2) |\Phi _N (\textbf{x})|^2 d\textbf{x} + C N^{-3/2} \Big ] \\&\qquad \times \Big [ \langle \Phi _N, H_N^\text {eff} \Phi _N \rangle - \frac{N(N-1)}{2} \langle \Phi _N, \big [ H_{N-2}^\text {eff} \otimes u_\ell (x_{N-1} - x_N) \big ] \Phi _N \rangle + C N^{-\varepsilon } \Big ] \\&\quad \le \langle \Phi _N, H_N^\text {eff} \Phi _N \rangle + \frac{N(N-1)}{2} \langle \Phi _N, H_N^\text {eff} \Phi _N \rangle \int |\Phi _N (\textbf{x})|^2 u_\ell (x_{N-1} - x_N) d\textbf{x}\\&\qquad - \frac{N(N-1)}{2} \langle \Phi _N, \big [ H_{N-2}^\text {eff} \otimes u_\ell (x_{N-1} - x_N) \big ] \Phi _N \rangle + C N^{-\varepsilon } \\&\quad \le \langle \Phi _N, H_N^\text {eff} \Phi _N \rangle - \frac{N(N-1)}{2}\\ {}&\qquad \Big \langle \Phi _N, \Big \{ \big [ H_{N-2}^\text {eff} - 4 \pi \mathfrak {a} N \big ] \otimes u_\ell (x_{N-1} - x_N) \Big \} \Phi _N \Big \rangle + C N^{-\varepsilon }. \end{aligned} \end{aligned}$$

$\square$

5 Properties of the Effective Hamiltonian

Motivated by the results of the last sections, in particular by (2.12), by Prop.  3.1 and by Prop. 4.1, we would like to choose $\Phi _N \in L^2_s (\Lambda ^N)$ as a good trial state for the effective Hamiltonian $H_N^\text {eff}$ defined in (4.3) (i. e. $\Phi _N$ should lead to a small expectation of $H_N^\text {eff}$ and, at the same time, it should satisfy the bounds (4.4)). Since $u_\ell = 1 - f_\ell ^2$ is small, unless particles are very close, we can think of $H_N^\text {eff}$ as a perturbation of

$$\begin{aligned} H_{\ell , N} = \sum _{j=1}^N -\Delta _{x_j} + 2\lambda _\ell \sum _{i<j}^N \chi _\ell (x_i - x_j) .\end{aligned}$$

(5.1)

Keeping in mind that, by (2.4), $\lambda _\ell \simeq 3\mathfrak {a} / N \ell ^3$ and that $1/N \ll \ell \ll 1$, (5.1) looks like the Hamilton operator of a Bose gas in an intermediate scaling regime, interpolating between mean-field and Gross–Pitaevskii limits. The validity of Bogoliubov theory in such regimes has been recently established in [6]. The goal of this section is to apply the strategy of [6] to the Hamilton operator (4.3). This will lead to bounds for the operator $H_N^\text {eff}$ and, eventually, to an ansatz for $\Phi _N$. While part of our analysis in this section can be taken over from [6], we need additional work to control the effect of the difference $u_\ell = 1- f_\ell ^2$, appearing in the kinetic and the potential energy in the effective Hamiltonian (4.3).

To determine the spectrum of (4.3), it is useful to factor out the condensate and to focus instead on its orthogonal excitations. To this end, following [25], we define a unitary map $U_N: L^2_s (\Lambda ^N) \rightarrow \mathcal {F}^{\le N}_+ = \bigoplus _{n=0}^N L^2_\perp (\Lambda )^{\otimes _s n}$, requiring that

$$\begin{aligned} U_N \psi = \{ \alpha _0, \alpha _1, \dots , \alpha _N \} \in \mathcal {F}^{\le N}_+ \end{aligned}$$

(5.2)

if

$$\begin{aligned} \psi = \alpha _0 \varphi _0^{\otimes N} + \alpha _1 \otimes _s \varphi _0^{\otimes (N-1)} + \cdots + \alpha _N. \end{aligned}$$

Here $\varphi _0 (x) \equiv 1$ for all $x \in \Lambda$ denotes the condensate wave function, and $L^2_\perp (\Lambda )$ is the orthogonal complement of $\varphi _0$ in $L^2 (\Lambda )$. The action of the unitary operator $U_N$ is determined by the rules

$$\begin{aligned} U_N \, a^*_0 a_0 \, U_N^*= & {} {} N- \mathcal {N}_+ \nonumber \\ U_N \, a^*_p a_0 \, U_N^*= & {} {} a^*_p \sqrt{N-\mathcal {N}_+ } = \sqrt{N} b_p^* \nonumber \\ U_N \, a^*_0 a_p \, U_N^*= & {} {} \sqrt{N-\mathcal {N}_+ } \, a_p = \sqrt{N} b_p \nonumber \\ U_N \, a^*_p a_q \, U_N^*= & {} {} a^*_p a_q. \end{aligned}$$

(5.3)

where $\mathcal {N}_+$ denotes the number of particles operator on $\mathcal {F}^{\le N}_+$ (it measures therefore the number of excitations of the condensate) and where we introduced modified creation and annihilation operators $b^*_p, b_p$ satisfying the commutation relations

$$\begin{aligned}{}[ b_p, b_q^* ] = \left( 1 - \frac{\mathcal {N}_+}{N} \right) \delta _{p,q} - \frac{1}{N} a_q^* a_p , \qquad [ b_p, b_q ] = [b_p^*, b_q^*] = 0 \end{aligned}$$

(5.4)

and

$$\begin{aligned} {[} a_r^* a_s, b^*_p ] = \delta _{p,s} b_r^*, \qquad [ a_r^* a_s, b_p ] = - \delta _{r,} b_s. \end{aligned}$$

(5.5)

On the truncated Fock space $\mathcal {F}_+^{\le N}$, we can define the excitation Hamiltonian $\mathcal {L}^\text {eff}_N = U_N H_N^\text {eff} U_N^*$. To compute $\mathcal {L}^\text {eff}_N$, we first rewrite (4.3) in momentum space, using the formalism of second quantization, as

$$\begin{aligned} H_N^\text{ eff }= & {} {} \sum _{p \in \Lambda ^*} p^2 a_p^* a_p - \quad \sum _{p,q,r \in \Lambda ^*} p \cdot (p+r) {\widehat{u}}_\ell (r) a_{p+r}^* a_{q-r}^* a_p a_q \nonumber \\{}{} & {} {} \quad + \lambda _\ell \sum _{p,q,r \in \Lambda ^*} \widehat{\chi _\ell f_\ell ^2} (r) a_{p+r}^* a_q^* a_{q+r} a_p. \end{aligned}$$

(5.6)

Then, we apply (5.3). This will produce a constant term, as well as contributions that are quadratic, cubic and quartic in (modified) creation and annihilation operators. Following Bogoliubov’s method, we would like to eliminate cubic and quartic terms. This would reduce $\mathcal {L}_N^\text {eff}$ to a quadratic expression, whose spectrum could be computed through diagonalization with a (generalized) Bogoliubov transformation. As explained in [6], though, cubic and quartic terms in $\mathcal {L}^\text {eff}_N$ are not negligible (they contribute to the energy to order $\ell ^{-1}$). Before proceeding with the diagonalization, we need to extract relevant contributions to the energy from cubic and quartic terms. As in [6], we do so by conjugating $\mathcal {L}_N^\text {eff}$ with a (generalized) Bogoliubov transformation removing short-distance correlations characterising low-energy states. To reach this goal, we fix $\ell _0 \gg \ell$, small, but of order one, independent of N. Similarly as in (2.1), we define $f_{\ell _0}$ to be the ground state solution of the Neumann problem for the hard sphere potential in the ball $B_{\ell _0}$. Extending $f_{\ell _0}$ to the box $\Lambda$, we find

$$\begin{aligned} -\Delta f_{\ell _0} (x) = \lambda _{\ell _0} \chi _{\ell _0} (x) f_{\ell _0} (x) \end{aligned}$$

with $f_{\ell _0} (x) = 0$ for $|x| = \mathfrak {a}/N$ (the eigenvalue $\lambda _{\ell _0}$ is approximately given by (2.4), of course with $\ell$ replaced by $\ell _0$). For $\mathfrak {a}/N \le |x| \le \ell _0$, we can then define $g_{\ell _0} (x) = f_{\ell _0} (x) / f_\ell (x)$. We can also extend $g_{\ell _0}$ to $\Lambda$, setting $g_{\ell _0} (x) = \lim _{|y| \downarrow \text{\AA} / N} g_{\ell _0} (y)$ for all $|x| \le \text{\AA} /N$ and $g_{\ell _0} (x) = 1$ for all $x \in \Lambda \backslash B_{\ell _0}$. A simple computation shows that $g_{\ell _0}$ solves the equation

$$\begin{aligned} -\nabla \big [ f_\ell ^2 \nabla g_{\ell _0} \big ] + \lambda _\ell \chi _\ell f_\ell ^2 g_{\ell _0} = \lambda _{\ell _0} \chi _{\ell _0} f_\ell ^2 g_{\ell _0} \end{aligned}$$

(5.7)

with the Neumann boundary condition $\partial _r g_{\ell _0} (x) = 0$ for $|x| = \ell _0$ (this follows easily from the observation that, for $\ell \le |x| \le \ell _0$, $g_{\ell _0} (x) = f_{\ell _0} (x)$). Conversely, it is interesting to observe that, integrating (5.7) against $g_{\ell _0}$, we find

$$\begin{aligned} \int f_\ell ^2 |\nabla g_{\ell _0}|^2 dx + \lambda _\ell \int \chi _\ell f_\ell ^2 g_{\ell _0}^2 dx = \lambda _{\ell _0} \int \chi _{\ell _0} f_\ell ^2 g_{\ell _0}^2 dx. \end{aligned}$$

(5.8)

With (2.1), we find

$$\begin{aligned} \int |\nabla (f_\ell g_{\ell _0})|^2 dx = \lambda _{\ell _0} \int \chi _{\ell _0} |(f_\ell g_{\ell _0})|^2 dx \end{aligned}$$

(5.9)

which implies that (5.7) is solved by $g_{\ell _0} = f_{\ell _0}/f_\ell$.

With $g_{\ell _0}$, we define ${\check{\eta }} (x): = - N (1 - g_{\ell _0} (x))$. Some properties of $g_{\ell _0}, {\check{\eta }}$ and of their Fourier coefficients are collected in the next lemma, whose proof is deferred to Appendix A. We introduce here the notation

$$\begin{aligned} V_\ell (x) = 2 N \lambda _\ell \chi _\ell (x) f_\ell ^2 (x). \end{aligned}$$

(5.10)

Lemma 5.1

We have ${\check{\eta }} (x) = 0$ for $|x| \ge \ell _0$. For $|x| \le \ell _0$, we have the bounds

$$\begin{aligned} |{{\check{\eta }}}(x)| \le \frac{ C \mathfrak {a}}{|x| +\ell }, \qquad |\nabla {\check{\eta }}(x)| \le \frac{C \mathfrak {a}}{(|x| + \ell )^2 }. \end{aligned}$$

(5.11)

Furthermore

$$\begin{aligned} \left| \int V_\ell (x) g_{\ell _0} (x) dx - 8 \pi \mathfrak {a}\right|= & {} \left| 2N \lambda _\ell \int \chi _{\ell } (x) f_\ell ^2 (x) g_{\ell _0} (x) dx - 8 \pi \mathfrak {a}\right| \le C N^{-1} \nonumber \\ \left| \int V_\ell (x) g_{\ell _0} (x) e^{-ip \cdot x} dx \right|= & {} \left| 2N \lambda _\ell \int \chi _{\ell } (x) f_\ell ^2 (x) g_{\ell _0} (x) e^{-ip \cdot x} dx \right| \le \frac{C }{\ell ^2 p^2}\nonumber \\ \end{aligned}$$

(5.12)

and, analogously,

$$\begin{aligned} \begin{aligned} \Big | 2N \lambda _{\ell _0} \int \chi _{\ell _0} (x) f_{\ell }^2 (x) g_{\ell _0} (x) dx - 8 \pi \mathfrak {a}\Big |&\le C N^{-1}\\ \Big | 2N \lambda _{\ell _0} \int \chi _{\ell _0} (x) f_\ell ^2 (x) g_{\ell _0} (x) e^{-ip \cdot x} dx \Big |&\le \frac{C }{ p^2}. \end{aligned} \end{aligned}$$

(5.13)

Recall the definition $u_\ell = 1 - f_\ell ^2$. For $p \in \Lambda ^*_+$, let

$$\begin{aligned} D_p = - \sum _{r \in \Lambda ^*} p \cdot (p+r) {\hat{u}}_\ell (r) \eta _{p+r} \end{aligned}$$

(5.14)

and denote by $\eta _p$ the Fourier coefficients of ${\check{\eta }}$. Then (5.7) takes the form

$$\begin{aligned} p^2 \eta _p + D_p + N \lambda _\ell \big ( \widehat{\chi _\ell f_\ell ^2} * {\widehat{g}}_{\ell _0} \big ) (p) = N \lambda _{\ell _0} \big (\widehat{\chi _{\ell _0} f_{\ell }^2} *{\widehat{g}}_{\ell _0} \big )(p). \end{aligned}$$

(5.15)

or, equivalently, with the definition (5.10),

$$\begin{aligned} p^2 \eta _p + D_p + \frac{1}{2} {\widehat{V}}_\ell (p) + \frac{1}{2N} \big ( {\widehat{V}}_\ell * \eta \big ) (p) = N \lambda _{\ell _0} \big (\widehat{\chi _{\ell _0} f_{\ell }^2} *{\widehat{g}}_{\ell _0} \big )(p). \end{aligned}$$

(5.16)

We have

$$\begin{aligned} \eta _0 = -\frac{2}{5} \pi \mathfrak {a}\ell _0^2 + \mathcal {O}\Big ( \frac{\mathfrak {a}^2\ell _0}{N}\Big )+\mathcal {O}\big (\mathfrak {a}\ell ^2\big ) \end{aligned}$$

(5.17)

and, for $p \in \Lambda ^*_+$,

$$\begin{aligned} \begin{aligned} |\eta _p|&\le C \min \left\{ \frac{1}{|p|^2}\,; \frac{1}{\ell ^2 |p|^4} \right\} \\ |D_p|&\le C \min \left\{ \,\frac{1}{N \ell }\,; \frac{1}{\ell ^2 |p|^2} \right\} . \end{aligned} \end{aligned}$$

(5.18)

In particular, this implies

$$\begin{aligned} \sum _{q\in \Lambda ^*_+} |q|^{r} |\eta _q|^2 \le C \ell ^{1-r} \end{aligned}$$

(5.19)

for all $1< r < 5$.

Using the coefficients $\eta _p$, for $p\in \Lambda ^*_+$, we define now

$$\begin{aligned} B(\eta ) = \frac{1}{2} \sum _{p\in \Lambda ^*_+} \eta _p \big ( b_p^* b_{-p}^* - b_p b_{-p} \big ) \, \end{aligned}$$

(5.20)

and we introduce the renormalized excitation Hamiltonian

$$\begin{aligned} \mathcal {G}^\text {eff}_{N,\ell } = e^{-B(\eta )} U_N H_N^\text {eff} U_N^* e^{B(\eta )}. \end{aligned}$$

(5.21)

As explained in [6], conjugation with the generalized Bogoliubov transformation $e^{B(\eta )}$ models correlations up to scales of order one (determined by the radius $\ell _0$ of the ball used to define $g_{\ell _0}$). It extracts important contributions to the energy from terms in $\mathcal {L}_N^\text {eff}$ that are quartic in creation and annihilation operators. This will allow us to approximate $\mathcal {G}_N^{\text {eff}}$ by the sum of a constant and of a quadratic expression in creation and annihilation operators, whose ground state energy will be computed by simple diagonalization (through a second Bogoliubov transformation). Unfortunately, conjugation with $e^{B(\eta )}$ also produces several error terms, which need to be bounded. For $1< r <5$, we consider the positive operator

$$\begin{aligned} \mathcal {P}^{(r)} = \sum _{p \in \Lambda ^*_+} |p|^r a_p^* a_p \end{aligned}$$

(5.22)

acting on $\mathcal {F}^{\le N}_+$. The growth of $\mathcal {P}^{(r)}$ (and of products of $\mathcal {P}^{(r)}$ with moments of the number fo particles operator) under the action of $B(\eta )$ is controlled by the next lemma.

Lemma 5.2

Let $B(\eta )$ be defined as in (5.20). Then, for every $n \in \mathbb {N}$ and $r \in (1; 5)$ there is $C > 0$ such that, for all $t\in [0;1]$,

$$\begin{aligned} \begin{aligned} e^{- tB(\eta )} (\mathcal {N}_+ +1)^{n} e^{t B(\eta )}&\le C (\mathcal {N}_+ +1)^{n} \\ e^{-t B(\eta )} \mathcal {P}^{(r)}(\mathcal {N}_+ +1)^{n} e^{tB(\eta )}&\le C \big ( \mathcal {P}^{(r)} + \ell ^{1-r} \big ) (\mathcal {N}_+ +1)^n. \end{aligned}\end{aligned}$$

(5.23)

Proof

The proof of the first bound in (5.23) is standard and can be found for example in [10, Lemma 6.1]. As for the second inequality, let us consider the case $n=0$. For any $\xi \in \mathcal {F}_+^{\le N}$ and $t \in [0;1]$ we write

$$\begin{aligned} \langle \xi , e^{-tB(\eta )} {\mathcal {P}}^{(r)} e^{tB(\eta )} \xi \rangle = \langle \xi , {\mathcal {P}}^{(r)} \xi \rangle +\int _0^t ds\, \langle \xi , e^{-sB (\eta )}\big [{\mathcal {P}}^{(r)},B(\eta )\big ]e^{sB(\eta )} \xi \rangle \nonumber \\ \end{aligned}$$

(5.24)

where

$$\begin{aligned}{}[{\mathcal {P}}^{(r)},B(\eta )]=\;\sum _{q\in \Lambda ^*_+} |q|^{r} \eta _q b^*_{q} b^*_{-q}+\mathrm {h.c.}. \end{aligned}$$

By Cauchy-Schwarz’s inequality and (5.19) we get

$$\begin{aligned} \begin{aligned} \big |\langle \xi , [{\mathcal {P}}^{(r)},B(\eta )] \xi \rangle \big |&\le C \sum _{q \in \Lambda ^*_+} |q|^r |\eta _q| \Vert a_q \xi \Vert \Vert a_q^* \xi \Vert \\ {}&\le C \sum _{q \in \Lambda ^*_+} |q|^r |\eta _q| \Vert a_q \xi \Vert \big [ \Vert a_q \xi \Vert + \Vert \xi \Vert \big ] \le C \langle \xi , {\mathcal {P}}^{(r)} \xi \rangle + \ell ^{1-r} \Vert \xi \Vert ^2 .\end{aligned} \end{aligned}$$

Inserting this into (5.24) and using Gronwall’s Lemma, we obtain the desired bound. The proof for $n\ge 1$ is similar, we omit further details. $\square$

With Lemma 5.2 we are ready to establish the form of $\mathcal {G}_{N,\ell }^\text {eff}$, up to errors which are negligible on our trial state. We use the notation (recall the definition (5.10) of $V_\ell$)

$$\begin{aligned} \mathcal {K}= \sum _{p\in \Lambda ^*_+} p^2 a^*_p a_p, \qquad \text{ and }\qquad \mathcal {V}_\ell = \frac{1}{2N} \sum _{\begin{array}{c} p,q \in \Lambda ^*_+,\, r \in \Lambda ^*:\\ r\ne -p, -q \end{array}} {{\widehat{V}}}_\ell (r) a^*_{p+r} a^*_q a_p a_{q+r}.\qquad \nonumber \\ \end{aligned}$$

(5.25)

Proposition 5.3

Let $\mathcal {G}^\text {eff}_{N,\ell }$ be defined as in (5.21), with $B(\eta )$ as in (5.20), with $\ell \ge N^{-1+\nu }$ for some $\nu > 0$ and $\ell _0 > 0$ small enough (but fixed, independent of N). Let $\mathcal {P}^{(r)}$ be defined as in (5.22). Then, for any $0< \kappa < \nu /2$ we have

$$\begin{aligned} \mathcal {G}^\text{ eff}_{N,\ell } \ge 4 \pi \mathfrak {a}N - C(\mathcal {N}_+ +1) - \frac{C}{N^\kappa }\, \mathcal {P}^{(2+\kappa )} (\mathcal {N}_++1). \end{aligned}$$

(5.26)

On the other hand, using the notation $\gamma _p=\cosh (\eta _p)$ and $\sigma _p=\sinh (\eta _p)$, let

$$\begin{aligned} C_{N,\ell }= & {} \, \frac{(N-1)}{2}{\widehat{V}}_\ell (0) +\sum _{p\in \Lambda ^*_+}\Big [ p^2\sigma _p^2+ \widehat{V}_\ell (p)(\sigma _p^2+\sigma _p\gamma _p)\nonumber \\ {}{} & {} +\frac{1}{2N}\sum _{q \ne 0}{{\widehat{V}}}_\ell (p-q)\eta _p\eta _q + D_p\eta _p \Big ] \end{aligned}$$

(5.27)

with $D_p$ defined in (5.14). Denote also

$$\begin{aligned} \mathcal {Q}_{N,\ell } = \sum _{p \in \Lambda ^*_+}\left[ F_p a^*_p a_p+\dfrac{1}{2}G_p\left( b^*_p b^*_{-p}+b_pb_{-p}\right) \right] \end{aligned}$$

(5.28)

with

$$\begin{aligned} \begin{aligned} F_p=\,&p^2(\sigma _p^2+\gamma _p^2)+ (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p) (\gamma _p +\sigma _p)^2\\ G_p=\,&2p^2 \gamma _p \sigma _p +(\widehat{V_\ell } *\widehat{g}_{\ell _0})(p) (\gamma _p+\sigma _p)^2+ 2D_p. \end{aligned} \end{aligned}$$

(5.29)

Then

$$\begin{aligned} \mathcal {G}^\text {eff}_{N, \ell } = C_{N,\ell } +\mathcal {Q}_{N,\ell }+\mathcal {E}_{N,\ell } \end{aligned}$$

(5.30)

where

$$\begin{aligned} \pm \mathcal {E}_{N,\ell } \le \frac{C}{\sqrt{N \ell }} \,(\mathcal {K}+ \mathcal {V}_{\ell } +\mathcal {P}^{(5/2)})(\mathcal {N}_++1)\end{aligned}$$

and $\mathcal {K}$ and $\mathcal {V}_\ell$ are defined in (5.25).

Proof

According to (5.6) we can decompose

$$\begin{aligned} \mathcal {G}^\text {eff}_{N,\ell } = \mathcal {G}_{N,\ell } + \mathcal {J}_{N,\ell } \end{aligned}$$

with

$$\begin{aligned} \mathcal {G}_{N,\ell } = e^{-B(\eta )} U_N \left[ \sum _{p \in \Lambda ^*} p^2 a_p^* a_p + \frac{1}{2N} \sum _{p,q,r \in \Lambda ^*} {\widehat{V}}_\ell (r) a^*_{p+r} a_q^* a_{q+r} a_p \right] U_N^* e^{B(\eta )}\nonumber \\ \end{aligned}$$

(5.31)

and

$$\begin{aligned} \mathcal {J}_{N,\ell } = - e^{-B(\eta )} U_N \left[ \sum _{p,q,r \in \Lambda ^*} p \cdot (p+r) {\widehat{u}}_\ell (r) a_{p+r}^* a_{q-r}^* a_p a_q \right] U_N^* e^{B(\eta )}. \end{aligned}$$

(5.32)

We can compute $\mathcal {G}_{N,\ell }^\text {eff}$ with tools developed in [6]. From Propositions 7.4–7.7 of [6], we obtain, on the one hand, the lower bound

$$\begin{aligned} \begin{aligned} \mathcal {G}_{N,\ell } \ge \,&\frac{(N-1)}{2} {{\widehat{V}}}_\ell (0) + \sum _{p \in \Lambda ^*_+} \big [ p^2 \eta _p + {{\widehat{V}}}_\ell (p) + \frac{1}{2N} ({\widehat{V}}_\ell * \eta ) (p) \big ] \eta _p \\&+ \sum _{p \in \Lambda ^*_+} \big [ p^2 \eta _p + \frac{1}{2} \widehat{V}_\ell (p) +\frac{1}{2N} ({\widehat{V}}_\ell * \eta ) (p) \big ] \big ( b_p b_{-p} + b^*_p b^*_{-p} \big ) - C (\mathcal {N}_+ + 1) \end{aligned}\nonumber \\ \end{aligned}$$

(5.33)

and, on the other hand, the approximation

$$\begin{aligned} \mathcal {G}_{N,\ell }= & {} \frac{(N-1)}{2} {{\widehat{V}}}_\ell (0) \nonumber \\{} & {} + \sum _{p \in \Lambda ^*_+} \big [ p^2 \sigma _p^2 + {{\widehat{V}}}_\ell (p) \sigma _p^2 + {{\widehat{V}}}_\ell (p) \gamma _p \sigma _p +\frac{1}{2N} \sum _{q \in \Lambda ^*_+}{{\widehat{V}}}_\ell (p-q) \eta _p \eta _q \big ] \nonumber \\{} & {} + \sum _{p \in \Lambda ^*_+} \big [2 p^2 \sigma _p^2 + {{\widehat{V}}}_\ell (p) (\gamma _p+\sigma _p)^2 \big ] b^*_p b_p + \mathcal {K}+ \mathcal {V}_\ell \nonumber \\{} & {} + \sum _{p \in \Lambda ^*_+} \big [p^2 \sigma _p \gamma _p +\frac{1}{2}\widehat{V}_\ell (p) (\gamma _p + \sigma _p)^2 +\frac{1}{2N}\sum _{q \in \Lambda ^*_+}{{\widehat{V}}}_\ell (p-q) \eta _q \big ]\nonumber \\{} & {} \times \big ( b_p b_{-p} + b^*_p b^*_{-p} \big ) + \mathcal {E}_\mathcal {G}\end{aligned}$$

(5.34)

where

$$\begin{aligned} \pm \mathcal {E}_\mathcal {G}\le \frac{C}{\sqrt{N \ell }} \,(\mathcal {K}+ \mathcal {V}_{\ell } +1)(\mathcal {N}_++1). \end{aligned}$$

Some care is required here when we apply results from [6]. First of all, the interaction potential considered in [6] has the form $N^{3\beta } W (N^\beta x)$, for some $0< \beta < 1$. The potential $V_\ell (x) = 2 N \lambda _\ell \chi _\ell (x) f_\ell ^2 (x)$ appearing in (5.31) has this form only if we approximate $f_\ell \simeq 1$ and $\lambda _\ell \simeq 3\mathfrak {a} / (N \ell ^3)$. A closer inspection to [6] shows, however, that (5.34) does not rely on the precise form of the interaction potential but instead only on the bounds

$$\begin{aligned} \sup _{q \in \Lambda ^*_+}\sum _{\begin{array}{c} r \in \Lambda ^*\\ r \ne -q \end{array}} \frac{|{{\widehat{V}}}_\ell (r)|}{|q+r|^2} \le C\ell ^{-1}, \qquad \sum _{\begin{array}{c} r \in \Lambda ^*,\, q \in \Lambda ^*_+\\ r \ne -q \end{array}} \frac{|{{\widehat{V}}}_\ell (r)|}{|q+r|^2 |q|^2} \le C\ell ^{-2} \end{aligned}$$

which are the analog of [6, Eq. (7.5) and (7.75)] and follow from $\Vert {{\widehat{V}}}_\ell \Vert _\infty \le C$ and $\Vert \widehat{V}_\ell \Vert _2 \le C \ell ^{-3/2}$. Moreover, the estimate (5.34) was proven in [6] under the assumption that $W = \lambda V$, for a sufficiently small $\lambda > 0$. This assumption was used in [6] to make sure that the $\ell ^2$-norm of $\eta$ is sufficiently small. As later shown in [8], smallness of $\Vert \eta \Vert$ can also be achieved by choosing the parameter $\ell _0$ small enough, with no restriction on the size of the interaction potential.¹ Finally, in [6], the choice of $\eta$ was slightly different from the definition given after (5.7) (the presence of the second term on the r.h.s. of (5.6) affects the choice of $\eta$, as we will see shortly). However, the derivation of (5.34) does not depend on the exact form of $\eta$, but rather on bounds, proven in Lemma 5.1, that holds for both choices of $\eta$. This explains why (5.34) holds true, for sufficiently small values of $\ell _0$.

Let us now consider (5.32). With (5.3) we find

$$\begin{aligned} U_N \Big [ - \sum _{p,q,r \in \Lambda ^*} p \cdot (p+r) {\widehat{u}}_\ell (r) a_{p+r}^* a_{q-r}^* a_p a_q \Big ] U_N^* = Z_1 + Z_2 + Z_3 \end{aligned}$$

with

$$\begin{aligned} Z_1= & {} {} - (N-\mathcal {N}_+)\, {{\widehat{u}}}_\ell (0) \sum _{p \in \Lambda ^*_+} p^2 a^*_p a_p \nonumber \\ Z_2= & {} {} - \sqrt{N} \sum _{\begin{array}{c} p, r \in \Lambda ^*_+: \nonumber \\ p+r \ne 0 \end{array}} p \cdot (p+r)\, {{\widehat{u}}}_\ell (r) \big ( b^*_{p+r} a^*_{-r} a_{p} + \text{ h.c. } \big ) \nonumber \\ Z_3= & {} {} - \sum _{\begin{array}{c} r \in \Lambda ^*,\, p,q \in \Lambda ^*_+: \\ r \ne -p,q \end{array}} p \cdot (p+r)\, {{\widehat{u}}}_\ell (r) a^*_{p+r} a^*_{q-r} a_{p}a_q. \end{aligned}$$

(5.35)

Using Lemma 2.1 to bound $\Vert {\widehat{u}}_\ell \Vert _\infty \le \Vert u_\ell \Vert _1 \le C \Vert \omega _\ell \Vert _1 \le C \ell ^2 N^{-1}$ and Lemma 5.2 (in particular, the second inequality in (5.23), with $r = 2$), we find

$$\begin{aligned} |\langle \xi , e^{-B(\eta )} Z_1 e^{B(\eta )}\xi \rangle | \le C N \Vert u_\ell \Vert _1 \Vert (\mathcal {K}+ 1)^{1/2} e^{B(\eta )}\xi \Vert ^2 \le C \ell \Vert (\mathcal {K}+1)^{1/2} \xi \Vert ^2 \end{aligned}$$

because $\mathcal {N}_+ \le C \mathcal {K}$. As for the term $Z_{2}$, we have, from $\Vert u_\ell \Vert _2 \le C \Vert \omega _\ell \Vert _2 \le C \ell ^{1/2} / N$ and by Lemma 5.2,

$$\begin{aligned} \begin{aligned}&|\langle e^{B(\eta )}\xi , Z_{2} e^{B(\eta )}\xi \rangle |\\&\,\le \sqrt{N} \Big ( \sum _{\begin{array}{c} p, r \in \Lambda ^*_+: \\ p+r \ne 0 \end{array}} | p+r|^2 \Vert a_{p+r}a_{-r} e^{B(\eta )}\xi \Vert ^2 \Big )^{1/2} \Big ( \sum _{\begin{array}{c} p, r \in \Lambda ^*_+: \\ p+r \ne 0 \end{array}} |{{\widehat{u}}}_\ell (r)|^2 |p|^2 \,\Vert a_p e^{B(\eta )}\xi \Vert ^2 \Big )^{1/2} \\&\,\le C \sqrt{N} \, \Vert u_\ell \Vert _2 \Vert \mathcal {K}^{1/2} \mathcal {N}_+^{1/2} e^{B(\eta )}\xi \Vert \Vert \mathcal {K}^{1/2} e^{B(\eta )} \xi \Vert \\ {}&\;\le C (\ell N)^{-1/2} \Vert (\mathcal {K}+1)^{1/2} (\mathcal {N}_++1)^{1/2}\xi \Vert ^2. \end{aligned}\end{aligned}$$

Hence, we obtain

$$\begin{aligned} \mathcal {J}_{N,\ell } = Z_3 + \int _0^1 e^{-tB(\eta )} [Z_3, B(\eta )] e^{t B(\eta )} dt + \mathcal {E}_1 \end{aligned}$$

with

$$\begin{aligned} \pm \mathcal {E}_{1} \le C (N\ell )^{-1/2} (\mathcal {K}+1) (\mathcal {N}_++1).\end{aligned}$$

Using (5.5) we find

$$\begin{aligned} {[}Z_3, B(\eta )] = \sum _{i=1}^3 W_i \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} W_1 =\,&\sum _{p \in \Lambda ^*_+} D_p \big (b^*_p b^*_{-p} + b_p b_{-p} \big )\\ W_2 =\,&- \sum _{\begin{array}{c} r \in \Lambda ^*,\, p,q \in \Lambda ^*_+: \\ p+r,\, q-r \ne 0 \end{array}} p \cdot (p+r)\, {{\widehat{u}}}_\ell (r) \eta _{q-r} \big ( b^*_{p} b^*_{r-q} a^*_{q}a_{p+r} +\text{ h.c. }\big ) \\ W_3 =\,&- \sum _{\begin{array}{c} r \in \Lambda ^*,\, p,q \in \Lambda ^*_+: \\ p+r,\, q-r \ne 0 \end{array}} p \cdot (p+r)\, {{\widehat{u}}}_\ell (r) \eta _{p+r} \big ( b^*_{-p-r} b^*_{q} a^*_{p}a_{q-r} +\text{ h.c. }\big ). \end{aligned}\end{aligned}$$

For any $t \in [0;1]$, we have (using again $\Vert u_\ell \Vert _2 \le C \ell ^{1/2}/N$ and $\Vert \eta \Vert _2 \le C$, from (5.18))

$$\begin{aligned} \begin{aligned} |\langle&e^{tB(\eta )}\xi , W_2 e^{tB(\eta )}\xi \rangle | \\&\le \Big [ \sum _{\begin{array}{c} r \in \Lambda ^*,\, p,q \in \Lambda ^*_+: \\ p+r,\, q-r \ne 0 \end{array}} | p|^2 \Vert a_q b_{r-q} b_{p} (\mathcal {N}_+ + 1)^{-1/2} e^{tB(\eta )}\xi \Vert \Big ]^{1/2} \\&\quad \times \Big [ \sum _{\begin{array}{c} r \in \Lambda ^*,\, p,q \in \Lambda ^*_+: \\ p+r,\, q-r \ne 0 \end{array}} | {{\widehat{u}}}_\ell (r)|^2 |\eta _{q-r}|^2 |p+r|^2 \Vert a_{p+r} (\mathcal {N}_++1)^{1/2}e^{tB(\eta )}\xi \Vert \Big ]^{1/2} \\&\le \Vert u_\ell \Vert _2 \Vert \eta \Vert _2 \Vert \mathcal {K}^{1/2} (\mathcal {N}_++1)^{1/2} e^{tB(\eta )}\xi \Vert ^2 \le C \ell ^{1/2} (N\ell )^{-1} \Vert \mathcal {K}^{1/2} (\mathcal {N}_++1)^{1/2}\xi \Vert ^2. \end{aligned}\end{aligned}$$

The contribution of $W_3$ can be bounded similarly. Hence,

$$\begin{aligned} \mathcal {J}_{N,\ell } = Z_3 + W_1 + \int _0^1 dt \int _0^t ds \, e^{-s B(\eta )} \big [ W_1, B(\eta ) \big ] e^{s B(\eta )} + \mathcal {E}_2 \end{aligned}$$

with

$$\begin{aligned} \pm \mathcal {E}_2 \le \frac{C}{\sqrt{N\ell }} \, (\mathcal {K}+ 1) (\mathcal {N}_++1).\end{aligned}$$

With (5.4), we compute

$$\begin{aligned} \big [ W_1, B(\eta ) \big ] = \sum _{i=1}^4 X_i + \text {h.c.} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} X_1 =\,&\sum _{p\in \Lambda ^*_+} D_p \eta _p \\ X_2 =\,&\sum _{p\in \Lambda ^*_+} D_p \eta _p \, \bigg [ \left( 1- \frac{\mathcal {N}_+}{N} \right) \left( 1- \frac{\mathcal {N}_++1}{N} \right) -1 \bigg ] \\ X_3 =\,&2\sum _{p\in \Lambda ^*_+} D_p \eta _p\, a^*_p \left( 1 - \frac{\mathcal {N}_++2}{N} \right) \left( 1 - \frac{\mathcal {N}_++1}{N} \right) a_p\\ X_4 =\,&- \frac{1}{N} \sum _{p,q \in \Lambda ^*_+} D_p \eta _q \, a^*_p a^*_{-p} \bigg [ 2 \left( 1- \frac{\mathcal {N}_+}{N} \right) - \frac{3}{N} \bigg ] a_{q} a_{-q}. \end{aligned}\end{aligned}$$

With (5.18), we find, for any $t \in [0;1]$,

$$\begin{aligned} \begin{aligned} |\langle e^{tB(\eta )}\xi , X_2 e^{tB(\eta )}\xi \rangle |&\le \frac{C}{N} \Big [ \sum _{\begin{array}{c} p \in \Lambda ^*_+ \\ |p|\le \ell ^{-1} \end{array}} \frac{1}{|p|^2} + \sum _{\begin{array}{c} p \in \Lambda ^*_+ \\ |p|\ge \ell ^{-1} \end{array}} \frac{1}{\ell ^2 |p|^4} \Big ] \Vert (\mathcal {N}_+ +1)^{1/2}e^{tB(\eta )}\xi \Vert ^2 \\&\le C (N \ell )^{-1} \Vert (\mathcal {N}_+ + 1)^{1/2}\xi \Vert ^2. \end{aligned}\end{aligned}$$

Again from (5.18), we have $|D_p \eta _p| \le C (N\ell )^{-1}$ for all $p \in \Lambda ^*_+$. Thus

$$\begin{aligned} \begin{aligned} |\langle e^{tB(\eta )}\xi , X_3 e^{tB(\eta )}\xi \rangle |&\le C (N\ell )^{-1} \Vert \mathcal {N}_+^{1/2} e^{t B (\eta )} \xi \Vert ^2 \le C (N\ell )^{-1} \Vert (\mathcal {N}_+ + 1)^{1/2} \xi \Vert ^2.\end{aligned} \end{aligned}$$

As for the expectation of $X_4$, using (5.18) we obtain

$$\begin{aligned}\begin{aligned}&|\langle e^{tB(\eta )}\xi , X_4 e^{tB(\eta )}\xi \rangle | \\&\qquad \le \frac{C}{N} \Big ( \sum _{p,q \in \Lambda ^*_+} \frac{|D_p|^2}{|p|^2} \Vert a_{-q} a_q\, e^{tB(\eta )} \xi \Vert ^2 \Big )^{1/2} \\&\qquad \quad \times \Big ( \sum _{p,q \in \Lambda ^*_+} |\eta _q |^2 |p|^2\Vert a_p a_{-p} \,e^{tB(\eta )}\xi \Vert ^2 \Big )^{1/2} \\&\qquad \le \frac{C}{N \ell ^{1/2}} \Vert \mathcal {K}^{1/2}\mathcal {N}_+^{1/2}e^{tB(\eta )}\xi \Vert \Vert \mathcal {N}_+ e^{tB(\eta )}\xi \Vert \le \frac{C}{N \ell }\, \Vert (\mathcal {K}+ 1)^{1/2} (\mathcal {N}_+ + 1) \xi \Vert ^2. \end{aligned}\end{aligned}$$

We conclude that

$$\begin{aligned} \mathcal {J}_{N,\ell } = Z_3 + W_1 + X_1 + \mathcal {E}_3 \end{aligned}$$

with

$$\begin{aligned} \pm \mathcal {E}_3 \le \frac{C}{\sqrt{N\ell }} \, (\mathcal {K}+ 1) (\mathcal {N}_++1). \end{aligned}$$

Let us now go back to control the term $Z_3$, as defined in (5.35). We can estimate, for any $\kappa > 0$,

$$\begin{aligned} \begin{aligned} | \langle \xi , Z_3 \xi \rangle |&\le C \Big ( \sum _{\begin{array}{c} r \in \Lambda ^*,\, p,q \in \Lambda ^*_+: \\ p+r,\, q-r \ne 0 \end{array}} |p|^{2+\kappa } \frac{|{\widehat{\omega }}_\ell (r)|}{|p+r|^\kappa } \, \Vert a_{p}a_q \xi \Vert ^2 \Big )^{1/2}\\ {}&\quad \times \Big ( \sum _{\begin{array}{c} r \in \Lambda ^*,\, p',q' \in \Lambda ^*_+: \\ p'-r,\, q'+r \ne 0 \end{array}} |p'|^{2+\kappa } \frac{|{\widehat{\omega }}_\ell (r)|}{|p'-r|^\kappa } \, \Vert a_{p'}a_{q'}\xi \Vert ^2 \Big )^{1/2} \\ {}&\le \frac{C}{N^{\kappa }}\, \langle \xi , \mathcal {P}^{(2+\kappa )} \mathcal {N}_+ \xi \rangle \end{aligned}\end{aligned}$$

where we used the change of variables $p'=p+r, q' = q-r$ and the bound

$$\begin{aligned} \sup _{p \in \Lambda ^*_+} \sum _{r \in \Lambda ^*: \,r\ne -p} \frac{|{\widehat{\omega }}_\ell (r)|}{|p-r|^\kappa } \le C N^{-\kappa } \end{aligned}$$

(5.36)

valid for any $\kappa > 0$. To prove (5.36), we use the bound (2.10) for $|\widehat{\omega }_\ell (r)|$. More precisely, we consider separately the sets where i) $|p-r| < N$ and $|r| < N$ (here we use $|\widehat{\omega }_\ell (r)| \le C/(N |r|^2)$ and we estimate $|r|^{-2} |p-r|^{-\kappa } \lesssim |r|^{-2-\kappa } + |p-r|^{-2-\kappa }$), ii) $|p-r| \ge N$ and $|r| \ge N$ (here we apply $|\widehat{\omega }_\ell (r)| \le C / |r|^3$ and we use $|r|^{-3} |p-r|^{-\kappa } \lesssim |r|^{-3-\kappa } + |p-r|^{-3-\kappa }$), iii) $|p-r| < N$ and $|r| \ge N$ (here we estimate $|\widehat{\omega }_\ell (r)| \le CN^{-3}$), iv) $|p-r| \ge N$ and $|r| < N$ (here we use $|\widehat{\omega }_\ell (r)| \le C / (N |r|^2)$ and we estimate $|p-r|^{-\kappa } \le C N^{-\kappa }$).

Thus, for any $\kappa >0$, we arrive at

$$\begin{aligned} \mathcal {J}_{N,\ell } = \sum _{p\in \Lambda ^*_+} D_p \eta _p + \sum _{p \in \Lambda ^*_+} D_p \big (b^*_p b^*_{-p} + b_p b_{-p} \big ) + \mathcal {E}_\mathcal {J}\end{aligned}$$

where

$$\begin{aligned} \pm \mathcal {E}_\mathcal {J}\le \frac{C}{\sqrt{N\ell }} \, (\mathcal {K}+ 1) (\mathcal {N}_++1) + \frac{C}{N^\kappa } \mathcal {P}^{(2 + \kappa )} (\mathcal {N}_+ +1). \end{aligned}$$

(5.37)

Combining the last estimate with (5.33), we obtain

$$\begin{aligned} \begin{aligned} \mathcal {G}_{N,\ell }^\text{ eff } \ge \;&\frac{(N-1)}{2} {{\widehat{V}}}_\ell (0) + \sum _{p \in \Lambda ^*_+} \big [ p^2 \eta _p + D_p + {{\widehat{V}}}_\ell (p) + \frac{1}{2N} ({\widehat{V}}_\ell * \eta ) (p) \big ] \eta _p \\ {}&+ \sum _{p \in \Lambda ^*_+} \big [ p^2 \eta _p + D_p + \frac{1}{2} {{\widehat{V}}}_\ell (p) +\frac{1}{2N} ({\widehat{V}}_\ell * \eta ) (p) \big ] \big ( b_p b_{-p} + b^*_p b^*_{-p} \big ) \\ {}&- C (\mathcal {N}_+ + 1) - \frac{C}{N^\kappa } \mathcal {P}^{(2+\kappa )} (\mathcal {N}_+ + 1) \end{aligned} \end{aligned}$$

now with the restriction $0< \kappa < \nu /2$ (from $\ell \ge N^{-1 + \nu }$, it then follows that $N\ell \ge N^{\nu } \ge N^{2\kappa }$; thus, the first term on the r.h.s. of (5.37) can be controlled by the second). With the scattering equation (5.16) and using the bound on the second line of (5.13), we obtain

$$\begin{aligned} \mathcal {G}_{N,\ell }^\text {eff} \ge \frac{N}{2} \int V_\ell (x) g_{\ell _0} (x) dx - C (\mathcal {N}_+ + 1) - \frac{C}{N^\kappa } \mathcal {P}^{(2+\kappa )} (\mathcal {N}_+ + 1) \end{aligned}$$

for any $0< \kappa < \nu /2$. With (5.12), we find (5.26).

On the other hand, combining (5.37) with (5.34), we arrive at

$$\begin{aligned} \begin{aligned} \mathcal {G}^\text {eff}_{N,\ell } =&\frac{(N-1)}{2} {{\widehat{V}}}_\ell (0) \\ {}&+ \sum _{p \in \Lambda ^*_+} \big [ p^2 \sigma _p^2 + {{\widehat{V}}}_\ell (p) \sigma _p^2 + {{\widehat{V}}}_\ell (p) \gamma _p \sigma _p +\frac{1}{2N} \sum _{q \in \Lambda ^*_+}{{\widehat{V}}}_\ell (p-q) \eta _p \eta _q + D_p \eta _p \big ] \\&\qquad + \sum _{p \in \Lambda ^*_+} \big [ p^2 \sigma _p \gamma _p +\frac{1}{2} {{\widehat{V}}}_\ell (p) (\gamma _p + \sigma _p)^2 +\frac{1}{2N} \sum _{q \in \Lambda ^*_+}{{\widehat{V}}}_\ell (p-q) \eta _q + D_p \big ]\\ {}&\times \big ( b_p b_{-p} + b^*_p b^*_{-p} \big ) \\&\qquad + \sum _{p \in \Lambda ^*_+} \big [2 p^2 \sigma _p^2 + \widehat{V}_\ell (p) (\gamma _p+\sigma _p)^2 \big ] b^*_p b_p + \mathcal {K}+ \mathcal {V}_\ell + \widetilde{\mathcal {E}}_{N,\ell } \end{aligned}\end{aligned}$$

where

$$\begin{aligned} \pm \, {{\widetilde{\mathcal {E}}}}_{N,\ell } \le \frac{C}{\sqrt{N \ell }} \,(\mathcal {K}+ \mathcal {V}_{\ell } +1) (\mathcal {N}_+ + 1) + \frac{C}{N^\kappa } \mathcal {P}^{(2+\kappa )} (\mathcal {N}_++1) \end{aligned}$$

for any $0< \kappa < 1$. Observing that

$$\begin{aligned} \begin{aligned} \Big |\frac{1}{N} \sum _{p \in \Lambda ^*_+,\, q \in \Lambda ^* } {{\widehat{V}}}_\ell (p-q) \eta _q (\sigma _p + \gamma _p)^2 \langle \xi , b^*_p b_{p}\xi \rangle \Big |&\le \frac{C}{N\ell } \Vert (\mathcal {N}_++1)^{1/2}\xi \Vert ^2 \\ \Big |\frac{1}{N} \sum _{p \in \Lambda ^*_+,\, q \in \Lambda ^* } {{\widehat{V}}}_\ell (p-q) \eta _q \big ((\sigma _p + \gamma _p)^2 -1 \big ) \langle \xi , b_p b_{-p}\xi \rangle \Big |&\le \frac{C}{N\ell } \Vert (\mathcal {N}_++1)^{1/2}\xi \Vert ^2 \\ \Big | \sum _{p\in \Lambda ^*_+} p^2 \langle \xi , (b^*_p b_p - a^*_p a_p)\xi \rangle \Big |&\le \frac{C}{N} \Vert \mathcal {K}^{1/2} \mathcal {N}_+^{1/2}\xi \Vert ^2 \end{aligned}\end{aligned}$$

and that

$$\begin{aligned} \begin{aligned} \langle \xi , \mathcal {V}_\ell \, \xi \rangle&= \frac{1}{2N} \sum _{\begin{array}{c} p,q \in \Lambda ^*_+, r \in \Lambda ^* \\ r \not = -p, -q \end{array}} {\widehat{V}}_\ell (r) \langle \xi , a_{p+r}^* a_q^* a_{q+r} a_p \xi \rangle \\ {}&\le \frac{1}{2N} \sum _{\begin{array}{c} p,q \in \Lambda ^*_+, r \in \Lambda ^* \\ r \not = -p, -q \end{array}} \frac{|{\widehat{V}}_\ell (r)}{|q+r|^2} |p+r|^2 \Vert a_{p+r} a_q \xi \Vert ^2 \le \frac{C}{N\ell } \Vert \mathcal {K}^{1/2} \mathcal {N}_+^{1/2} \xi \Vert ^2 \end{aligned} \end{aligned}$$

we arrive at (5.30), choosing $\kappa = 1/2$. $\square$

6 Diagonalization of the Effective Hamiltonian

According to Prop. 5.3, we need to find a good ansatz for the ground state of the quadratic Hamiltonian $\mathcal {Q}_{N,\ell }$, defined in (5.28). To this end, we are going to conjugate $\mathcal {G}_{N,\ell }^\text {eff}$ with a second generalized Bogoliubov transformation, diagonalizing $\mathcal {Q}_{N,\ell }$. In order to define the appropriate Bogoliubov transformation, we first need to establish some properties of the coefficients $F_p, G_p$, defined in (5.29).

Lemma 6.1

Suppose $\ell \ge N^{-1+\nu }$, for some $\nu > 0$. Then there exists a constant $C > 0$ such that

$$\begin{aligned} p^2 /2 \le F_p \le C (1+p^2), \, \qquad |G_p| \le \frac{C }{p^2}, \qquad |G_p| < F_p \end{aligned}$$

for all $N \in \mathbb {N}$ large enough.

Proof

Recall the notations $\gamma _p=\cosh (\eta _p)$ and $\sigma _p=\sinh (\eta _p)$. With $(\sigma ^2_p + \gamma ^2_p) \le C$ (from the boundedness of $\eta _p$) and (5.13) in Lemma 5.1, we immediately obtain $F_p \le C (1+ p^2)$. To prove the lower bound for $F_p$, let us first consider $|p| > \ell ^{-1/2}$. With $(\sigma ^2_p + \gamma ^2_p)=\cosh (2\eta _p) \ge 1$, we find $F_p \ge p^2 - C \ge p^2/2$, if N is large enough (so that $\ell$ is small enough). For $|p| \le \ell ^{-1/2}$, we use $(\widehat{\chi _\ell f_\ell ^2} * {\widehat{g}}_{\ell _0} ) (0) > 0$ to estimate

$$\begin{aligned} \left( \widehat{\chi _\ell f_\ell ^2} * {\widehat{g}}_{\ell _0} \right) \left( p\right) > \left( \widehat{\chi _\ell f_\ell ^2} * {\widehat{g}}_{\ell _0} \right) \left( p\right) - \left( \widehat{\chi _\ell f_\ell ^2} * {\widehat{g}}_{\ell _0} \right) \left( 0\right) . \end{aligned}$$

With

$$\begin{aligned} \left| \left( \widehat{\chi _\ell f_\ell ^2} * {\widehat{g}}_{\ell _0} \right) \left( p\right) - \left( \widehat{\chi _\ell f_\ell ^2} * {\widehat{g}}_{\ell _0} \right) \left( 0\right) \right| \le C |p| \int |x| \chi _\ell (x) f_\ell ^2 ( x) g_{\ell _0} (x) dx \le C \ell ^{\frac{7}{2}} \end{aligned}$$

we conclude that

$$\begin{aligned} F_p \ge p^2 - C\ell ^{1/2} \ge \frac{p^2}{2}. \end{aligned}$$

Next, we show $|G_p| \le C/p^2$. With the scattering equation (5.16), we obtain

$$\begin{aligned} G_p = 2N \lambda _{\ell _0} (\widehat{\chi _{\ell _0} f_\ell ^2} * {\widehat{g}}_{\ell _0}) (p) + 2p^2 (\gamma _p\sigma _p - \eta _p) + ({\widehat{V}}_\ell *{\widehat{g}}_{\ell _0} )(p) \big [ (\gamma _p+\sigma _p)^2-1\big ]. \end{aligned}$$

Since

$$\begin{aligned}{} & {} {} \left| \gamma _p \sigma _p - \eta _p \right| =\Big |\frac{1}{2}\sinh (2\eta _p)-\eta _p\Big | \le C \left| \eta _p\right| ^3\le \frac{C}{|p|^6} \nonumber \\{}{} & {} {} \big | (\gamma _p + \sigma _p)^2-1 \big |=\Big |\sinh (2\eta _p)+\cosh (2\eta _p)-1\Big | \le C |\eta _p| \le \frac{C}{p^2} \end{aligned}$$

(6.1)

and using (5.13) we obtain $|G_p| \le C / p^2$, as claimed.

It remains to show $| G_p | \le F_p$. To this end, we write

$$\begin{aligned}&F_p-G_p =p^2\left( \gamma _p-\sigma _p\right) ^2 -2D_p \\ {}&F_p+G_p= \big [ p^2 + 2 ({\widehat{V}}_\ell * {\widehat{g}}_{\ell _0}) (p) \big ] (\gamma _p+\sigma _p )^2 + 2 D_p\,. \end{aligned}$$

By Lemma 5.1 we have $\left| D_p\right| \le C/ (N\ell )$. Hence, we find, for N large enough, $F_p - G_p \ge p^2 - C /(N\ell ) \ge 0$ and, similarly as in the proof of $F_p \ge p^2/2$ (distinguishing small and large |p|), $F_p + G_p \ge C p^2 - C/ (N\ell ) > 0$. This shows that $F_p > |G_p|$ and concludes the proof of the lemma. $\square$

With Lemma 6.1, using in particular the bound $|G_p| < F_p$, we can define, for every $p \in \Lambda ^*_+$, $\tau _p \in \mathbb {R}$ through the identity

$$\begin{aligned} \tanh (2\tau _p) = - \frac{G_p}{F_p}. \end{aligned}$$

Equivalently,

$$\begin{aligned} \tau _p = \frac{1}{4} \log \frac{1- G_p/F_p}{1+G_p/F_p}\,. \end{aligned}$$

(6.2)

From Lemma 6.1 we obtain

$$\begin{aligned} |\tau _p| \le C \frac{|G_p|}{F_p} \le \frac{C}{|p|^{4}} \end{aligned}$$

(6.3)

for all $p \in \Lambda ^*_+$. With the coefficients $\tau _p$, we define the antisymmetric operator

$$\begin{aligned} B(\tau ) = \frac{1}{2} \sum _{p \in \Lambda ^*_+} \tau _p \big ( b_p^* b_{-p}^* - b_p b_{-p} \big ) \end{aligned}$$

(6.4)

and we consider the generalized Bogoliubov transformation $e^{B(\tau )}$.

Lemma 6.2

Let $\tau _p$ be defined as in (6.2). Then, for every $n \in \mathbb {N}$ and any $r\in (0; 5)$ there exists a constant $C>0$ such that

$$\begin{aligned} e^{B(\tau )} (\mathcal {K}+\mathcal {V}_\ell +\mathcal {P}^{(r)}+1)(\mathcal {N}_+ +1)^n e^{B(\tau )} \le C (\mathcal {K}+\mathcal {V}_\ell + \mathcal {P}^{(r)} + 1)(\mathcal {N}_+ +1)^n\,. \end{aligned}$$

(6.5)

Proof

Proceeding as in [6, Lemma 5.4] and using that, by (6.3), $\Vert \tau \Vert _1$, $\Vert \tau \Vert _2$ and $\Vert \tau \Vert _{H^2}$ are all bounded uniformly in $\ell$ and N, we find

$$\begin{aligned} e^{B(\tau )} (\mathcal {K}+ \mathcal {V}_\ell +1)(\mathcal {N}_+ +1) e^{B(\tau )} \le C (\mathcal {K}+\mathcal {V}_\ell + 1)(\mathcal {N}_+ +1). \end{aligned}$$

The growth of $\mathcal {P}^{(r)}(\mathcal {N}_++1)$ can be controlled as in Lemma 5.2, with the only difference that now $\sum _{q \in \Lambda ^*_+} |q|^{r} |\tau _q|^2 \le C$, for all $0< r < 5$. For $n\ge 1$, we can proceed similarly.

$\square$

The reason why we are interested in the Bogoliubov transformation $e^{B(\tau )}$ is that it diagonalizes the quadratic operator $\mathcal {Q}_{N,\ell }$ defined as in Prop. 5.3.

Lemma 6.3

Let $\mathcal {Q}_{N,\ell }$ be defined as in (5.28), and $\tau _p$ as in (6.2). Then, we have

$$\begin{aligned} e^{-B(\tau )} \mathcal {Q}_{N,\ell } e^{B(\tau )} = \frac{1}{2} \sum _{p \in \Lambda ^*_+} \left[ -F_p + \sqrt{F_p^2 - G_p^2} \right] + \sum _{p \in \Lambda ^*_+} \sqrt{F_p^2 - G_p^2} \; a_p^* a_p + \delta _{N, \ell } \end{aligned}$$

where

$$\begin{aligned} \pm \delta _{N,\ell } \le \frac{C}{N} (\mathcal {K}+1)(\mathcal {N}+1)\,. \end{aligned}$$

Proof

The proof of Lemma 6.3 follows exactly as in [8, Lemma 5.3], using Lemma 6.1 (which implies $\Vert \tau \Vert _1 \le C$), Lemma 5.2 and Lemma 6.2. $\square$

With the generalized Bogoliubov transformation $e^{B(\eta )}$, we define a new excitation Hamiltonian $\mathcal {M}^\text {eff}_{N,\ell }: \mathcal {F}_+^{\le N} \rightarrow \mathcal {F}_+^{\le N}$, setting²

$$\begin{aligned} \mathcal {M}^\text {eff}_{N, \ell } = e^{-B(\tau )} \mathcal {G}_{N,\ell }^\text {eff} e^{B(\tau )}. \end{aligned}$$

(6.6)

Since the generalized Bogoliubov transformation $e^{B(\tau )}$ diagonalizes the quadratic part of $\mathcal {G}_{N,\ell }^\text {eff}$, the vacuum vector $\Omega \in \mathcal {F}_+^{\le N}$ is a good trial state for $\mathcal {M}^\text {eff}_{N,\ell }$. This correspond to the trial state $\Phi _N = U_N^* e^{B(\eta )} e^{B(\tau )}\Omega \in L^2_s (\Lambda ^N)$ for the Hamiltonian $H_N^\text {eff}$.

Proposition 6.4

Let $\mathcal {M}^\text {eff}_{N,\ell }$ be as defined in (6.6), with $B(\tau )$ as in (6.4) and $\mathcal {G}_{N,\ell }^\text {eff}$ as in (5.21), with $\ell \ge N^{-1+\nu }$ for some $\nu > 0$ and $\ell _0 > 0$ small enough. Then, we have

$$\begin{aligned} \begin{aligned} \langle \Omega , \mathcal {M}^\text{ eff}_{N,\ell }\Omega \rangle = \,&4 \pi \mathfrak {a}(N -1) \, +\, e_\Lambda \mathfrak {a}^2 \\ {}&-\frac{1}{2} \sum _{p \in \Lambda ^*_+} \bigg [ p^2 + 8 \pi \mathfrak {a}- \sqrt{|p|^4 + 16 \pi \mathfrak {a}p^2} - \frac{(8 \pi \mathfrak {a})^2}{2p^2}\bigg ] + \mathcal {O}(N^{-\nu /2}) \end{aligned} \end{aligned}$$

with $e_\Lambda$ defined as in (1.5).

Proof

With (5.30) and Lemma 6.2, we have

$$\begin{aligned} \mathcal {M}^\text {eff}_{N,\ell } = C_{N,\ell } + e^{-B(\tau )} \mathcal {Q}_{N,\ell } e^{B(\tau )} + \mathcal {E}'_{N,\ell } \end{aligned}$$

with

$$\begin{aligned} \pm \mathcal {E}'_{N,\ell } \le \frac{C}{(N\ell )^{1/2}} (\mathcal {K}+ \mathcal {V}_\ell + \mathcal {P}^{(5/2)} + 1) (\mathcal {N}_++1)\,. \end{aligned}$$

With Lemma 6.3 and the assumption $\ell \ge N^{-1+\nu }$, we obtain

$$\begin{aligned} \langle \Omega , \mathcal {M}^\text{ eff}_{N,\ell }\Omega \rangle = C_{N,\ell } + \frac{1}{2} \sum _{p \in \Lambda ^*_+} \left[ - F_p + \sqrt{F_p^2 - G_p^2} \right] + \mathcal {O}( N^{-\nu /2}) \end{aligned}$$

(6.7)

with $C_{N,\ell }$, $F_p$ and $G_p$ defined as in (5.27) and (5.29). We rewrite

$$\begin{aligned} C_{N,\ell }= & {} {} \, \frac{(N-1)}{2}{\widehat{V}}_\ell (0) +\sum _{p\in \Lambda ^*_+} \Big [ p^2 \eta _p^2 + {\widehat{V}}_\ell (p) \eta _p + \frac{1}{2N} ({\widehat{V}}_\ell * \eta ) (p) \eta _p +D_p \eta _p \Big ] \nonumber \\{}{} & {} {} + \sum _{p \in \Lambda ^*_+} \Big [ p^2 (\sigma _p^2 -\eta _p^2) + {{\widehat{V}}}_\ell (p) \big ( \sigma ^2_p +\gamma _p \sigma _p - \eta _p \big ) - \frac{1}{2N} {{\widehat{V}}}_\ell (p)\eta _p \eta _0 \Big ].\nonumber \\ \end{aligned}$$

(6.8)

With the scattering equation (5.16) we find

$$\begin{aligned} \begin{aligned} C_{N,\ell } =\,&\frac{(N-1)}{2}{\widehat{V}}_\ell (0) + \sum _{p \in \Lambda ^*_+} \Big [\,\frac{1}{2}{{\widehat{V}}}_\ell (p)\eta _p + N \lambda _{\ell _0} \big (\widehat{(\chi _{\ell _0}f_\ell ^2)} *{{\widehat{g}}}_{\ell _0}\big )(p) \eta _p \,\Big ] \\&+ \sum _{p \in \Lambda ^*_+} \Big [ p^2 (\sigma _p^2 -\eta _p^2) + {{\widehat{V}}}_\ell (p) \big ( \sigma ^2_p +\gamma _p \sigma _p - \eta _p \big ) - \frac{1}{2N} {{\widehat{V}}}_\ell (p)\eta _p \eta _0 \Big ].\end{aligned} \end{aligned}$$

Recalling that $V_\ell = 2N \lambda _\ell \chi _\ell f_\ell ^2$ we obtain, switching to position space,

$$\begin{aligned}\begin{aligned} C_{N,\ell } =\,&N (N-1) \lambda _\ell \int \chi _\ell (x) f_\ell ^2 (x) dx + N \lambda _\ell \int \chi _\ell (x) f_\ell ^2 (x) {\check{\eta }} (x) dx \\ {}&+ N \lambda _{\ell _0} \int \chi _{\ell _0} (x) f_\ell ^2 (x) g_{\ell _0} (x) {\check{\eta }} (x) dx - N \lambda _\ell \widehat{(\chi _{\ell } f_\ell ^2)}(0)\eta _0 \\ {}&-N \lambda _{\ell _0} \big (\widehat{\chi _{\ell _0}f_\ell ^2} *{{\widehat{g}}}_{\ell _0}\big )(0) \eta _0 \\&+ \sum _{p \in \Lambda ^*_+} \Big [ p^2 (\sigma _p^2 -\eta _p^2) + {{\widehat{V}}}_\ell (p) (\sigma ^2_p+\gamma _p \sigma _p - \eta _p) - \frac{1}{2N} {{\widehat{V}}}_\ell (p)\eta _p \eta _0 \Big ]. \end{aligned} \end{aligned}$$

With ${\check{\eta }} = N (g_{\ell _0} - 1)$, we arrive at

$$\begin{aligned} \begin{aligned} C_{N,\ell } =\,&N (N-1) \lambda _\ell \int \chi _\ell (x) f_\ell ^2 (x) dx + N^2 \lambda _\ell \int \chi _\ell (x) f_\ell ^2 (x) g_{\ell _0} (x) dx \\ {}&- N^2 \lambda _\ell \int \chi _\ell (x) f_\ell ^2 (x) dx+ N^2 \lambda _{\ell _0} \int \chi _{\ell _0} (x) f_\ell ^2 (x) g^2_{\ell _0} (x) dx \\ {}&- N^2 \lambda _{\ell _0} \int \chi _{\ell _0} (x) f_\ell ^2 (x) g_{\ell _0} (x) dx -N \lambda _\ell \widehat{(\chi _{\ell } f_\ell ^2)}(0)\eta _0 \\ {}&-N \lambda _{\ell _0} \big (\widehat{\chi _{\ell _0}f_\ell ^2} *{{\widehat{g}}}_{\ell _0}\big )(0) \eta _0 \\&+ \sum _{p \in \Lambda ^*_+} \Big [ p^2 (\sigma _p^2 -\eta _p^2) + {{\widehat{V}}}_\ell (p) (\sigma ^2_p+\gamma _p \sigma _p - \eta _p) - \frac{1}{2N} {{\widehat{V}}}_\ell (p)\eta _p \eta _0 \Big ]. \end{aligned}\end{aligned}$$

With (5.7) and since $g_{\ell _0}$ satisfies Neumann boundary conditions, we notice that

$$\begin{aligned}&{} \lambda _{\ell } \int _{B_{\ell _0}} \quad \chi _{\ell }(x) f_\ell ^2(x) g_{\ell _0}(x) dx -\, \lambda _{\ell _0} \int _{B_{\ell _0}} \quad \chi _{\ell _0}(x) f_\ell ^2(x) g_{\ell _0}(x)dx \\{}&{} \qquad \quad = \int _{B_{\ell _0}} \quad \nabla \big ( f_\ell ^2(x) \nabla g_{\ell _0}(x) \big ) dx = 0\,. \end{aligned}$$

Thus, using $f_{\ell _0} = f_\ell g_{\ell _0}$, we conclude that³

$$\begin{aligned} C_{N,\ell }= & {} N (N-1) \lambda _{\ell _0} \int \chi _{\ell _0} (x) f^2_{\ell _0} (x) dx \nonumber \\{} & {} + N \lambda _{\ell _0} \int \chi _{\ell _0} (x) f^2_{\ell _0} (x) dx - N \lambda _\ell \int \chi _\ell (x) f_\ell ^2 (x) dx \nonumber \\{} & {} -N \lambda _\ell \widehat{(\chi _{\ell } f_\ell ^2)}(0)\eta _0 -N \lambda _{\ell _0} \big (\widehat{\chi _{\ell _0}f_\ell ^2} *{{\widehat{g}}}_{\ell _0}\big )(0) \eta _0 \nonumber \\{} & {} + \sum _{p \in \Lambda ^*_+} \Big [ p^2 (\sigma _p^2 -\eta _p^2) + {{\widehat{V}}}_\ell (p) (\sigma ^2_p+\gamma _p \sigma _p - \eta _p) - \frac{1}{2N} {{\widehat{V}}}_\ell (p)\eta _p \eta _0 \Big ].\qquad \end{aligned}$$

(6.9)

To bound the terms on the second line of (6.9), we use Lemma 2.1 to show that

$$\begin{aligned} \begin{aligned} \Big | N \lambda _{\ell _0} \int \chi _{\ell _0}(x) f_{\ell _0}^2 (x) dx - 4 \pi \mathfrak {a}\Big |&\le \frac{C}{N\ell _0} \\ \Big | N \lambda _{\ell } \int \chi _{\ell }(x) f_\ell ^2(x) dx - 4 \pi \mathfrak {a}\Big |&\le \frac{C}{N\ell }\,. \end{aligned}\end{aligned}$$

Similarly, we find

$$\begin{aligned} - N \lambda _\ell \widehat{(\chi _{\ell } f_\ell ^2)}(0)\eta _0 -N \lambda _{\ell _0} \big (\widehat{\chi _{\ell _0}f_\ell ^2} *\widehat{g}_{\ell _0}\big )(0) \eta _0 = - 8\pi \mathfrak {a}\eta _0 + \mathcal {O}((N\ell )^{-1})\,.\end{aligned}$$

As for the terms on the fourth line, the last contribution can be bounded, using that $|\eta _0 | \le C$, by

$$\begin{aligned} \Big | \frac{1}{2N} \sum _{p \in \Lambda ^*_+} {\widehat{V}}_\ell (p) \eta _p \eta _0 \Big | \le \frac{C}{N\ell }. \end{aligned}$$

To handle the other terms on the fourth line of (6.9), we combine them with the first term in the sum on the r.h.s. of (6.7). Recalling (5.29), we find (using again $\ell \ge N^{-1+\nu }$)

$$\begin{aligned}{} & {} \sum _{p \in \Lambda ^*_+} \bigg [ p^2 (\sigma _p^2 -\eta _p^2) + {{\widehat{V}}}_\ell (p) (\sigma ^2_p +\gamma _p \sigma _p - \eta _p) - \frac{1}{2} F_p \bigg ] \nonumber \\{} & {} \qquad = -\sum _{p \in \Lambda ^*_+} \left[ \frac{p^2}{2} +\frac{1}{2} \,(\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p) + p^2 \eta _p^2 +(\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p) \eta _p \right] + \mathcal {O}(N^{-\nu })\nonumber \\ \end{aligned}$$

(6.10)

where we bounded, using $|\sigma _p^2 + \gamma _p \sigma _p - \eta _p|\le C|\eta _p|^2 \le C /|p|^4$ (see (6.1)),

$$\begin{aligned}\Big | \sum _{p \in \Lambda ^*_+} \big ( {{\widehat{V}}}_\ell (p) - (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p) \big )(\sigma ^2_p +\gamma _p \sigma _p - \eta _p ) \Big | \le \frac{C}{N\ell }\,. \end{aligned}$$

As for the remaining term on the r.h.s. of (6.7), we can write

$$\begin{aligned} F_p^2 - G_p^2= |p|^4 +2p^2 (\widehat{V_\ell } *\widehat{g}_{\ell _0})(p) + A_p\end{aligned}$$

with the notation

$$\begin{aligned} A_p = -4 D_p \Big ( (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p) (\gamma _p+\sigma _p)^2 + D_p + 2 p^2 \gamma _p \sigma _p \Big )\,. \end{aligned}$$

From (5.18), we have $|D_p|\le C/ (N\ell )$. Thus, with $(\gamma _p + \sigma _p)^2 \le C$ and $|\gamma _p \sigma _p|\le C|p|^{-2}$, we obtain $|A_p| \le C / (N\ell )$. Using this bound and the observation that $|p|^4 +2p^2 (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p)$ and $|p|^4 +2p^2 (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p) + A_p$ are positive and bounded away from zero we write

$$\begin{aligned} \begin{aligned} \sqrt{F_p^2 - G_p^2}&= \sqrt{|p|^4 +2p^2 (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p)} \\ {}&\quad + \frac{A_p}{ \sqrt{|p|^4 +2p^2 (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p) + A_p} + \sqrt{|p|^4 +2p^2 (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p)} }\,. \end{aligned}\end{aligned}$$

Expanding the square roots in the denominator around $p^2$, we easily find (using again $|A_p| \le C / (N\ell )$),

$$\begin{aligned}{} & {} {} \sum _{p \in \Lambda ^*_+} \frac{A_p}{ \sqrt{|p|^4 +2p^2 (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p) + A_p} + \sqrt{|p|^4 +2p^2 (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p)} } \\ {}{}{} & {} {} \qquad \quad \quad = \sum _{p \in \Lambda ^*_+} \frac{A_p}{2p^2} + \mathcal {O}(N^{-\nu })\,.\end{aligned}$$

Combining the last two equations with (6.7), (6.9), (6.10), we find

$$\begin{aligned} \begin{aligned} \langle \Omega ,&\mathcal {M}^\text{ eff}_{N,\ell }\Omega \rangle =\, N (N-1) \lambda _{\ell _0} \int \chi _{\ell _0} (x) f^2_{\ell _0} (x) dx - 8\pi \mathfrak {a}\eta _0 \\ {}&+ \frac{1}{2} \sum _{p\in \Lambda ^*_+} \Big [ -p^2 -(\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p) + \sqrt{|p|^4 +2p^2 (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0}) (p) } + \frac{\big ((\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p) \big )^2}{2p^2} \Big ] \\ {}&- \sum _{p \in \Lambda ^*_+} \left[ p^2 \eta ^2_p + (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p)\eta _p - \frac{A_p}{ 4p^2} + \frac{\big ( (\widehat{V_\ell } *\widehat{g}_{\ell _0})(p)\big )^2}{4p^2}\right] +\mathcal {O}(N^{-\nu /2})\,.\end{aligned}\nonumber \\ \end{aligned}$$

(6.11)

Estimating $|(\gamma _p + \sigma _p)^2 -1 | \le C |\eta _p| \le C/|p|^2$ and $|\gamma _p \sigma _p - \eta _p| \le C \eta _p^3 \le C / |p|^6$ (see (6.1)), we obtain

$$\begin{aligned} \sum _{p \in \Lambda ^*_+} \frac{A_p}{4p^2} = - \sum _{p \in \Lambda ^*_+} \frac{D_p}{p^2} \left[ 2p^2 \eta _p + ({\widehat{V}}_\ell * {\widehat{g}}_{\ell _0}) (p) + D_p \right] +\mathcal {O}(N^{-\nu })\,.\end{aligned}$$

Solving the scattering equation (5.16) for $D_p$, we obtain

$$\begin{aligned} \sum _{p \in \Lambda ^*_+} \frac{A_p}{4p^2} = - \sum _{p \in \Lambda ^*_+} \frac{D_p}{p^2} \left[ p^2 \eta _p + \frac{1}{2} ({\widehat{V}}_\ell * {\widehat{g}}_{\ell _0}) (p) + N \lambda _{\ell _0} \big ( \widehat{\chi _{\ell _0} f_\ell ^2} * {\widehat{g}}_{\ell _0} \big ) (p) \right] +\mathcal {O}(N^{-\nu })\, . \end{aligned}$$

Inserting this bound in the last line of (6.11), we get

$$\begin{aligned} \begin{aligned}&\sum _{p \in \Lambda ^*_+} \left[ p^2 \eta ^2_p + (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p)\eta _p - \frac{A_p}{ 4p^2} + \frac{\big ( (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p)\big )^2}{4p^2}\right] \\ {}&= \, \sum _{p \in \Lambda ^*_+} \Big [ p^2 \eta _p +\frac{1}{2} ({\widehat{V}}_\ell * {\widehat{g}}_{\ell _0}) (p) + D_p \Big ] \eta _p \\ {}&\qquad + \sum _{p \in \Lambda ^*_+} \frac{1}{2} ({\widehat{V}}_\ell * {\widehat{g}}_{\ell _0}) (p) \Big [ \eta _p + \frac{({\widehat{V}}_\ell * {\widehat{g}}_{\ell _0}) (p)}{2p^2} + \frac{D_p}{p^2} \Big ] \\ {}&\qquad + \sum _{p \in \Lambda ^*_+} \frac{N \lambda _{\ell _0} \big ( \widehat{\chi _{\ell _0} f_\ell ^2} * {\widehat{g}}_{\ell _0} \big ) (p)}{p^2} D_p +\mathcal {O}(N^{-\nu })\,.\end{aligned} \end{aligned}$$

With the scattering equation (5.16), we find

$$\begin{aligned} \begin{aligned}&\sum _{p \in \Lambda ^*_+} \Big [ p^2 \eta ^2_p + (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p)\eta _p - \frac{A_p}{ 4p^2} + \frac{\big ( (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p)\big )^2}{4p^2}\Big ] \\ {}&\quad = \sum _{p \in \Lambda ^*_+} \frac{N \lambda _{\ell _0} \big ( \widehat{\chi _{\ell _0} f_\ell ^2} * {\widehat{g}}_{\ell _0} \big ) (p)}{p^2} \left[ p^2 \eta _p + \frac{1}{2} (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p) + D_p \right] +\mathcal {O}(N^{-\nu }) \\ {}&\quad = \sum _{p \in \Lambda ^*_+} \frac{\big [ N \lambda _{\ell _0} \big ( \widehat{\chi _{\ell _0} f_\ell ^2} * {\widehat{g}}_{\ell _0} \big )(p)\big ]^2}{p^2} = \frac{9 \mathfrak {a}^2}{\ell _0^6} \sum _{p \in \Lambda ^*_+} \frac{\widehat{\chi }_{\ell _0}^2 (p)}{p^2} +\mathcal {O}(N^{-\nu })\end{aligned} \end{aligned}$$

where in the last step we used Lemma 2.1 and Lemma 5.1. From (6.11), we conclude that

$$\begin{aligned} \langle \Omega , \mathcal {M}^\text{ eff}_{N,\ell }\Omega \rangle= & {} {} N (N-1) \lambda _{\ell _0} \int \chi _{\ell _0} (x) f^2_{\ell _0} (x) dx - 8\pi \mathfrak {a}\eta _0 - \frac{9\mathfrak {a}^2}{\ell _0^6} \,\sum _{p \in \Lambda ^*_+} \frac{\widehat{\chi }_{\ell _0}(p)^2}{p^2} \nonumber \\{}{} & {} {} -\frac{1}{2} \sum _{p \in \Lambda ^*_+} e_{N}(p) +\mathcal {O}(N^{-\nu /2}) \end{aligned}$$

(6.12)

where we introduced the notation

$$\begin{aligned} e_{N}(p) = p^2 +(\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p) - \sqrt{|p|^4 +2p^2 (\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0}) (p) } - \frac{\big ((\widehat{V_\ell } *{{\widehat{g}}}_{\ell _0})(p) \big )^2}{2p^2}\,. \end{aligned}$$

Expanding the square root, we find that $|e_{N}(p)|\le C |p|^{-4}$, uniformly in N and $\ell$. This allows us to cut the sum to $|p| \le \ell ^{-1}$, with a negligible error. For $|p| \le \ell ^{-1}$, we can then compare $({\widehat{V}}_\ell * {\widehat{g}}_{\ell _0}) (p)$ with $({\widehat{V}}_\ell * {\widehat{g}}_{\ell _0}) (0)$ and then with ${\widehat{V}}_\ell (0)$. Proceeding similarly to [8, Eq. (5.26)-(5.27)], we conclude that

$$\begin{aligned} \sum _{p \in \Lambda ^*_+} e_{N}(p) = \sum _{p \in \Lambda ^*_+} \Big [ p^2 + 8\pi \mathfrak {a} - \sqrt{|p|^4 + 16 \pi \mathfrak {a} p^2} - \frac{(8\pi \mathfrak {a})^2}{2p^2} \Big ] + \mathcal {O}(\ell \log \ell )\,. \qquad \end{aligned}$$

(6.13)

Finally, let us compute the last term on the first line on the r.h.s. of (6.12). Using the expressions (see [8, Eq. (5.5), (5.29) and (5.33)]):

$$\begin{aligned} \begin{aligned} {\widehat{\chi }}_{\ell _0}(p)&= \frac{4 \pi \ell _0}{|p|^2} \left( \frac{\sin (\ell _0 |p|)}{\ell _0|p|} - \cos (\ell _0 p)\right) \\ \widehat{(\chi _{\ell _0} |\cdot |^2)}(p)&= \frac{4 \pi \ell _0^3}{|p|^2} \left( - \frac{6\sin (\ell _0 |p|)}{\ell _0^3 |p|^3} + \frac{6\cos (\ell _0 p)}{\ell _0^2|p|^2} +\frac{3\sin (\ell _0 |p|)}{\ell _0 |p|} - \cos (\ell _0 p)\right) \\ \widehat{(\chi _{\ell _0}|\cdot |^{-1})}(p)&= \frac{4 \pi }{|p|^2} \Big ( 1- \cos (\ell _0 p)\Big ) \\ \end{aligned}\end{aligned}$$

we can rewrite

$$\begin{aligned} \begin{aligned} - \frac{9\mathfrak {a}^2}{\ell _0^6} \sum _{p\in \Lambda ^*_+} \frac{ \widehat{\chi }_{\ell _0}(p)^2}{p^2} =\,&- 12 \pi \frac{\mathfrak {a}^2}{\ell _0^3} \sum _{p\in \Lambda ^*_+} \frac{\widehat{\chi }_{\ell _0}(p)}{|p|^2} + \frac{3\mathfrak {a}^2}{2\ell _0^6} \sum _{p\in \Lambda ^*_+}{\widehat{\chi }}_{\ell _0}(p) \cdot \widehat{(\chi _{\ell _0}|\cdot |^2)}(p) \\ {}&- \frac{9}{2} \frac{\mathfrak {a}^2}{\ell _0^4} \sum _{p\in \Lambda ^*_+} \widehat{\chi }_{\ell _0}(p)^2 + \frac{3\mathfrak {a}^2}{\ell _0^3} \sum _{p\in \Lambda ^*_+}{\widehat{\chi }}_{\ell _0}(p) \cdot \widehat{(\chi _{\ell _0}|\cdot |^{-1})}(p)\,.\qquad \end{aligned} \end{aligned}$$

(6.14)

From [8, Eq. (5.31)] we have

$$\begin{aligned} - 12 \pi \frac{\mathfrak {a}^2}{\ell _0^3} \sum _{p\in \Lambda ^*_+}\frac{\widehat{\chi }_{\ell _0}(p)}{|p|^2} = 6 \pi \mathfrak {a}^2 \big ( I_0- \frac{1}{\ell _0} - \frac{4}{15}\pi \ell _0^2 \big ) \end{aligned}$$

(6.15)

where

$$\begin{aligned} I_0 = \frac{1}{3 \pi } - \frac{2}{3\pi } \lim _{M \rightarrow \infty } \sum _{\begin{array}{c} p \in \Lambda ^*_+: \\ |p_i|\le M \end{array}} \frac{\cos (|p|)}{p^2}\,. \end{aligned}$$

Computing the different terms on the r.h.s. of (6.14) and using (6.15) we obtain

$$\begin{aligned} - \frac{9\mathfrak {a}^2}{\ell _0^6} \sum _{p\in \Lambda ^*_+} \frac{ \widehat{\chi }_{\ell _0}(p)^2}{p^2} = 6 \pi \mathfrak {a}I_0 - \frac{24}{5} \pi \frac{\mathfrak {a}^2}{\ell _0} - \frac{16}{5} \pi ^2 \mathfrak {a}^2 \ell _0^2\,. \end{aligned}$$

Inserting (6.13), (2.6), (5.17) and the last equation in (6.12), we conclude that

$$\begin{aligned} \begin{aligned} \langle \Omega , \mathcal {M}^\text {eff}_{N,\ell } \Omega \rangle&= 4 \pi \mathfrak {a}(N-1) + e_\Lambda \mathfrak {a}^2 \\ {}&\quad -\frac{1}{2} \sum _{p \in \Lambda ^*_+} \big [ p^2 + 8\pi \mathfrak {a} - \sqrt{|p|^4 + 16 \pi \mathfrak {a} p^2} - \frac{(8\pi \mathfrak {a})^2}{2p^2} \Big ] + \mathcal {O}(N^{-\nu /2})\, \end{aligned} \end{aligned}$$

with $e_\Lambda$ defined as in (1.5). $\square$

7 Bounds on the Trial State

We introduce some operators to control the regularity of our trial state. First of all, we recall the definition of the operator $\mathcal {P}^{(r)}$, defined in (5.23) for $1< r < 5$. Furthermore, we need some observables acting of several particles. For $n \in \mathbb {N}$, we define

$$\begin{aligned} \mathcal {T}_n =\sum _{p_1, \dots , p_n \in \Lambda ^*_+} p_1^2 \dots p_n^2 \, a_{p_1}^* \dots a_{p_n}^* a_{p_n} \dots a_{p_1}. \end{aligned}$$

(7.1)

Since $\eta$ has limited decay in momentum space (see (5.18)), we will only be able to control the expectation of $\mathcal {T}_n$ for $n = 2,3,4$. To control some error terms, it is also important to use less derivatives on each particle. We define, for $\delta > 0$ small enough (we will later impose the condition $\delta \in (0;1/6)$),

$$\begin{aligned} \mathcal {A}^{(\delta )}_{n} = \sum _{p_1, \dots , p_n \in \Lambda ^*_+} |p_1|^{3/2+ \delta } \dots |p_n|^{3/2+ \delta } a_{p_1}^* \dots a_{p_n}^* a_{p_n} \dots a_{p_1}. \end{aligned}$$

(7.2)

We will be able to control the expectation of $\mathcal {A}^{(\delta )}_n$, for all $n \in \mathbb {N}$. Additionally, we will also need the observable

$$\begin{aligned} \mathcal {S}^{(\varepsilon ,\delta )}_n = \sum _{p_1, \dots , p_n \in \Lambda ^*_+} |p_1|^{2+ \varepsilon } |p_2|^{3/2+\delta } \dots |p_n|^{3/2+ \delta } a_{p_1}^* \dots a_{p_n}^* a_{p_n} \dots a_{p_1}. \end{aligned}$$

(7.3)

All these operators act on the excitation Fock space $\mathcal {F}^{\le N}_+$. In order to bound their expectation on our trial state, we need to control their growth under the action of $B(\eta )$, similarly as we did in Lemma 5.2 for $\mathcal {P}^{(r)}$.

Lemma 7.1

For $n \in \mathbb {N}\backslash \{ 0 \}$ and $0< \delta < 1/6$, we consider $\mathcal {A}_n^{(\delta )}$ as in (7.2). We define recursively the sequence $\alpha _n$ (depending on the parameter $\delta$) by setting $\alpha _1 = 1/2 + \delta$, $\alpha _2 = 2+2\delta$ and

$$\begin{aligned} \alpha _n = \big [ \alpha _{n-1} + \alpha _{n-2} \big ] /2 + 7/4 + 3\delta /2\,.\end{aligned}$$

(7.4)

Then, for every $k \in \mathbb {N}$, there exists a constant $C > 0$ (depending also on n and $\delta$) such that

$$\begin{aligned} \langle e^{B(\eta )} \xi , \mathcal {A}^{(\delta )}_{n} (\mathcal {N}_+ + 1)^k e^{B(\eta )} \xi \rangle \le C \ell ^{-\alpha _{n}} \Big \{ \Vert (\mathcal {N}_+ + 1)^{k/2} \xi \Vert ^2 + \sum _{j=1}^n \langle \xi , {\mathcal {A}}_{j}^{(\delta )} (\mathcal {N}_+ + 1)^k \xi \rangle \Big \}\nonumber \\ \end{aligned}$$

(7.5)

for all $\xi \in \mathcal {F}^{\le N}_+$.

For $n \in {\mathbb {N}}\backslash \{ 0 \}$, let

$$\begin{aligned} \mathcal {I}_n = \{ ( \varepsilon , \delta ) \in (-1;3) \times (0;1/6): \varepsilon +2\delta < 3/2^{(n-1)} \}\,. \end{aligned}$$

For $(\varepsilon , \delta ) \in \mathcal {I}_n$, we consider $\mathcal {S}^{(\varepsilon ,\delta )}_n$ as in (7.3). Moreover, we define the sequence $\beta ^\varepsilon _n = \alpha _n + 1/2 + \varepsilon - \delta$, with $\alpha _n$ as in (7.4) (the sequence $\beta _n^\varepsilon$ depends also on $\delta$; since this dependence does not play an important role in the proof, we do not make it explicit in the notation). Then, for every $k \in \mathbb {N}$, there exists a constant $C > 0$ (depending also on $n,\varepsilon ,\delta$) such that

$$\begin{aligned} \begin{aligned} \langle e^{B(\eta )} \xi ,&\mathcal {S}^{(\varepsilon ,\delta )}_{n} e^{B(\eta )} \xi \rangle \le C \ell ^{-\beta ^\varepsilon _{n}} \Big \{ \Vert \xi \Vert ^2 + \sum _{j=1}^n \sup _{\varepsilon ,\delta \in \mathcal {I}_j} \; \langle \xi , {\mathcal {S}}_{j}^{(\varepsilon ,\delta )} \xi \rangle \Big \} \end{aligned} \end{aligned}$$

(7.6)

for all $\xi \in \mathcal {F}^{\le N}_+$.

For $n \in \{2,3,4 \}$, we can also control the growth of the operator $\mathcal {T}_n$, defined in (7.2). We find

$$\begin{aligned} \begin{aligned} \langle e^{B(\eta )} \xi , \mathcal {T}_2 e^{B(\eta )} \xi \rangle&\le C \ell ^{-3} \langle \xi , \big ( 1 + \mathcal {P}^{(4)} + \mathcal {T}_2 \big ) \xi \rangle \\ \langle e^{B(\eta )} \xi , \mathcal {T}_3 e^{B(\eta )} \xi \rangle&\le C \ell ^{-4} \langle \xi , \big ( 1 + \mathcal {P}^{(4)} + \mathcal {Z}_{4,2} + \mathcal {T}_3 \big ) \xi \rangle \\ \langle e^{B(\eta )} \xi , \mathcal {T}_4 e^{B(\eta )} \xi \rangle&\le C \ell ^{-6} \langle \xi , \big ( 1 + \mathcal {P}^{(4)} + \mathcal {Z}_{4,4} + \mathcal {Z}_{4,2,2} + \mathcal {T}_4 \big ) \xi \rangle \end{aligned} \end{aligned}$$

(7.7)

for every $\xi \in \mathcal {F}^{\le N}_+$. Here we introduced the notation (for $m =2,4$)

$$\begin{aligned} \begin{aligned} \mathcal {Z}_{4, 2}&= \sum _{\begin{array}{c} p_1, p_2 \in \Lambda ^*_+: \\ p_1 \not = \pm p_2 \end{array}} |p_1|^4 p_2^2 \, a^*_{p_1} a_{p_2}^* a_{p_2} a_{p_1}, \qquad \mathcal {Z}_{4, 4} = \sum _{\begin{array}{c} p_1, p_2 \in \Lambda ^*_+: \\ p_1 \not = \pm p_2 \end{array}} |p_1|^4 |p_2|^{4} \, a^*_{p_1} a_{p_2}^* a_{p_2} a_{p_1} \\ \mathcal {Z}_{4,2,2}&= \sum _{\begin{array}{c} p_1, p_2, p_3 \in \Lambda ^*_+:\\ p_1 \not = \pm p_2, \pm p_3 \end{array}} |p_1|^4 p_2^2 p_3^2 \, a^*_{p_1} a_{p_2}^* a_{p_3}^* a_{p_3} a_{p_2} a_{p_1}. \end{aligned}\nonumber \\ \end{aligned}$$

(7.8)

Finally, we will also need an improvement of (7.6), for $n=3$. For $\varepsilon > -1$, $0< \delta < 1/6$ with $\varepsilon + \delta < 1$, we find

$$\begin{aligned} \langle e^{B(\eta )} \xi , \mathcal {S}_3^{(\varepsilon ,\delta )} e^{B(\eta )} \xi \rangle \le C \ell ^{-3-\varepsilon -2\delta } \Big \{ \langle \xi , \big [ 1 + \mathcal {P}^{(4)} + \mathcal {Z}_{4,2} \big ] \xi \rangle + \sup _{(\varepsilon , \delta ) \in \mathcal {I}_3} \langle \xi , S^{(\varepsilon ,\delta )}_3 \xi \rangle \Big \}\nonumber \\ \end{aligned}$$

(7.9)

for all $\xi \in \mathcal {F}_+^{\le N}$ (observe that, in (7.6), $\beta _3^\varepsilon = 7/2+\varepsilon + 2\delta$).

Remark

The sequence $\alpha _n$ defined in (7.4) is given explicitly by

$$\begin{aligned} \alpha _n = \left( \frac{7}{6} + \delta \right) n - \frac{4}{9} \left( 1- \left( -\frac{1}{2} \right) ^n \right) \,. \end{aligned}$$

(7.10)

Proof

We begin with (7.5). We consider $k=0$; the case $k > 0$ can be handled similarly. For $n \ge 1$ and $0< \delta < 1/6$, we set

$$\begin{aligned} F^{(\delta )}_n (t) = \langle e^{t B(\eta )} \xi , \mathcal {A}_{n}^{(\delta )} e^{t B(\eta )} \xi \rangle \,.\end{aligned}$$

For $n \ge 2$, we compute

$$\begin{aligned} \begin{aligned} \frac{dF^{(\delta )}_n}{dt} (t)&= \langle e^{t B(\eta )} \xi , \big [ \mathcal {A}_{n}^{(\delta )}, B(\eta ) \big ] e^{t B(\eta )} \xi \rangle \\ {}&= \sum _{p_1, \dots ,p_n \in \Lambda ^*_+} |p_1|^{3/2+\delta } \dots |p_n|^{3/2+\delta } \\ {}&\quad \times \langle e^{t B(\eta )} \xi , \big [ a_{p_1}^* \dots a_{p_n}^* a_{p_n} \dots a_{p_1}, B(\eta ) \big ] e^{t B(\eta )} \xi \rangle . \end{aligned} \end{aligned}$$

With the identity

$$\begin{aligned} \left[ a^*_{p_1}\dots a^*_{p_n} a_{p_n}\dots a_{p_1}, b^*_q \right] = \sum _{j=1}^n \delta _{q,p_j} b^*_{p_j} a^*_{p_1}\dots a^*_{p_{j-1}}a^*_{p_{j+1}}\dots a^*_{p_n} a_{p_n}\dots a_{p_{j+1}} a_{p_{j-1}}\dots a_{p_1} \end{aligned}$$

we find

$$\begin{aligned} \begin{aligned} \big [ a^*_{p_1}\dots a^*_{p_n}&a_{p_n}\dots a_{p_1}, b^*_q b_{-q}^* \big ] \\ = \,&\sum _{j=1}^n \delta _{p_j, -q} b_q^* b_{p_j}^* a_{p_1}^* \dots a_{p_{j-1}}^* a_{p_{j+1}}^* \dots a_{p_n}^* a_{p_n} \dots a_{p_{j+1}} a_{p_{j-1}} \dots a_{p_1} \\ {}&+\sum _{j=1}^n \delta _{p_j, q} b_{p_j}^* a_{p_1}^* \dots a_{p_{j-1}}^* a_{p_{j+1}}^* \dots a_{p_n}^* a_{p_n} \dots a_{p_{j+1}} a_{p_{j-1}} \dots a_{p_1} b_{-q}^*\,.\end{aligned} \end{aligned}$$

Thus

$$\begin{aligned}{} & {} \big [ a^*_{p_1} \dots a^*_{p_n} a_{p_n}\dots a_{p_1}, b^*_q b_{-q}^* \big ] \nonumber \\ {}{} & {} \qquad = \, \sum _{j=1}^n (\delta _{p_j, -q} + \delta _{p_j, q}) \, b_q^* b_{-q}^* a_{p_1}^* \dots a_{p_{j-1}}^* a_{p_{j+1}}^* \dots a_{p_n}^* a_{p_n} \dots a_{p_{j+1}} a_{p_{j-1}} \dots a_{p_1} \nonumber \\ {}{} & {} \qquad + \sum _{j=1}^n \sum _{i \not = j} \delta _{p_j, q} \delta _{p_i, -q} b_q^* b_{-q}^* a_{p_1}^* \dots a_{p_{j-1}}^* a_{p_{j+1}}^* \dots a_{p_{i-1}}^* a_{p_{i+1}}^* \dots a_{p_n}^* \nonumber \\{} & {} \quad \qquad \times a_{p_n} \dots a_{p_{j+1}} a_{p_{j-1}} \dots a_{p_{i+1}} a_{p_{i-1}} \dots a_{p_1}. \end{aligned}$$

(7.11)

Therefore, we can bound

$$\begin{aligned} \Big | \frac{dF^{(\delta )}_n}{dt} (t) \Big |\le & {} {} C \sum _{p_1, \dots , p_{n-1}, q \in \Lambda ^*_+} |\eta _q | |q|^{3/2+\delta } |p_1|^{3/2 + \delta } \dots |p_{n-1}|^{3/2+ \delta } \nonumber \\{}{} & {} {} \times \Vert a_q a_{p_1} \dots a_{p_{n-1}} e^{tB(\eta )} \xi \Vert \Vert a_{-q}^* a_{p_1} \dots a_{p_{n-1}} e^{tB(\eta )} \xi \Vert \nonumber \\ {}{}{} & {} {} + C \sum _{p_1, \dots , p_{n-2}, q \in \Lambda ^*_+} |\eta _q | |q|^{3+2\delta } |p_1|^{3/2 + \delta } \dots |p_{n-2}|^{3/2+ \delta } \nonumber \\{}{} & {} {} \times \Vert a_q a_{p_1} \dots a_{p_{n-2}} e^{tB(\eta )} \xi \Vert \Vert a_{-q}^* a_{p_1} \dots a_{p_{n-2}} e^{tB(\eta )} \xi \Vert \end{aligned}$$

(7.12)

for a constant C depending on n. Estimating $\Vert a^*_{-q} \zeta \Vert \le \Vert a_{-q} \zeta \Vert + \Vert \zeta \Vert$ and applying Cauchy-Schwarz’s inequality, we obtain, for any $n \ge 3$,

$$\begin{aligned} \begin{aligned} \Big | \frac{dF^{(\delta )}_n}{dt} (t) \Big | \le \,&C \Vert \eta \Vert _\infty F^{(\delta )}_n (t) +C F^{(\delta )}_n (t)^{\frac{1}{2}} F^{(\delta )}_{n-1} (t)^{\frac{1}{2}} \Big [ \sum _{q \in \Lambda ^*_+} |q|^{3/2+\delta } \eta _q^2 \Big ]^{\frac{1}{2}} \\ {}&+ C \big [ \sup _{q\in \Lambda ^*_+} |q|^{3/2+ \delta } \eta _q \big ] \, F^{(\delta )}_{n-1} (t) + C F^{(\delta )}_{n-1} (t)^{\frac{1}{2}} F^{(\delta )}_{n-2} (t)^{\frac{1}{2}} \Big [ \sum _{q \in \Lambda ^*_+} |q|^{9/2 + 3\delta } \eta _q^2 \Big ]^{\frac{1}{2}}.\end{aligned} \end{aligned}$$

With Lemma 5.1, we arrive at

$$\begin{aligned} \Big | \frac{dF^{(\delta )}_n}{dt} (t) \Big | \le C F^{(\delta )}_n (t) + C \ell ^{-1/2-\delta } F^{(\delta )}_{n-1} (t) + C \ell ^{-7/4-3\delta /2} F^{(\delta )}_{n-1} (t)^{\frac{1}{2}} \, F^{(\delta )}_{n-2} (t)^{\frac{1}{2}}.\nonumber \\ \end{aligned}$$

(7.13)

This bound is also valid for $n=2$, setting $F^{(\delta )}_0 (t) = \Vert \xi \Vert ^2$. If $n=1$, we can use (5.23) to estimate

$$\begin{aligned} F^{(\delta )}_1 (t) \le C \ell ^{-1/2-\delta } \big [ \Vert \xi \Vert ^2 + \langle \xi , \mathcal {A}_1^{(\delta )} \xi \rangle \big ] \end{aligned}$$

for all $t \in [0;1]$. Inserting this bound on the r.h.s. of (7.13) (with $n=2$), we obtain

$$\begin{aligned} F^{(\delta )}_2 (t) \le C F^{(\delta )}_2 (0) + C \ell ^{-2-2\delta } \langle \xi ,\big [ \Vert \xi \Vert ^2 + \mathcal {A}_1^{(\delta )} \big ] \xi \rangle \le C \ell ^{-2-2\delta } \langle \xi , \big [ 1 + \mathcal {A}_1^{(\delta )} + \mathcal {A}_2^{(\delta )} \big ] \xi \rangle \,. \end{aligned}$$

Defining the coefficients $\alpha _n$ iteratively, as in (7.4), by simple induction we conclude from (7.13) that, for all $n \in \mathbb {N}$, there exists a constant $C > 0$ such that

$$\begin{aligned} F^{(\delta )}_n (t) \le C \ell ^{-\alpha _n} \big \langle \xi , \big [ 1 + \sum _{j=1}^n \mathcal {A}_j^{(\delta )} \big ] \xi \big \rangle \,.\end{aligned}$$

(7.14)

Let us consider (7.6), again for $k=0$. For $n \ge 1$, $(\varepsilon ;\delta ) \in \mathcal {I}_n$, $t \in [0;1]$, we define

$$\begin{aligned} G^{(\varepsilon ,\delta )}_n (t) = \langle e^{t B(\eta )} \xi , \mathcal {S}^{(\varepsilon ,\delta )}_n e^{t B(\eta )} \xi \rangle \,. \end{aligned}$$

Proceeding similarly to (7.13) we find, for $n \ge 2$ (with the convention that $G_0^{(\varepsilon ,\delta )} (t) = 0$ and $F_0^{(\delta )} (t) = \Vert \xi \Vert ^2$ for all $t \in [0;1]$),

$$\begin{aligned} \begin{aligned} \Big | \frac{dG_n^{(\varepsilon ,\delta )} (t)}{dt} \Big | \le \,&C G_n^{(\varepsilon ,\delta )} (t) + C \ell ^{-1/2-\delta } G_{n-1}^{(\varepsilon ,\delta )} (t) + C \ell ^{-7/4-3\delta /2} \, G_{n-1}^{(\varepsilon ,\delta )} (t)^{\frac{1}{2}} \, G_{n-2}^{(\varepsilon ,\delta )} (t)^{\frac{1}{2}} \\&\quad + C \ell ^{-1-\varepsilon } F_{n-1}^{(\delta )} (t) + C \ell ^{-2-\delta -\varepsilon /2+\theta /2} \, G_{n-1}^{(\varepsilon +\theta , \delta )} (t)^{\frac{1}{2}} \, F_{n-2}^{(\delta )} (t)^{\frac{1}{2}} \end{aligned}\nonumber \\ \end{aligned}$$

(7.15)

for a $\theta > \varepsilon + 2\delta$. The second line arises from the contributions to the commutator (7.11) where q coincides with the variable raised to the power $2+\varepsilon$. In fact, the contribution from the first term in (7.11) can be estimated by

$$\begin{aligned}{} & {} {} \sum _{q, p_1, \dots ,p_{n-1} \in \Lambda ^*_+} |\eta _q| |q|^{2+\varepsilon } |p_1|^{3/2+\delta } \dots |p_{n-1}|^{3/2 + \delta } \\{}{} & {} {} \qquad \quad \qquad \quad \times \Vert a_q a_{p_1} \dots a_{p_{n-1}} e^{t B(\eta )} \xi \Vert \Vert a_{-q}^* a_{p_1} \dots a_{p_{n-1}} e^{t B(\eta )} \xi \Vert \\{}{} & {} {} \qquad \quad \le C \Vert \eta \Vert _\infty G_n^{(\varepsilon , \delta )} (t) + C G_n^{(\varepsilon ,\delta )} (t)^{\frac{1}{2}} F_{n-1}^{(\delta )} (t)^{\frac{1}{2}} \Big [ \sum _q \eta _q^2 |q|^{2+ \varepsilon } \Big ]^{\frac{1}{2}} \\{}{} & {} {} \qquad \quad \le C G_n^{(\varepsilon ,\delta )} (t) + C \ell ^{-1-\varepsilon } F_{n-1}^{(\delta )} (t)\,. \end{aligned}$$

The contribution from the second term on the r.h.s. of (7.11), on the other hand, can be bounded by

$$\begin{aligned} \begin{aligned}&\sum _{q, p_1, \dots ,p_{n-2} \in \Lambda ^*_+} |\eta _q | |q|^{7/2+\varepsilon + \delta } |p_1|^{3/2+\delta } \dots |p_{n-2}|^{3/2 + \delta } \\ {}&\quad \times \Vert a_q a_{p_1} \dots a_{p_{n-2}} e^{t B(\eta )} \xi \Vert \Vert a_{-q}^* a_{p_1} \dots a_{p_{n-2}} e^{t B(\eta )} \xi \Vert \\ {}&\le C \Big [ \sup _q |\eta _q| |q|^{3/2+\delta } \Big ] G_{n-1}^{(\varepsilon ,\delta )} (t) + C G_{n-1}^{(\varepsilon + \theta , \delta )} (t)^{\frac{1}{2}} F_{n-2}^{(\delta )} (t)^{\frac{1}{2}} \Big [ \sum |q|^{5+\varepsilon + 2 \delta -\theta } \eta _q^2 \Big ]^{\frac{1}{2}} \\ {}&\le C G_{n-1}^{(\varepsilon ,\delta )} (t) + C \ell ^{-2-\delta -\varepsilon /2 +\theta /2} G_{n-1}^{(\varepsilon + \theta , \delta )} (t)^{\frac{1}{2}} F_{n-2}^{(\delta )} (t)^{\frac{1}{2}}\end{aligned} \end{aligned}$$

for a $\theta > \varepsilon + 2\delta$ (this condition is needed to apply (5.19), in Lemma 5.1).

If $n=1$, we use again (5.23) to estimate

$$\begin{aligned} G_1^{(\varepsilon ,\delta )} (t) \le C \ell ^{-1-\varepsilon } \Big \{ \Vert \xi \Vert ^2 + \langle \xi , \mathcal {S}^{(\varepsilon ,\delta )}_1 \xi \rangle \Big \} \le C \ell ^{-\beta ^\varepsilon _1} \Big \{ \Vert \xi \Vert ^2 + \sup _{(\varepsilon ,\delta ) \in \mathcal {I}_1} \langle \xi , \mathcal {S}^{(\varepsilon ,\delta )}_1 \xi \rangle \Big \} \end{aligned}$$

for all $\varepsilon < 3$ ($G_1^{(\varepsilon ,\delta )}$ does not depend on $\delta$). Inserting this bound in (7.15), we arrive at

$$\begin{aligned}\Big | \frac{dG_2^{(\varepsilon ,\delta )} (t)}{dt} \Big | \le C G_2^{(\varepsilon ,\delta )} (t) + C \ell ^{-5/2-\varepsilon -\delta } \Big \{ \Vert \xi \Vert ^2 + \sup _{(\varepsilon ,\delta ) \in \mathcal {I}_1} \langle \xi , \mathcal {S}^{(\varepsilon ,\delta )}_1 \xi \rangle \Big \} \end{aligned}$$

if we can find $\theta > 0$ such that $\theta > \varepsilon + 2\delta$ and $\varepsilon + \theta < 3$, i.e. if $\varepsilon + \delta < 3/2$ (this condition is certainly true, if $\varepsilon + 2\delta < 3/2$). By Gronwall’s lemma (noticing that $\beta _2^\varepsilon = 5/2+\varepsilon +\delta$), we conclude that

$$\begin{aligned} \begin{aligned} G^{(\varepsilon ,\delta )}_2 (t)&\le C \ell ^{-\beta ^\varepsilon _2}\Big \{ \Vert \xi \Vert ^2 + \sup _{(\varepsilon ,\delta ) \in \mathcal {I}_1} \langle \xi , \mathcal {S}^{(\varepsilon ,\delta )}_1 \xi \rangle + \sup _{(\varepsilon ,\delta ) \in \mathcal {I}_2} \langle \xi , \mathcal {S}^{(\varepsilon ,\delta )}_2 \xi \rangle \Big \}\end{aligned} \end{aligned}$$

for all $\delta \in (0;1/6)$, $\varepsilon \in (-1;3)$ such that $\varepsilon + 2\delta < 3/2$. Now, we proceed by induction. We fix $n \in \mathbb {N}$ and we assume that for all $j \le n-1$ there exists a constant $C > 0$ such that

$$\begin{aligned} G_j^{(\varepsilon , \delta )} (t) \le C \ell ^{-\beta ^\varepsilon _j} \Big \{ \Vert \xi \Vert ^2 + \sum _{i=1}^j \sup _{(\varepsilon ,\delta ) \in \mathcal {I}_i} \langle \xi , \mathcal {S}^{(\varepsilon ,\delta )}_i \xi \rangle \Big \} \end{aligned}$$

for all $\delta \in (0;1/6)$ and all $\varepsilon \in (-1;3)$ with $\varepsilon + \delta < 3/2^{(j-1)}$ and all $t \in [0;1]$. Then, using also (7.14), (7.15) implies that

$$\begin{aligned} \Big | \frac{dG_n^{(\varepsilon ,\delta )} (t)}{dt} \Big | \le C G_n^{(\varepsilon ,\delta )} (t) + C \ell ^{-\beta ^\varepsilon _n} \Big \{ \Vert \xi \Vert ^2 + \sum _{i=1}^{n-1} \sup _{(\varepsilon ,\delta ) \in \mathcal {I}_i} \langle \xi , \mathcal {S}^{(\varepsilon ,\delta )}_i \xi \rangle \Big \} \end{aligned}$$

(7.16)

if we can show that

$$\begin{aligned} \left\{ \begin{array}{llll} \beta _n^{\varepsilon } \ge \beta _{n-1}^{\varepsilon } + \delta -1/2 \\ \beta _n^{\varepsilon } \ge 7/4 + 3\delta /2 + (\beta ^{\varepsilon }_{n-1} + \beta ^{\varepsilon }_{n-2})/2 \\ \beta _n^{\varepsilon } \ge 1 + \varepsilon + \alpha _{n-1} \\ \beta _n^{\varepsilon } \ge 2 + \delta + \varepsilon /2 - \theta /2 + \beta ^{\varepsilon + \theta }_{n-1}/2 + \alpha _{n-2}/2 \\ \end{array} \right. \end{aligned}$$

(7.17)

and if we can find $\theta \in \mathbb {R}$ such that $\theta > \varepsilon + 2 \delta$ and $\varepsilon + \theta + 2\delta < 3/2^{(n-2)}$, i. e. if $\varepsilon + 2\delta < 3/2^{(n-1)}$. To verify (7.17), we use that $\beta _n^{\varepsilon } = \alpha _n + 1/2 + \varepsilon - \delta$. The first and the third conditions in (7.17) are equivalent to

$$\begin{aligned}\alpha _n \ge \alpha _{n-1} + 1/2 +\delta \end{aligned}$$

which follows easily from the explicit formula (7.10). The second and the fourth conditions are immediate consequences of the recursive definition (7.4) of the coefficients $\alpha _n$. From (7.16), by Gronwall’s lemma we conclude that

$$\begin{aligned} G_n^{(\varepsilon ,\delta )} (t) \le C \ell ^{-\beta ^\varepsilon _n} \Big \{ \Vert \xi \Vert ^2 + \sum _{i=1}^{n} \sup _{(\varepsilon ,\delta ) \in \mathcal {I}_i} \langle \xi , \mathcal {S}^{(\varepsilon ,\delta )}_i \xi \rangle \Big \} \end{aligned}$$

(7.18)

for all $\delta \in (0;1/6), \varepsilon \in (-1;3)$ with $\varepsilon + 2\delta < 3/2^{(n-1)}$.

Next, we show (7.7). For $t \in [0;1]$ and for $n = 2,3,4$, we set

$$\begin{aligned} H_n (t) = \langle e^{t B(\eta )} \xi , \mathcal {T}_n e^{t B(\eta )} \xi \rangle \,. \end{aligned}$$

Proceeding as in the proof of (7.12), we find

$$\begin{aligned} \begin{aligned} \Big | \frac{dH_2 (t)}{dt} \Big | \le \,&C \sum _{p,q \in \Lambda ^*_+} |\eta _q| \, q^2 p^2 \, \Vert a_q a_p e^{t B(\eta )} \xi \Vert \Vert a_{-q}^* a_p e^{t B(\eta )} \xi \Vert \\ {}&+ C \sum _{q \in \Lambda ^*_+} |\eta _q| \, |q|^4 \Vert a_q e^{t B(\eta )} \xi \Vert \Vert a_{-q}^*e^{t B(\eta )} \xi \Vert \,. \end{aligned} \end{aligned}$$

(7.19)

Using $\Vert a_{-q}^* \zeta \Vert \le \Vert a_{-q} \zeta \Vert + \Vert \zeta \Vert$ and Cauchy-Schwarz’s inequality we obtain, with (5.23) and (5.19),

$$\begin{aligned}\Big | \frac{dH_2 (t)}{dt} \Big | \le C H_2 (t) + C \ell ^{-2} \langle \xi , (1+\mathcal {P}^{(2)} )\xi \rangle + C \ell ^{-3} \langle \xi , (1+ \mathcal {P}^{(4)} ) \xi \rangle \,. \end{aligned}$$

By Gronwall’s lemma, we conclude that

$$\begin{aligned} H_2 (t) \le C \ell ^{-3} \langle \xi , \big ( 1 + \mathcal {T}^{(2)} + \mathcal {P}^{(4)} \big ) \xi \rangle \end{aligned}$$

(7.20)

for all $t \in [0;1]$.

Analogously to (7.19), we find

$$\begin{aligned} \begin{aligned} \Big | \frac{dH_3 (t)}{dt} \Big | \le \,&C \sum _{q,p_1, p_2 \in \Lambda ^*_+} |\eta _q | \, q^2 p_1^2 p_2^2 \, \Vert a_q a_{p_1} a_{p_2} e^{t B(\eta )} \xi \Vert \Vert a_{-q}^* a_{p_1} a_{p_2} e^{t B(\eta )} \xi \Vert \\ {}&+ C \sum _{q, p \in \Lambda ^*_+} |\eta _q| \, |q|^4 p^2 \Vert a_q a_p e^{t B(\eta )} \xi \Vert \Vert a_{-q}^* a_p e^{t B(\eta )} \xi \Vert \,. \end{aligned} \end{aligned}$$

Thus

$$\begin{aligned} \begin{aligned} \Big | \frac{dH_3 (t)}{dt} \Big | \le \,&C H_3 (t) + C \ell ^{-1} H_2 (t) + C \sum _{q,p \in \Lambda ^*_+} |\eta _q| |q|^4 p^2 \Vert a_q a_p e^{t B(\eta )} \xi \Vert \Vert a_p e^{t B(\eta )} \xi \Vert \,. \end{aligned}\nonumber \\ \end{aligned}$$

(7.21)

To control the last term, we distinguish the contribution

$$\begin{aligned} \sum _{q\in \Lambda ^*_+} |\eta _q| |q|^6 \Vert a_q^2 e^{t B(\eta )} \xi \Vert \Vert a_{q} e^{t B(\eta )} \xi \Vert\le & {} {} C \langle e^{t B(\eta )} \xi , \mathcal {P}^{(4)} (\mathcal {N}_+ + 1) e^{t B(\eta )} \xi \rangle \nonumber \\ {}\le & {} {} C \ell ^{-3} \langle \xi , \mathcal {P}^{(4)} (\mathcal {N}_+ + 1) \xi \rangle \end{aligned}$$

(7.22)

arising from terms with $p= q$, a similar contribution from terms with $p=-q$ and the contribution arising from terms with $p \not = -q, q$, which can be bounded, with Cauchy-Schwarz’s inequality, by

$$\begin{aligned}{} & {} {} \sum _{q,p \in \Lambda ^*_+:\, p \not = -q,q} |\eta _q| |q|^4 p^2 \Vert a_q a_p e^{t B(\eta )} \xi \Vert \Vert a_p e^{t B(\eta )} \xi \Vert \nonumber \\{}{} & {} {} \qquad \le C \ell ^{-3/2} W_{4,2}^{1/2} (t) \langle e^{t B(\eta )} \xi , \mathcal {P}^{(2)} e^{t B(\eta )} \xi \rangle ^{1/2} \nonumber \\ {}{}{} & {} {} \qquad \le W_{4,2} (t) + C \ell ^{-4} \langle \xi , ( 1 +\mathcal {P}^{(2)}) \xi \rangle \end{aligned}$$

(7.23)

where we applied (5.23) and we defined

$$\begin{aligned} W_{4,2} (t) = \sum _{p_1, p_2 \in \Lambda ^*_+:\, p_1 \not = -p_2, p_2} |p_1|^4 p_2^2 \, \Vert a_{p_1} a_{p_2} e^{t B(\eta )} \xi \Vert ^2.\end{aligned}$$

(7.24)

To compute the derivative of $W_{4,2}$, we proceed once again as in (7.12), noticing however that, because of the restriction to $p_1 \not = -p_2, p_2$, the contribution from the second term on the r.h.s. of (7.11) vanishes. We find, with (5.23),

$$\begin{aligned} \begin{aligned} \Big | \frac{dW_{4,2} (t)}{dt} \Big |&\le C \sum _{q,p \in \Lambda ^*_+:\, p \not = -q} | \eta _q | |q|^4 p^2 \Vert a_q a_p e^{t B(\eta )} \xi \Vert \Vert a_{-q}^* a_p e^{tB(\eta )} \xi \Vert \\ {}&\quad + C \sum _{q,p \in \Lambda ^*_+:\, p \not = -q} | \eta _q | q^2 |p|^4 \Vert a_q a_p e^{t B(\eta )} \xi \Vert \Vert a_{-q}^* a_p e^{tB(\eta )} \xi \Vert \\ {}&\le C W_{4,2} (t) + C \ell ^{-3} \langle e^{t B(\eta )} \xi , \mathcal {P}^{(2)} e^{t B(\eta )} \xi \rangle + C \ell ^{-1} \langle e^{t B(\eta )} \xi , \mathcal {P}^{(4)} e^{t B(\eta )} \xi \rangle \\ {}&\le C W_{4,2} (t) + C \ell ^{-4} \langle \xi , \big (1 + \mathcal {P}^{(4)} \big ) \xi \rangle \,. \end{aligned} \end{aligned}$$

By Gronwall’s lemma, we conclude (recalling (7.8)) that

$$\begin{aligned} W_{4,2} (t) \le C \ell ^{-4} \langle \xi , \big ( 1 + \mathcal {P}^{(4)} + \mathcal {Z}_{4,2} \big ) \xi \rangle \end{aligned}$$

(7.25)

for all $t \in [0;1]$. Inserting this estimate in (7.23), and then, together with (7.22), in (7.21), we obtain (using also that $\mathcal {P}^{(4)} \mathcal {N}_+ \le \mathcal {Z}_{4,2}$)

$$\begin{aligned}\Big | \frac{dH_3 (t)}{dt} \Big | \le C H_3 (t) + C \ell ^{-1} H_2 (t) + C \ell ^{-4} \langle \xi , \big ( 1 + \mathcal {P}^{(4)} + \mathcal {Z}_{4,2} \big ) \xi \rangle \,. \end{aligned}$$

With (7.20) and Gronwall’s lemma, we conclude that

$$\begin{aligned} H_3 (t) \le C \ell ^{-4} \langle \xi , \big ( 1 + \mathcal {T}^{(3)} + \mathcal {Z}_{4,2} + \mathcal {P}^{(4)} \big ) \xi \rangle \,. \end{aligned}$$

(7.26)

To control $H_4$, we proceed again as we did to show (7.21) and we bound

$$\begin{aligned} \begin{aligned} \Big | \frac{dH_4 (t)}{dt} \Big | \le \,&C H_4 (t) + C \ell ^{-1} H_3 (t) \\ {}&+ C \sum _{q, p_1, p_2 \in \Lambda ^*_+} |\eta _q| |q|^4 p_1^2 p_2^2 \Vert a_q a_{p_1} a_{p-2} e^{t B(\eta )} \xi \Vert \Vert a_{p_1} a_{p_2} e^{t B(\eta )} \xi \Vert \,.\end{aligned} \end{aligned}$$

In the last term, if $q = \pm p_1$ or $q = \pm p_2$, we find terms that can be bounded using (5.18) and (7.26) (and the trivial estimate $\mathcal {T}_2 \mathcal {N}_+ \le \mathcal {T}_3$) by

$$\begin{aligned}{} & {} {} \sum _{q, p \in \Lambda ^*_+} |\eta _q| |q|^6 p^2 \Vert a_q a_{\pm q} a_p e^{t B(\eta )} \xi \Vert \Vert a_{\pm q} a_p e^{t B(\eta )} \xi \Vert \nonumber \\{}{} & {} {} \qquad \le C \ell ^{-2} \sum _{q, p \in \Lambda ^*_+} q^2 p^2 \Vert a_q a_p \mathcal {N}_+^{1/2} e^{t B(\eta )} \xi \Vert \Vert a_{\pm q} a_p e^{t B(\eta )} \xi \Vert \nonumber \\ {}{}{} & {} {} \qquad \le C \ell ^{-2} \langle e^{t B(\eta )} \xi , \mathcal {T}_2 ( \mathcal {N}_+ + 1) e^{t B(\eta )} \xi \rangle \le C \ell ^{-6} \langle \xi , (1 + \mathcal {T}_3 + \mathcal {Z}_{4,2} + \mathcal {P}^{(4)} ) \xi \rangle \,.\nonumber \\ \end{aligned}$$

(7.27)

Contributions from terms with $q \not = \pm p_1, \pm p_2$, on the other hand, can be estimated (with (7.20)) by

$$\begin{aligned}{} & {} {} \sum _{q,p_1,p_2:\, q \not = \pm p_1, \pm p_2} |\eta _q| |q|^4 p_1^2 p_2^2 \Vert a_q a_{p_1} a_{p_2} e^{t B(\eta )} \xi \Vert \Vert a_{p_1} a_{p_2} e^{t B(\eta )} \xi \Vert \nonumber \\ {}{}{} & {} {} \qquad \le C \ell ^{-3/2} W_{4,2,2}^{1/2} (t) \langle e^{t B(\eta )} \xi , \mathcal {T}_2 \, e^{t B(\eta )} \xi \rangle ^{1/2}\nonumber \\ {}{}{} & {} {} \qquad \le C W_{4,2,2} (t) + C \ell ^{-6} \langle \xi , ( 1 + \mathcal {T}_2 + \mathcal {P}^{(4)} ) \xi \rangle \end{aligned}$$

(7.28)

where we defined

$$\begin{aligned} W_{4,2,2} (t) = \sum _{p_1,p_2,p_3 \in \Lambda ^*_+:\, p_1 \not = \pm p_2, \pm p_3} |p_1|^4 p_2^2 \, p_3^2 \, \Vert a_{p_1} a_{p_2} a_{p_3} e^{t B (\eta )} \xi \Vert ^2.\end{aligned}$$

We compute

$$\begin{aligned} \begin{aligned}&\Big | \frac{dW_{4,2,2} (t)}{dt} \Big |\\ {}&\qquad \quad \le \, C \sum _{q, p_2,p_3 \in \Lambda _+^*:\, q \not = \pm p_2, \pm p_3} |\eta _q| |q|^4 p_2^2 p_3^2 \Vert a_q a_{p_1} a_{p_2} e^{t B(\eta )} \xi \Vert \Vert a_{-q}^* a_{p_1} a_{p_2} e^{t B(\eta )} \xi \Vert \\ {}&\qquad \qquad + C \sum _{q,p_1, p_2 \in \Lambda ^*_+:\, p_1 \not = \pm q, \pm p_2} q^2 |\eta _q| |p_1|^4 p_2^2 \Vert a_q a_{p_1} a_{p_2} e^{t B(\eta )} \xi \Vert \Vert a_{-q}^* a_{p_1} a_{p_2} e^{t B (\eta )} \xi \Vert \\ {}&\qquad \qquad + C \sum _{q,p \in \Lambda ^*_+:\, q \not = \pm p} |p|^4 |q|^4 |\eta _q| \, \Vert a_q a_p e^{t B(\eta )} \xi \Vert \Vert a_{-q}^* a_p e^{t B(\eta )} \xi \Vert \end{aligned} \end{aligned}$$

which leads to

$$\begin{aligned} \Big | \frac{dW_{4,2,2} (t)}{dt} \Big | \le C W_{4,2,2} (t) + C \ell ^{-6} \langle \xi , (1 + \mathcal {P}^{(4)} + \mathcal {Z}_{4,2}) \xi \rangle + C W_{4,4} (t) \end{aligned}$$

(7.29)

where

$$\begin{aligned}W_{4,4} (t) = \sum _{q,p \in \Lambda ^*_+:\, q \not = \pm p} |q|^4 |p|^4 \Vert a_q a_p e^{t B(\eta )} \xi \Vert ^2 \end{aligned}$$

satisfies the estimate

$$\begin{aligned} \begin{aligned} \Big | \frac{dW_{4,4} (t)}{dt} \Big |&\le C \sum _{q,p \in \Lambda ^*_+:\,q \not = p} |p|^4 |q|^4 \eta _q \Vert a_q a_p e^{t B(\eta )} \xi \Vert \Vert a_{_q}^* a_p e^{t B(\eta )} \xi \Vert \\ {}&\le C W_{4,4} (t) + C \ell ^{-6} \langle \xi , (1 + \mathcal {P}^{(4)}) \xi \rangle \,.\end{aligned} \end{aligned}$$

Thus, recalling the definition (7.8), we find

$$\begin{aligned}W_{4,4} (t) \le C \ell ^{-6} \langle \xi , (1 + \mathcal {P}^{(4)}+ \mathcal {Z}_{4,4}) \xi \rangle \,.\end{aligned}$$

Inserting this bound in (7.29), we obtain

$$\begin{aligned}W_{4,2,2} (t) \le C \ell ^{-6} \langle \xi , (1 + \mathcal {P}^{(4)} + \mathcal {Z}_{4,2} + \mathcal {Z}_{4,2,2} ) \xi \rangle \,. \end{aligned}$$

Plugging the last equation in (7.28) and using (7.27), we arrive at

$$\begin{aligned} H_4 (t) \le C \ell ^{-6} \langle \xi , (1 + \mathcal {P}^{(4)} + \mathcal {Z}_{4,4} + \mathcal {Z}_{4,2,2} + \mathcal {T}_4 ) \xi \rangle \,. \end{aligned}$$

Finally, we prove (7.9). For $\varepsilon > -1$, $\delta \in (0;1/6)$ with $\varepsilon + \delta < 1$, we define

$$\begin{aligned} J^{(\varepsilon ,\delta )} (t) = \langle e^{t B(\eta )} \xi , \mathcal {S}^{(\varepsilon ,\delta )}_3 e^{t B(\eta )} \xi \rangle \,.\end{aligned}$$

Proceeding as in the proof of (7.15), we find

$$\begin{aligned}\begin{aligned}\Big | \frac{dJ^{(\varepsilon ,\delta )} (t)}{dt} \Big | \le \,&C J^{(\varepsilon ,\delta )} (t) + C \ell ^{-1-\varepsilon } F^{(\delta )}_2 (t) + C \ell ^{-1/2-\delta } G_2^{(\varepsilon ,\delta )} (t) \\ {}&+ \sum _{p,q \in \Lambda ^*_+} |\eta _q| |q|^{7/2+\varepsilon + \delta } |p|^{3/2+\delta } \Vert a_p a_q e^{t B(\eta )} \xi \Vert \Vert a_p e^{tB(\eta )} \xi \Vert \\ {}&+ \sum _{p,q \in \Lambda ^*_+} |\eta _q| |q|^{3+ 2\delta } |p|^{2+\varepsilon } \Vert a_p a_q e^{t B(\eta )} \xi \Vert \Vert a_p e^{tB(\eta )} \xi \Vert \,.\end{aligned} \end{aligned}$$

Recalling the definition (7.24), we can estimate (distinguishing $p = q$ from $p \not = q$)

$$\begin{aligned} \begin{aligned}\sum _{p,q \in \Lambda ^*_+}&|\eta _q| |q|^{7/2+\varepsilon + \delta } |p|^{3/2+\delta } \Vert a_p a_q e^{t B(\eta )} \xi \Vert \Vert a_p e^{tB(\eta )} \xi \Vert \\ \le \,&C \langle e^{t B(\eta )} \xi , \mathcal {P}^{(3+\varepsilon +2\delta )} (\mathcal {N}_+ + 1) e^{tB(\eta )} \xi \rangle \\ {}&+ C W_{4,2} (t)^{\frac{1}{2}} \Big ( \sum _{q \in \Lambda ^*_+} \eta _q^2 |q|^{3+2\varepsilon +2\delta } \Big )^{\frac{1}{2}} \langle e^{t B(\eta )} \xi , \mathcal {P}^{(1+2\delta )} e^{t B(\eta )} \xi \rangle ^{\frac{1}{2}} \\ \le \;&C \langle e^{t B(\eta )} \xi , \mathcal {P}^{(3+\varepsilon +2\delta )} e^{tB(\eta )} \xi \rangle + C \ell ^{-1-\varepsilon -\delta } W_{4,2} (t)^{\frac{1}{2}} \langle e^{t B(\eta )} \xi , \mathcal {P}^{(1+2\delta )} e^{t B(\eta )} \xi \rangle ^{\frac{1}{2}} \end{aligned} \end{aligned}$$

and, similarly,

$$\begin{aligned} \begin{aligned} \sum _{p,q \in \Lambda ^*_+}&|\eta _q| |q|^{3+ 2\delta } |p|^{2+\varepsilon } \Vert a_p a_q e^{t B(\eta )} \xi \Vert \Vert a_p e^{tB(\eta )} \xi \Vert \\ \le \;&C \langle e^{t B(\eta )} \xi , \mathcal {P}^{(3+\varepsilon +2\delta )} (\mathcal {N}_+ + 1) e^{tB(\eta )} \xi \rangle + C \ell ^{-\frac{1}{2} - 2\delta } W_{4,2} (t)^{\frac{1}{2}} \langle e^{t B(\eta )} \xi , \mathcal {P}^{(2+2\varepsilon )} e^{t B(\eta )} \xi \rangle ^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

With Lemma 5.2, with (7.14), (7.18) and (7.25), we conclude that

$$\begin{aligned} \Big | \frac{dJ^{(\varepsilon ,\delta )} (t)}{dt} \Big | \le \; C J^{(\varepsilon ,\delta )} (t) + C \ell ^{-3-\varepsilon - 2\delta } \Big \{ \langle \xi , \big (1+ \mathcal {P}^{(4)} + \mathcal {Z}_{4,2} \big ) \xi \rangle + \sup _{(\varepsilon , \delta ) \in \mathcal {I}_2} \langle \xi , \mathcal {S}_2^{(\varepsilon ,\delta )} \xi \rangle \Big \} \end{aligned}$$

for all $t \in [0;1]$. By Gronwall’s lemma, we obtain (7.9). $\square$

8 Proof of Theorem 1.1

With the unitary operator $U_N$ as in (5.2), with $\eta$ as introduced after (5.7) and $\tau$ as in (6.2), we define $\Phi _N \in L^2_s (\Lambda ^N)$ setting

$$\begin{aligned} \Phi _N = U_N^* e^{B(\eta )} e^{B(\tau )} \Omega . \end{aligned}$$

(8.1)

We recall that we assumed $N^{-1+\nu }\le \ell \le N^{-3/4-\nu }$ (see Prop. 4.1) and $\ell _0>0$ small enough (independent of N). From Prop. 6.4, we find that

$$\begin{aligned} \begin{aligned} \langle \Phi _N, H_N^\text {eff} \Phi _N \rangle&= \langle \Omega , \mathcal {M}^\text {eff}_{N,\ell } \Omega \rangle \\&= 4\pi \mathfrak {a} (N-1) + e_\Lambda \mathfrak {a}^2 \\&\quad -\frac{1}{2} \sum _{p \in \Lambda ^*_+} \Big [ p^2 + 8\pi \mathfrak {a} - \sqrt{|p|^4 + 16 \pi \mathfrak {a} p^2} - \frac{(8\pi \mathfrak {a})^2}{2p^2} \Big ] + \mathcal {O}(N^{-\varepsilon }) \end{aligned} \end{aligned}$$

(8.2)

for a sufficiently small $\varepsilon > 0$.

Additionally, with Lemma 7.1 we obtain important regularity estimates for $\Phi _N$. From (7.7) (and from (5.23) in Lemma 5.2), we find $C > 0$ such that

$$\begin{aligned} \langle \Phi _N, (-\Delta _{x_1}) \Phi _N \rangle \le&{} \frac{C}{N\ell } \nonumber \\ \langle \Phi _N, (-\Delta _{x_1})(-\Delta _{x_2}) \Phi _N \rangle \le&{} \frac{C}{N^2 \ell ^3} \nonumber \\ \langle \Phi _N, (-\Delta _{x_1})(- \Delta _{x_2})(- \Delta _{x_3}) \Phi _N \rangle \le&{} \frac{C}{N^3 \ell ^4} \nonumber \\ \langle \Phi _N, (-\Delta _{x_1})(- \Delta _{x_2})(- \Delta _{x_3})(-\Delta _{x_4}) \Phi _N \rangle \le&{} \frac{C}{N^4 \ell ^6}\,. \end{aligned}$$

(8.3)

From (7.5) we find, for $n \in \mathbb {N}$ and $0< \delta < 1/6$, a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned} \langle \Phi _N, (-\Delta _{x_1})^{3/4+\delta /2} \dots (-\Delta _{x_n})^{3/4+ \delta /2} \Phi _N \rangle&\le \frac{C}{N^n \ell ^{\alpha _n}}\,. \end{aligned} \end{aligned}$$

(8.4)

From (7.6) in Lemma 7.1, we find, for $n \in \mathbb {N}$ and for every $\varepsilon \in (-1;3)$, $\delta \in (0;1/6)$ such that $\varepsilon + 2\delta < 3/2^{n-1}$, a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned} \langle \Phi _N, (-\Delta _{x_1})^{1+\varepsilon /2} (- \Delta _{x_2})^{3/4+\delta /2} \dots (-\Delta _{x_n})^{3/4+\delta /2} \Phi _N \rangle&\le \frac{C}{N^n \ell ^{\beta ^\varepsilon _n}}\,. \end{aligned} \end{aligned}$$

(8.5)

Let us prove (8.5), the other bounds can be shown similarly. First of all, we symmetrize the expectation on the l.h.s. of (8.5), writing

$$\begin{aligned}\begin{aligned}\langle \Phi _N,&(-\Delta _{x_1})^{1+\varepsilon /2} (- \Delta _{x_2})^{3/4+\delta /2} \dots (-\Delta _{x_n})^{3/4+\delta /2} \Phi _N \rangle \\ {}&= \frac{1}{{N \atopwithdelims ()n}} \sum _{1 \le i_1< \cdots < i_n \le N} \langle \Phi _N, (-\Delta _{x_{i_1}})^{1+\varepsilon /2} (- \Delta _{x_{i_2}})^{3/4+\delta /2} \dots (-\Delta _{x_{i_n}})^{3/4+\delta /2} \Phi _N \rangle \,. \end{aligned} \end{aligned}$$

Next, we express the observable in second quantized form and we apply the rules (5.3). We find

$$\begin{aligned}{} & {} {} \langle \Phi _N, (-\Delta _{x_1})^{1+\varepsilon /2} (- \Delta _{x_2})^{3/4+\delta /2} \dots (-\Delta _{x_n})^{3/4+\delta /2} \Phi _N \rangle \\ {}{}{} & {} {} \qquad \qquad \le \frac{C}{N^n} \sum _{p_1, \dots , p_n \in \Lambda ^*_+} |p_1|^{2+\varepsilon } |p_2|^{3/2+\delta } \dots |p_n|^{3/2+\delta } \\{}{} & {} {} \qquad \qquad \quad \times \langle e^{B(\eta )} e^{B(\tau )} \Omega , a_{p_1}^* \dots a_{p_n}^* a_{p_n} \dots a_{p_1} e^{B(\eta )} e^{B(\tau )} \Omega \rangle \,. \end{aligned}$$

With (7.6), we conclude that

$$\begin{aligned}{} & {} {} \langle \Phi _N, (-\Delta _{x_1})^{1+\varepsilon /2} (- \Delta _{x_2})^{3/4+\delta /2} \dots (-\Delta _{x_n})^{3/4+\delta /2} \Phi _N \rangle \\{}{} & {} {} \qquad \qquad \le \frac{C}{N^n \ell ^{\beta _n^\varepsilon }} \Big \{ 1 + \sum _{j=1}^n \sup _{\varepsilon , \delta \in \mathcal {I}_j} \langle e^{B(\tau )} \Omega , S_j^{(\varepsilon ,\delta )} e^{B(\tau )} \Omega \rangle \Big \}\,. \end{aligned}$$

To control the growth of $S_j^{(\varepsilon ,\delta )}$, we can proceed exactly as in the proof of Lemma 7.1; the difference is that, by (6.3), $|\tau _p| \le C /|p|^4$, uniformly in $N,\ell$ (this should be compared with the bound (5.18), for the coefficients $\eta _p$). As a consequence, for $0< r< 5$, we find

$$\begin{aligned} \sum _{p \in \Lambda ^*_+} |p|^r |\tau _p|^2 \le C \end{aligned}$$

and thus the analog of the bounds in Lemma 7.1, with $B(\eta )$ replaced by $B(\tau )$, holds uniformly in $\ell$. This observation leads to (8.5).

With $\Phi _N$ as in (8.1), we define the trial function $\Psi _N \in L^2_s (\Lambda ^N)$ by

$$\begin{aligned}\Psi _N ({\textbf {x}}) = \Phi _N ({\textbf {x}}) \cdot \prod _{i<j}^N f_\ell (x_i - x_j)\,. \end{aligned}$$

The presence of the Jastrow factor guarantees that $\Psi _N$ satisfies the hard-sphere condition (1.8). Combining (2.12), Prop.  3.1 and Prop. 4.1, we obtain

$$\begin{aligned} \frac{\langle \Psi _N, \sum _{j=1}^N -\Delta _{x_j} \Psi _N \rangle }{\Vert \Psi _N \Vert ^2}\le & {} \langle \Phi _N, H_N^\text {eff} \Phi _N \rangle - \frac{N(N-1)}{2}\nonumber \\{} & {} \quad \Big \langle \Phi _N, \Big \{ \big [ H_{N-2}^\text {eff} - 4 \pi \mathfrak {a} N \big ] \otimes u_\ell (x_{N-1} - x_N) \Big \} \Phi _N \Big \rangle + C N^{-\varepsilon }.\nonumber \\ \end{aligned}$$

(8.6)

Here we used (8.3), (8.4) and (8.5) to verify the assumption (3.1) of Prop. 3.1 and the assumption (4.4) for Prop. 4.1. Moreover, we used (8.2) to verify the condition $\langle \Phi _N, H_N^\text {eff} \Phi _N \rangle \le 4 \pi \mathfrak {a} N + C$ in Prop. 4.1.

Inserting (8.2) on the r.h.s. of (8.6), we arrive at

$$\begin{aligned} \begin{aligned}&\frac{\langle \Psi _N, \sum _{j=1}^N -\Delta _{x_j} \Psi _N \rangle }{\Vert \Psi _N \Vert ^2} \\&\quad \le 4\pi \mathfrak {a} (N-1) + e_\Lambda \mathfrak {a}^2 -\frac{1}{2} \sum _{p \in \Lambda ^*_+} \Big [ p^2 + 8\pi \mathfrak {a} - \sqrt{|p|^4 + 16 \pi \mathfrak {a} p^2} - \frac{(8\pi \mathfrak {a})^2}{2p^2} \Big ] \\&\qquad - \frac{N(N-1)}{2} \langle \Phi _N, \Big \{ \big [ H_{N-2}^\text {eff} - 4 \pi \mathfrak {a} N \big ] \otimes u_\ell (x_{N-1} - x_N) \Big \} \Phi _N \Big \rangle + C N^{-\varepsilon }. \end{aligned} \end{aligned}$$

(8.7)

To conclude the proof of Theorem 1.1, we still have to show that the contribution on the last line is negligible, in the limit $N \rightarrow \infty$.

From (5.26) in Prop. 5.3, we find

$$\begin{aligned} H_{N-2}^\text{ eff } - 4 \pi \mathfrak {a} N \ge U_{N-2}^* e^{B(\eta )} \Big \{ - C (\mathcal {N}_+ + 1) - C N^{-\kappa } \mathcal {P}^{(2+\kappa )} (\mathcal {N}_+ + 1) \Big \} e^{-B(\eta )} U_{N-2}\nonumber \\ \end{aligned}$$

(8.8)

for $0< \kappa < \nu /2$. Notice here that both sides of the equation are operators on the Hilbert space $L^2_s (\Lambda ^{N-2})$ describing states with $(N-2)$ particles.

For $\mu > 0$ to be chosen small enough, we can estimate

$$\begin{aligned} (\mathcal {N}_+ + 1) \le C N^\mu + C (\mathcal {N}_+ + 1) \chi (\mathcal {N}_+ \ge N^\mu ) \le C N^\mu + C N^{-m\mu } (\mathcal {N}_+ + 1)^{m+1} \qquad \end{aligned}$$

(8.9)

for any $m \in \mathbb {N}$. Thus, the contribution arising from the first term in the parenthesis on the r.h.s. of (8.8) can be bounded by

$$\begin{aligned} \begin{aligned}&\frac{N (N-1)}{2} \Big \langle \Phi _N, \Big \{ \Big [ U_{N-2}^* e^{B(\eta )} (\mathcal {N}_+ + 1) e^{-B(\eta )} U_{N-2} \Big ] \otimes u_\ell (x_{N-1} - x_N) \Big \} \Phi _N \Big \rangle \\ {}&\le C N^{2+\mu } \langle \Phi _N, u_\ell (x_{N-1} - x_N) \Phi _N \rangle \\ {}&\quad + C N^{2-m\mu } \Big \langle \Phi _N, \Big \{ \Big [ U_{N-2}^* e^{B(\eta )} (\mathcal {N}_+ + 1)^{m+1} e^{-B(\eta )} U_{N-2} \Big ] \otimes u_\ell (x_{N-1} - x_N) \Big \} \Phi _N \Big \rangle \,. \end{aligned} \end{aligned}$$

Using $\Vert u_\ell \Vert _1 \le C\ell ^2 / N$ and (3.3) in the first and $\Vert u_\ell \Vert _\infty \le C$ in the second term (by Lemma 2.1), we obtain

$$\begin{aligned} \begin{aligned} \frac{N (N-1)}{2} \Big \langle \Phi _N, \Big \{ \Big [ U_{N-2}^* e^{B(\eta )}&(\mathcal {N}_+ + 1) e^{-B(\eta )} U_{N-2} \Big ] \otimes u_\ell (x_{N-1} - x_N) \Big \} \Phi _N \Big \rangle \\ {}&\le C N^{1+\mu } \ell ^2 \langle \Phi _N, (1-\Delta _{x_1}) ( 1-\Delta _{x_2}) \Phi _N \rangle \\ {}&\quad + C N^{2-m\mu } \Big \langle e^{B(\eta )} e^{B(\tau )} \Omega , (\mathcal {N}_+ + 1)^{m+1} e^{B(\eta )} e^{B(\tau )} \Omega \Big \rangle \,. \end{aligned} \end{aligned}$$

Here we used Lemma 5.2 to control the growth of $(\mathcal {N}_+ +1)^{m+1}$ under the action of $B(\eta )$. Moreover, with $\mathfrak {q} = 1 - |\varphi _0 \rangle \langle \varphi _0|$ denoting the projection onto the orthogonal complement to the condensate wave function $\varphi _0$ in $L^2 (\Lambda )$ and with $\mathfrak {q}_j = 1 \otimes \cdots \otimes \mathfrak {q} \otimes \cdots \otimes 1$ acting as $\mathfrak {q}$ on the j-th particle, we estimated, on the N-particle space $L^2_s (\Lambda ^N)$,

$$\begin{aligned} U_{N-2}^* \, \mathcal {N}_+ U_{N-2} \otimes 1 = \sum _{j=1}^{N-2} \mathfrak {q}_j \le \sum _{j=1}^N \mathfrak {q}_j = U^*_N \mathcal {N}_+ U_N \end{aligned}$$

(8.10)

(with a slight abuse of notation, $\mathcal {N}_+$ denotes the number of particles operators on $\mathcal {F}^{\le (N-2)}_+$ on the l.h.s. and the number of particles operator on $\mathcal {F}^{\le N}_+$ on the r.h.s.). Using again Lemma 5.2 (and Lemma 6.2, for the action of $B(\tau )$), together with the bounds in (8.3), we conclude that

$$\begin{aligned} \begin{aligned}&\frac{N (N-1)}{2} \Big \langle \Phi _N, \Big \{ \Big [ U_{N-2}^* e^{B(\eta )} (\mathcal {N}_+ + 1) e^{-B(\eta )} U_{N-2} \Big ] \otimes u_\ell (x_{N-1} - x_N) \Big \} \Phi _N \Big \rangle \\&\qquad \qquad \qquad \quad \le N^{1+\mu } \ell ^2 \Big ( 1 + \frac{1}{N^2\ell ^3} \Big ) + C N^{2-m\mu } \le C N^{-\varepsilon } \end{aligned}\nonumber \\ \end{aligned}$$

(8.11)

choosing first $\mu > 0$ small enough and then $m \in \mathbb {N}$ large enough.

Let us now focus on the contribution of the second term in the parenthesis on the r.h.s. of (8.8). Also here, we use (8.9) to estimate

$$\begin{aligned} \begin{aligned}&\frac{N^{1-\kappa } (N-1)}{2} \Big \langle \Phi _N, \Big \{ \Big [ U_{N-2}^* e^{B(\eta )} \mathcal {P}^{(2+\kappa )} (\mathcal {N}_+ + 1) e^{-B(\eta )} U_{N-2} \Big ] \\ {}&\qquad \otimes u_\ell (x_{N-1} - x_N) \Big \} \Phi _N \Big \rangle \\ {}&\qquad \quad \le C N^{2-\kappa +\mu } \big \langle \Phi _N, \big \{ \big [ U_{N-2}^* e^{B(\eta )} \mathcal {P}^{(2+\kappa )} e^{-B(\eta )} U_{N-2} \big ] \otimes u_\ell (x_{N-1} - x_N) \big \} \Phi _N \big \rangle \\ {}&\qquad \qquad + C N^{2-\kappa -m \mu } \big \langle \Phi _N, \big \{ \big [ U_{N-2}^* e^{B(\eta )} \mathcal {P}^{(2+\kappa )} (\mathcal {N}_+ + 1)^{m+1} e^{-B(\eta )} U_{N-2} \big ] \\ {}&\qquad \otimes u_\ell (x_{N-1} - x_N) \big \} \Phi _N \big \rangle \\ {}&\qquad \quad = \text{ R}_1 + \text{ R}_2\,.\end{aligned} \end{aligned}$$

To bound $\text {R}_2$, we can estimate $\Vert u_\ell \Vert _\infty \le C$, we can apply Lemma 5.2 to control the growth of $\mathcal {P}^{(2+\kappa )} (\mathcal {N}_+ + 1)^{m+1}$ under conjugation with $e^{B(\eta )}$ and we can proceed similarly as in (8.10) to replace $U_{N-2}$ with $U_N$. We find

$$\begin{aligned} \text{ R}_2 \le CN^{2-\kappa -m\mu } \big \langle e^{B(\eta )} e^{B(\tau )} \Omega , \big ( \mathcal {P}^{(2+\kappa )} + \ell ^{-1-\kappa } \big ) (\mathcal {N}_+ + 1)^{m+1} e^{B(\eta )} e^{B(\tau )} \Omega \big \rangle \,. \end{aligned}$$

Applying again Lemma 5.2 (and then Lemma 6.2 for the action of $B(\tau )$), we conclude that

$$\begin{aligned} \text{ R}_{2} \le C N^{2-\kappa -m\mu } \ell ^{-1-\kappa }\,. \end{aligned}$$

(8.12)

As for the term $\text {R}_1$, we first use (3.3) in Lemma 3.2 to estimate, for $\delta > 0$ small enough,

$$\begin{aligned} \text{ R}_1\le & {} {} C N^{2-\kappa +\mu } \Vert u_\ell \Vert _1 \big \langle \Phi _N, \big \{ \big [ U_{N-2}^* e^{B(\eta )} \mathcal {P}^{(2+\kappa )} e^{-B(\eta )} U_{N-2} \big ] \otimes 1 \big \} \Phi _N \big \rangle \nonumber \\{}{} & {} {} + C N^{2-\kappa +\mu } \Vert u_\ell \Vert _1 \nonumber \\{}{} & {} {} \times \big \langle \Phi _N, \big \{ \big [ U_{N-2}^* e^{B(\eta )} \mathcal {P}^{(2+\kappa )} e^{-B(\eta )} U_{N-2} \big ] \otimes (\Delta _{x_{N-1}}\Delta _{x_N})^{3/4+\delta /2} \big \} \Phi _N \big \rangle \nonumber \\ {}= & {} {} \text{ R}_{11} + \text{ R}_{12}\,. \end{aligned}$$

(8.13)

To control $\text {R}_{12}$, we apply Lemma 5.2 to bound

$$\begin{aligned} \begin{aligned} U_{N-2}^* e^{B(\eta )} \mathcal {P}^{(2+\kappa )} e^{-B(\eta )} U_{N-2}&\le C U_{N-2}^* \big [ \mathcal {P}^{(2+\kappa )} + \ell ^{-1-\kappa } \big ] U_{N-2} \\ {}&= C \Big [ \sum _{j=1}^{N-2} (-\Delta _{x_j})^{1+\kappa /2} + \ell ^{-1-\kappa } \Big ]\,. \end{aligned} \end{aligned}$$

Thus

$$\begin{aligned} \begin{aligned} \text{ R}_{12} \le C N^{-1-\kappa +\mu } \ell ^2 \big \langle e^{B(\eta )} e^{B(\tau )} \Omega , \big [ S^{(\kappa , \delta )}_3 + \ell ^{-1-\kappa } \mathcal {A}^{(\delta )}_2 \big ] e^{B(\eta )} e^{B(\tau )} \Omega \big \rangle \,.\end{aligned} \end{aligned}$$

With (7.5) and with (7.9) from Lemma 7.1, we conclude that

$$\begin{aligned} \text {R}_{12} \le C N^\mu N^{-\widetilde{\varepsilon }} \end{aligned}$$

(8.14)

for some $\widetilde{\varepsilon } > 0$, if $\delta$ is chosen small enough, and $0< \kappa < \nu /2$.

It turns out that the term $\text {R}_{11}$ is more subtle; here we cannot afford the error arising from conjugation of $\mathcal {P}^{(2+\kappa )}$ with $e^{-B(\eta )}$. Instead, we have to use the fact that we conjugate back with $e^{B(\eta )}$ when we take expectation in the state $\Phi _N = e^{B(\eta )} e^{B(\tau )} \Omega$. The two generalized Bogoliubov transformations do not cancel identically (because one acts on $(N-2)$ particles, the other on N), but of course their combined action produces a much smaller error. We will make use of the following lemma.

Lemma 8.1

For $r\in (1;4]$ we have

$$\begin{aligned} e^{-B(\eta )} {\mathcal {P}}^{(r)}e^{B(\eta )}= {\mathcal {P}}^{(r)}+ \sum _{p \in \Lambda ^*_+} |p|^r \eta _p\left( b^*_p b^*_{-p}+\mathrm {h.c.} \right) +\sum _{p \in \Lambda ^*_+} |p|^r \eta _p^2 + {\mathcal {X}}_1\end{aligned}$$

(8.15)

with

$$\begin{aligned} \pm {\mathcal {X}}_1 \le C (\mathcal {N}_+ + 1) + C N^{-1} \big ( \mathcal {P}^{(r)} + \ell ^{1-r} \big ) (\mathcal {N}_+ + 1). \end{aligned}$$

Moreover,

$$\begin{aligned}{} & {} e^{-B(\eta )} \sum _{p \in \Lambda ^*_+} |p|^r \eta _p(b^*_p b_{-p}^* + \mathrm {h.c.}) e^{B(\eta )}\nonumber \\ {}{} & {} \quad = \sum _{p \in \Lambda ^*_+} |p|^r \eta _p \big [ b^*_p b_{-p}^*+b_p b_{-p} \big ]+2 \sum _{p \in \Lambda ^*_+} |p|^r \eta _p^2+ {\mathcal {X}}_2 \end{aligned}$$

(8.16)

with

$$\begin{aligned} \pm {\mathcal {X}}_2 \le C (\mathcal {N}_+ + 1) + C N^{-1} \big ( \mathcal {P}^{(r)} + \ell ^{1-r} \big ) (\mathcal {N}_+ + 1). \end{aligned}$$

We defer the proof of Lemma 8.1 to the end of the section, showing first how it can be used to estimate the error $\text {R}_{11}$ and to conclude the proof of Theorem 1.1. Notice first that $2+\kappa \le 4$ since $\kappa <\nu /2$ and $\nu$ is small enough. We can therefore apply Lemma 8.1 to find

$$\begin{aligned} \begin{aligned} \langle \Phi _N,&\big \{ \big [ U_{N-2}^* e^{B(\eta )} \mathcal {P}^{(2+\kappa )} e^{-B(\eta )} U_{N-2} \big ] \otimes 1 \big \} \Phi _N \rangle \\ \le \;&\big \langle \Phi _N \big \{ U_{N-2}^* \mathcal {P}^{(2+\kappa )} U_{N-2} \otimes 1 \big \} \Phi _N \rangle + \sum _{p \in \Lambda ^*_+} |p|^{2+\kappa } \eta _p^2 \\&\quad + \sum _{p \in \Lambda ^*_+} |p|^{2+\kappa } \eta _p \big \langle \Phi _N \big \{ U_{N-2}^* \big [ b_p^* b_{-p}^* + \text {h.c.} \big ] U_{N-2} \otimes 1 \big \} \Phi _N \rangle \\&\quad + C \langle \Phi _N, \big \{ U_{N-2}^* \big [ 1+N^{-1}\big ({\mathcal {P}}^{(2+\kappa )} + \ell ^{-1-\kappa }\big )\big ] ({\mathcal {N}}_++1) U_{N-2} \otimes 1\big \} \Phi _N \rangle . \end{aligned} \end{aligned}$$

We observe that

$$\begin{aligned} U_{N-2}^* \mathcal {P}^{(2+\kappa )} U_{N-2} \otimes 1 = \sum _{j=1}^{N-2} (-\Delta _{x_j})^{2+\kappa } \le \sum _{j=1}^{N} (-\Delta _{x_j})^{2+\kappa } = U_N^* \mathcal {P}^{(2+\kappa )} U_N \end{aligned}$$

and that, similarly,

$$\begin{aligned}\begin{aligned} U_{N-2}^*\big [ 1+N^{-1}\big ( {\mathcal {P}}^{(2+\kappa )}&+ \ell ^{-1-\kappa }\big )\big ]({\mathcal {N}}_++1) U_{N-2}\\\le \;&U_N^*\big [ 1+N^{-1}\big ({\mathcal {P}}^{(2+\kappa )} + \ell ^{-1-\kappa } \big )\big ]({\mathcal {N}}_++1) U_N.\end{aligned} \end{aligned}$$

Moreover, we find

$$\begin{aligned} \begin{aligned} \sum _{p \in \Lambda ^*_+} |p|^{2+\kappa } \eta _p U_{N-2}^* \big [ b_p^* b_{-p}^* + \text {h.c.} \big ] U_{N-2}&= \frac{1}{N-2} \sum _{p \in \Lambda ^*_+} |p|^{2+\kappa } \eta _p \big [ a_p^* a_{-p}^* a_0 a_0 + \text {h.c.} \big ] \\&= \frac{1}{N-2} \sum _{i<j}^{N-2} \big [ \theta (x_i - x_j) (\mathfrak {p}_i \otimes \mathfrak {p}_j) + \text {h.c.} \big ] \\ {}&= \frac{1}{N-2} \sum _{i<j}^{N} \big [ \theta (x_i - x_j) (\mathfrak {p}_i \otimes \mathfrak {p}_j) + \text {h.c.} \big ] \\&\quad - \frac{1}{N-2} \sum _{{i< j: j = N-1, N}} \big [ \theta (x_i - x_j) (\mathfrak {p}_i \otimes \mathfrak {p}_j) + \text {h.c.} \big ] \end{aligned} \end{aligned}$$

with $\theta$ defined by the Fourier coefficients ${\hat{\theta }}_p = |p|^{2+\kappa } \eta _p$, and with $\mathfrak {p}_j$ denoting the orthogonal projection $\mathfrak {p} = |\varphi _0 \rangle \langle \varphi _0|$ on the condensate wave function acting on the j-particle. Rewriting the first term in second quantized form (but now, on the N-particle space), we find

$$\begin{aligned} \begin{aligned} \sum _{p \in \Lambda ^*_+} |p|^{2+\kappa } \eta _p U_{N-2}^* \big [&b_p^* b_{-p}^* + \text {h.c.} \big ] U_{N-2} \\&= \frac{N}{N-2} \sum _{p \in \Lambda ^*_+} |p|^{2+\kappa } \eta _p U_N^* \big [ b_p^* b_{-p}^* + \text {h.c.} \big ] U_N \\&\quad - \frac{1}{N-2} \sum _{i < j: j=N-1, N} \big [ \theta (x_i - x_j) (\mathfrak {p}_i \otimes \mathfrak {p}_j) + \text {h.c.} \big ]. \end{aligned} \end{aligned}$$

Therefore, we find

$$\begin{aligned}{} & {} \big \langle \Phi _N, \big \{ \big [ U_{N-2}^* e^{B(\eta )} \mathcal {P}^{(2+\kappa )} e^{-B(\eta )} U_{N-2} \big ] \otimes 1 \big \} \Phi _N \big \rangle \nonumber \\ {}{} & {} \qquad \qquad \le \; \big \langle e^{B(\eta )} e^{B(\tau )} \Omega , \mathcal {P}^{(2+\kappa )} e^{B(\eta )} e^{B(\tau )} \Omega \big \rangle + \sum _{p \in \Lambda ^*_+} |p|^{2+\kappa } \eta _p^2 \nonumber \\ {}{} & {} \qquad \qquad \qquad - \frac{N}{N-2} \sum _{p \in \Lambda ^*_+} |p|^{2+\kappa } \eta _p \big \langle e^{B(\eta )} e^{B(\tau )} \Omega , \big ( b_p^* b_{-p}^* + b_p b_{-p} \big ) e^{B(\eta )} e^{B(\tau )} \Omega \big \rangle \nonumber \\ {}{} & {} \qquad \qquad \qquad + \frac{1}{N-2} \sum _{i < j: j=N-1, N} \big \langle \Phi _N, \big [ \theta (x_i - x_j) (\mathfrak {p}_i \otimes \mathfrak {p}_j) + \text {h.c.} \big ] \Phi _N \rangle \nonumber \\{} & {} \qquad \qquad \quad +C \langle e^{B(\eta )} e^{B(\tau )} \Omega ,\big [ 1+N^{-1}\big ( {\mathcal {P}}^{(2+\kappa )} + \ell ^{-1-\kappa } \big )\big ]({\mathcal {N}}_++1)e^{B(\eta )} e^{B(\tau )} \Omega \rangle .\nonumber \\ \end{aligned}$$

(8.17)

Applying again Lemma 8.1 to the first and third terms on the r.h.s. of (8.17), and Lemma 5.2 to the last, we obtain

$$\begin{aligned} \begin{aligned} \big \langle \Phi _N, \big \{ \big [ U_{N-2}^* e^{B(\eta )}&\mathcal {P}^{(2+\kappa )} e^{-B(\eta )} U_{N-2} \big ] \otimes 1 \big \} \Phi _N \big \rangle \\ \le \;&\big \langle e^{B(\tau )} \Omega , \mathcal {P}^{(2+\kappa )} e^{B(\tau )} \Omega \big \rangle + \left[ 2 - \frac{2N}{N-2} \right] \sum _{p \in \Lambda ^*_+} |p|^{2+\kappa } \eta _p^2 \\ {}&- \frac{2}{N-2} \sum _{p \in \Lambda ^*_+} |p|^{2+\kappa } \eta _p \langle e^{B(\tau )} \Omega , \big [ b_p^* b_{-p}^* + b_p b_{-p} \big ] e^{B(\tau )} \Omega \rangle \\&+ \frac{1}{N-2} \sum _{i < j: j=N-1, N} \big \langle \Phi _N, \big [ \theta (x_i - x_j) (\mathfrak {p}_i \otimes \mathfrak {p}_j) + \text {h.c.} \big ] \Phi _N \big \rangle \\&+ C \langle e^{B(\tau )} \Omega , \big [ 1+N^{-1}\big ({\mathcal {P}}^{(2+\kappa )} + \ell ^{-1-\kappa } \big )\big ]({\mathcal {N}}_++1) e^{B(\tau )} \Omega \rangle . \end{aligned}\nonumber \\ \end{aligned}$$

(8.18)

With the properties of $\tau$ (see Lemma 6.2) it is easy to check that all expectations in the state $e^{B(\tau )} \Omega$ are bounded, uniformly in $N,\ell$. Moreover, by (5.19), we find

$$\begin{aligned} \left[ 2 - \frac{2N}{N-2} \right] \sum _{p \in \Lambda ^*_+} |p|^{2+\kappa } \eta _p^2 \le \frac{C}{N \ell ^{1+\kappa }} . \end{aligned}$$

Finally, we can estimate the term on the fourth line in (8.18) by

$$\begin{aligned}{} & {} \Big | \frac{1}{N-2} \sum _{i < j: j=N-1, N} \big \langle \Phi _N, \big [ \theta (x_i - x_j) (\mathfrak {p}_i \otimes \mathfrak {p}_j) + \text {h.c.} \big ] \Phi _N \rangle \Big |\\ {}{} & {} \quad \le C \big | \langle \Phi _N, \theta (x_1 - x_2) (\mathfrak {p}_1 \otimes \mathfrak {p}_2) \Phi _N \rangle \big |.\end{aligned}$$

Since $\mathfrak {p}_1 \theta (x_1 - x_2) \mathfrak {p}_1 = \mathfrak {p}_1 \hat{\theta }_0 = 0$ and, similarly, $\mathfrak {p}_2 \theta (x_1 - x_2) \mathfrak {p}_2 = 0$, we have

$$\begin{aligned} \begin{aligned} \big | \langle \Phi _N, \theta (x_1 - x_2) (\mathfrak {p}_1 \otimes \mathfrak {p}_2) \Phi _N \rangle \big |&= \big | \langle \Phi _N, (\mathfrak {q}_1 \otimes \mathfrak {q}_2) \theta (x_1 - x_2) (\mathfrak {p}_1 \otimes \mathfrak {p}_2) \Phi _N \rangle \big | \\ {}&\le \Vert \theta \Vert _2 \Vert (\mathfrak {q}_1 \otimes \mathfrak {q}_2) \Phi _N \Vert \Vert \Phi _N \Vert .\end{aligned} \end{aligned}$$

With $\Vert \theta \Vert _2 = \Vert \hat{\theta } \Vert _2 \le C \ell ^{-3/2 -\kappa }$, for $0< \kappa < 1/2$, and with

$$\begin{aligned} \begin{aligned} \Vert (\mathfrak {q}_1 \otimes \mathfrak {q}_2) \Phi _N \Vert ^2&\le C N^{-2} \big \langle \Phi _N, \big [ \sum _{i=1}^N \mathfrak {q}_i \big ]^2 \Phi _N \big \rangle \\ {}&= C N^{-2} \langle e^{B(\eta )} e^{B(\tau )} \Omega , (\mathcal {N}_+ + 1)^2 e^{B(\eta )} e^{B(\tau )} \Omega \rangle \le C N^{-2}\end{aligned} \end{aligned}$$

we conclude that

$$\begin{aligned} \Big | \frac{1}{N-2} \sum _{i < j: j=N-1, N} \big \langle \Phi _N, \big [ \theta (x_i - x_j) (\mathfrak {p}_i \otimes \mathfrak {p}_j) + \text {h.c.} \big ] \Phi _N \rangle \Big | \le \frac{C}{N \ell ^{3/2+\kappa }}. \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} \Big | \big \langle \Phi _N, \big \{ \big [ U_{N-2}^* e^{B(\eta )} \mathcal {P}^{(2+\kappa )} e^{-B(\eta )} U_{N-2} \big ] \otimes 1 \big \} \Phi _N \big \rangle \Big | \le \frac{C}{N\ell ^{3/2+\kappa }} \end{aligned}$$

for $\ell \le N^{-2/3}$. Since $\Vert u_\ell \Vert _1 \le C \ell ^2 /N$ by Lemma 2.1, the error term $\text {R}_{11}$ introduced in (8.13) is bounded by

$$\begin{aligned} \text {R}_{11} \le C N^{-\kappa + \mu } \ell ^{1/2-\kappa } \le C N^\mu \ell ^{1/2}. \end{aligned}$$

With (8.14), we find

$$\begin{aligned} \text {R}_1 \le C N^\mu N^{-\widetilde{\varepsilon }} \end{aligned}$$

for $\widetilde{\varepsilon } > 0$ small enough. Combining this bound with (8.12) we conclude, choosing first $\mu > 0$ small enough and then $m \in \mathbb {N}$ sufficiently large, that

$$\begin{aligned}{} & {} \frac{N(N-1)}{2} \Big | \big \langle \Phi _N, \big \{ \big [ U_{N-2}^* e^{B(\eta )} \mathcal {P}^{(2+\kappa )} (\mathcal {N}_+ + 1) e^{-B(\eta )} U_{N-2} \big ] \otimes u_\ell (x_{N-1} - x_N) \big \} \Phi _N \big \rangle \Big | \\ {}{} & {} \quad \le C N^{-\varepsilon } \end{aligned}$$

for a sufficiently small $\varepsilon > 0$. Together with (8.8) and (8.11), this estimate implies that

$$\begin{aligned} - \frac{N(N-1)}{2} \big \langle \Phi _N, \big \{ \big [ H_{N-2}^\text {eff} - 4 \pi \mathfrak {a} N \big ] \otimes u_\ell (x_{N-1} - x_N) \big \} \Phi _N \big \rangle \le C N^{-\varepsilon } .\end{aligned}$$

From (8.7), we obtain

$$\begin{aligned} \begin{aligned}&\frac{\langle \Psi _N, \sum _{j=1}^N -\Delta _{x_j} \Psi _N \rangle }{\Vert \Psi _N \Vert ^2} \le 4\pi \mathfrak {a} (N-1) + e_\Lambda \mathfrak {a}^2 \\&\quad -\frac{1}{2} \sum _{p \in \Lambda ^*_+} \Big [ p^2 + 8\pi \mathfrak {a} - \sqrt{|p|^4 + 16 \pi \mathfrak {a} p^2} - \frac{(8\pi \mathfrak {a})^2}{2p^2} \Big ] + C N^{-\varepsilon }.\end{aligned} \end{aligned}$$

We conclude the proof of Theorem 1.1 by giving the proof of Lemma 8.1.

Proof (of Lemma 8.1)

With (5.5), we can compute $[{\mathcal {P}}^{(r)},B(\eta )]$ to show that

$$\begin{aligned} e^{-B(\eta )} {\mathcal {P}}^{(r)} e^{B(\eta )} = {\mathcal {P}}^{(r)}+ \int _0^1 ds \,e^{-sB(\eta )} \sum _{p \in \Lambda ^*_+} |p|^r \eta _p \Big [ b^*_p b_{-p}^*+b_p b_{-p} \Big ]e^{sB(\eta )}.\nonumber \\ \end{aligned}$$

(8.19)

Furthermore, expanding the integrand on the r.h.s. of (8.19), we write

$$\begin{aligned} \begin{aligned} e^{-sB(\eta )} \sum _{p \in \Lambda ^*_+} |p|^r&\eta _p \Big [ b^*_p b_{-p}^*+b_p b_{-p} \Big ]e^{sB(\eta )} \\ =\;&\sum _{p \in \Lambda ^*_+} |p|^r \eta _p \Big [ b^*_p b_{-p}^*+b_p b_{-p} \Big ] \\&+ \int _0^s dt\, e^{-tB(\eta )} \sum _{p \in \Lambda ^*_+} |p|^r \eta _p \Big [ b^*_p b^*_{-p}+b_p b_{-p},B(\eta ) \Big ]e^{tB(\eta )}. \end{aligned}\nonumber \\ \end{aligned}$$

(8.20)

Let us compute the last commutator. With (5.4), we find

$$\begin{aligned} \begin{aligned} \sum _{p \in \Lambda ^*_+} |p|^r \eta _p \Big [ b^*_p b^*_{-p} + b_p b_{-p},B(\eta ) \Big ]=\;&\frac{1}{2} \sum _{p,q \in \Lambda ^*_+} |p|^r \eta _p \eta _q \big [ b_p b_{-p},b^*_q b^*_{-q} \big ]+\mathrm {h.c.}\\ =\;&2\sum _{p \in \Lambda ^*_+}|p|^r \eta _p^2+ \Xi \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} \Xi =\;&4 \sum _{p\in \Lambda ^*_+} |p|^r \eta _p^2 a^*_p \left[ \left( 1-\frac{{\mathcal {N}}_++2}{N} \right) \left( 1-\frac{ {\mathcal {N}}_++1}{N} \right) -\frac{1}{2N^2}\right] a_p\\&+ 2\sum _{p \in \Lambda ^*_+}|p|^r \eta _p^2 \left[ \left( 1-{{\mathcal {N}}_++1\over N} \right) \left( 1-{{\mathcal {N}}_+\over N} \right) -1\right] \\&-\frac{1}{N} \sum _{p,q \in \Lambda ^*_+}|p|^r \eta _p \eta _q a^*_q a^*_{-q} \left[ 2\left( 1-\frac{{\mathcal {N}}_+}{N}\right) -\frac{3}{N}\right] a_p a_{-p}+\mathrm {h.c.}. \end{aligned} \end{aligned}$$

To control the last term, we write $a_q^* a_{-q}^* a_p a_{-p} = a_q^* a_p a_{-q}^* a_{-p} - \delta _{-q,p} a_q^* a_{-p}$ and we bound, for an arbitrary $\xi \in \mathcal {F}_+^{\le N}$,

$$\begin{aligned} \begin{aligned} \Big | \frac{1}{N} \sum _{p,q \in \Lambda ^*_+} |p|^r&\eta _p \eta _q \langle \xi , a_q^* a_p a_{-q}^* a_{-p} \xi \rangle \Big | \\ {}&\le \frac{1}{N} \sum _{p,q \in \Lambda ^*_+} |p|^r |\eta _p| |\eta _q| \Vert a_p^* a_q \xi \Vert \Vert a_{-q}^* a_{-p} \xi \Vert \\ {}&\le \frac{1}{N} \sum _{p,q \in \Lambda ^*_+} |p|^r |\eta _p| |\eta _q| \big [ \Vert a_p a_q \xi \Vert + \Vert a_q \xi \Vert \big ] \big [ \Vert a_{-q} a_{-p} \xi \Vert + \Vert a_{-p} \xi \Vert \big ] .\end{aligned} \end{aligned}$$

With Cauchy-Schwarz’s inequality and with the bounds $r \le 4$, $|\eta _p|\le C|p|^{-2}$, we find

$$\begin{aligned} \pm \Xi \le C (\mathcal {N}_+ + 1) + C N^{-1} \left( \mathcal {P}^{(r)} + \ell ^{1-r} \right) (\mathcal {N}_+ + 1). \end{aligned}$$

Inserting this back in (8.20) and using (5.23) we obtain

$$\begin{aligned} e^{-sB(\eta )} \sum _{p \in \Lambda ^*_+} |p|^r \eta _p \Big [ b^*_p b_{-p}^*+b_p b_{-p} \Big ]e^{sB(\eta )}= & {} \sum _{p \in \Lambda ^*_+} |p|^r \eta _p \Big [ b^*_p b_{-p}^*+b_p b_{-p} \Big ]\\{} & {} +2s \sum _{p \in \Lambda ^*_+} |p|^r \eta _p^2+ \widetilde{\Xi } \end{aligned}$$

again with

$$\begin{aligned} \pm \widetilde{\Xi } \le C (\mathcal {N}_+ + 1) + C N^{-1} \left( \mathcal {P}^{(r)} + \ell ^{1-r} \right) (\mathcal {N}_+ + 1). \end{aligned}$$

Setting $s=1$, this proves (8.16). Plugging now (8.16) in (8.19) and integrating over s we find (8.15). $\square$

Appendix A: Properties of One-Particle Scattering Equations

In this section we provide the proof of Lemma 2.1 and Lemma 5.1. We start with Lemma 2.1, where we describe properties of the solution of the eigenvalue equation (2.2).

Proof of Lemma 2.1

By standard arguments, the ground state solution of (2.1) is radial. Thus, we consider the ansatz $f_\ell (x) = m_\ell (|x|) / |x|$, which leads to the equation

$$\begin{aligned} m'' (r) + \lambda _\ell m (r) = 0 \end{aligned}$$

for $r \in [\text{\AA} /N; \ell ]$, with the boundary conditions $m (\mathfrak {a}/N) = 0$, $m' (\ell ) = 1$ and $m(\ell ) = \ell$. From $m (\text{\AA} /N) = 0$ and $m (\ell ) = \ell$, we obtain

$$\begin{aligned} m (r)= \frac{\ell \, \sin (\sqrt{\lambda }_\ell (r-\mathfrak {a}/N))}{\sin (\sqrt{\lambda }_\ell (\ell -\mathfrak {a}/N))} \end{aligned}$$

for all $r \in [\text{\AA} /N; \ell ]$. This proves (2.5). Imposing $m' (\ell ) = 1$, we arrive at

$$\begin{aligned} \tan \big ( \sqrt{\lambda _\ell }\, (\ell - \mathfrak {a}/N) \big ) = \sqrt{\lambda _\ell } \,\ell \end{aligned}$$

(A.1)

which shows (2.3). This equation allows us to estimate the eigenvalue $\lambda _\ell$. As already shown in [18, Lemma A.1], we find

$$\begin{aligned} \lambda _\ell = \frac{3\mathfrak {a}}{N \ell ^3} \big ( 1 + \mathcal {O}(\mathfrak {a}/ N \ell ) \big ) \end{aligned}$$

(A.2)

which implies that $\sqrt{\lambda _\ell } (\ell -\text{\AA} /N) \simeq \sqrt{\lambda _\ell } \ell \simeq (N \ell )^{-1/2} \ll 1$. With $\tan s = s + s^3 /3 + 2\,s^5 /15 + \mathcal {O}(s^7)$, we obtain

$$\begin{aligned} \sqrt{\lambda _\ell } \,\ell = \sqrt{\lambda _\ell } \big ( \ell - \mathfrak {a}/N \big ) + \frac{1}{3} \lambda _\ell ^{3/2} \big ( \ell - \mathfrak {a}/N \big )^3 + \frac{2}{15} \lambda _\ell ^{5/2} \big ( \ell - \mathfrak {a}/N \big )^5 + \mathcal {O}\big ( (N\ell )^{-7/2} \big ) \end{aligned}$$

which leads to (2.4).

With (A.2) for $\lambda _\ell$, we can expand the expression (2.5). We find, for $\mathfrak {a}/N \le |x| \le \ell$,

$$\begin{aligned} f_\ell (x)&= 1 - \frac{\mathfrak {a}}{N|x|} +\frac{3\mathfrak {a}}{2N\ell } - \frac{\mathfrak {a}^2}{2N^2 \ell |x|} - \frac{\mathfrak {a}|x|^2}{2N\ell ^3} +\mathcal {O}\Big (\frac{\mathfrak {a}^2}{N^2 \ell ^2}\Big )\nonumber \\ \partial _r f_\ell (x)&= \frac{\mathfrak {a}}{N |x|^2} - \frac{\mathfrak {a}|x|}{N \ell ^3} +\mathcal {O}\Big (\frac{\mathfrak {a}^2}{N^2 \ell ^2 |x|}\Big ). \end{aligned}$$

(A.3)

With these approximations, we obtain (2.6), (2.7), (2.8) and (2.9). Finally, we show (2.10). An explicit computation (using also the eigenvalue equation (A.1)) gives

$$\begin{aligned} \begin{aligned} \widehat{\omega }_p&= \widehat{\chi }_\ell (p) - \frac{2\pi \ell }{\sin \big ( \sqrt{\lambda _\ell } \, (\ell - \mathfrak {a}/N) \big )} \int _{\mathfrak {a}/ N}^\ell dr \, r \sin \big ( \sqrt{\lambda _\ell } \, (r-\mathfrak {a}/N) \big ) \\ {}&\quad \times \int _0^\pi d\theta \sin \theta \, e^{-i|p| r \cos \theta } \\ {}&= \frac{\lambda _\ell }{\lambda _\ell - p^2} \frac{4\pi \ell }{p^2} \Big [ \frac{\sin (|p| \ell )}{|p| \ell } - \cos |p| \ell \Big ] - \frac{4\pi }{|p| (\lambda _\ell - p^2)} \frac{\sin (|p| \mathfrak {a}/N)}{\cos (\sqrt{\lambda _\ell } (\ell - \mathfrak {a}_N))}. \end{aligned} \end{aligned}$$

For $|p| \ge \ell ^{-1}$, we have $|\lambda _\ell - p^2| \ge c p^2$. With (A.2), we easily find $|\widehat{\omega }_p| \le C / (Np^2)$, if $\ell ^{-1} \le |p| \le N$, and $|\widehat{\omega }_p| \le C / |p|^3$, if $|p| > N$. From (2.8), we also have $|\widehat{\omega }_p| \le \Vert \omega \Vert _1 \le C \ell ^2 / N$ for all $p \in \Lambda ^*$; this implies (2.10). $\square$

Next, we show Lemma 5.1, devoted to the properties of the solution of (5.7).

Proof of Lemma 5.1

We begin with (5.11). From the definition $g_{\ell _0}(x)=f_{\ell _0}(x)/f_\ell (x)$ and the explicit expression (2.5) we have

$$\begin{aligned} g_{\ell _0}(x)= \frac{\ell _0}{\ell }\, \frac{\sin (\sqrt{\lambda }_{\ell _0} (|x|-\mathfrak {a}/N))}{\sin (\sqrt{\lambda }_{\ell _0} (\ell _0-\mathfrak {a}/N))} \, \frac{\sin (\sqrt{\lambda }_\ell (\ell -\mathfrak {a}/N))}{\sin (\sqrt{\lambda }_\ell (|x|-\mathfrak {a}/N))}. \end{aligned}$$

(A.4)

for all $\text{\AA} / N \le |x| \le \ell$. Expanding, we find $g_{\ell _0} (x) = 1 + \mathcal {O}(\text{\AA} / N\ell )$ and thus $|{\check{\eta }} (x)| \le C \text{\AA} / \ell \le C \text{\AA} / (|x| + \ell )$ for all $\text{\AA} / N \le |x| \le \ell$. For $|x| \ge \ell$, $g_{\ell _0} (x) = f_{\ell _0} (x)$ and (2.7) implies that $|{\check{\eta }} (x)| \le C \text{\AA} / |x| \le C \text{\AA} /(|x| + \ell )$. Finally, for $|x| \le \text{\AA} / N$, we defined

$$\begin{aligned} g_{\ell _0} (x) = \lim _{|y| \downarrow \mathfrak {a}/ N} g_{\ell _0} (y) = 1 - \frac{3\mathfrak {a}}{2N \ell } + \mathcal {O}\Big ( \frac{1}{N^2 \ell ^2} \Big ) \end{aligned}$$

which gives $|{\check{\eta }} (x)| \le C \text{\AA} / \ell \le C \text{\AA} / (|x| + \ell )$. This shows the first estimate in (5.11). To bound $\nabla {\check{\eta }}$, we proceed similarly. For $\text{\AA} / N \le |x| \le \ell$, we find

$$\begin{aligned} \partial _r f_\ell (x) = f_\ell (x) \left( \frac{\sqrt{\lambda _\ell }}{\tan \big (\sqrt{\lambda _\ell }(r -\mathfrak {a}/N)\big )} - \frac{1}{r} \right) \end{aligned}$$

(A.5)

and thus

$$\begin{aligned} \partial _r {\check{\eta }}(x) = N g_{\ell _0}(x) \left( \frac{\sqrt{\lambda _\ell }}{\tan \big (\sqrt{\lambda _\ell }(|x| -\mathfrak {a}/N)\big )}- \frac{\sqrt{\lambda _{\ell _0}}}{\tan \big (\sqrt{\lambda _{\ell _0}}(|x| -\mathfrak {a}/N)\big )} \right) . \end{aligned}$$

(A.6)

With $|g_{\ell _0}(x)| \le C$ and expanding $\tan s = s + \mathcal {O}(s^3)$, we find $|\nabla {\check{\eta }} (x)| \le C \text{\AA} / \ell ^2 \le C \text{\AA} / (|x| + \ell )^2$, for all $\text{\AA} / N \le |x| \le \ell$. For $|x| \ge \ell$, we have $g_{\ell _0} (x) = f_{\ell _0} (x)$ and the estimate $|\nabla {\check{\eta }} (x)| \le C \text{\AA} / (|x| + \ell )^2$ follows from (2.7).

Next, we show (5.12). With (5.7) (noticing that the flux of $f_\ell ^2 \nabla g_{\ell _0}$ through the spheres $|x| = \text{\AA} / N$ and $|x| = \ell _0$ vanishes), we have

$$\begin{aligned}\begin{aligned}&2 N \lambda _\ell \int \chi _{\ell }(x) f_\ell ^2 (x) g_{\ell _0}(x) dx = 2N \lambda _{\ell _0} \int \chi _{\ell _0}(x) f_\ell ^2(x) g_{\ell _0}(x) dx \\&\quad = 2N \lambda _{\ell _0} \int \chi _{\ell _0}(x) dx + 2N \lambda _{\ell _0} \int \chi _{\ell _0}(x) (f_\ell f_{\ell _0}-1) (x) \end{aligned}\end{aligned}$$

since $g_{\ell _0}(x)=f_{\ell _0}(x)/f_\ell (x)$. With Lemma 2.1 we have

$$\begin{aligned} \Big | 2N \lambda _{\ell _0} \int \chi _{\ell _0}(x) dx - 8 \pi \mathfrak {a}\Big | \le C N^{-1} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \Big | 2N \lambda _{\ell _0} \int \chi _{\ell _0}(x) f_{\ell }(x) (f_{\ell _0}(x)-1) dx\Big |&\le C \Vert \omega _{\ell _0}\Vert _1 \le C /N\\ \Big | 2N \lambda _{\ell _0} \int \chi _{\ell _0}(x) (f_\ell (x)-1) dx\Big |&\le C \Vert \omega _\ell \Vert _1 \le C \ell ^2 /N. \end{aligned}\end{aligned}$$

This proves the first bound in (5.12) and also in (5.13). To show the second bound in (5.12), we compute (with a slight abuse of notation we write here, for $r > 0$, $f_\ell (r), g_{\ell _0} (r)$ to indicate the values of $f_\ell (x), g_{\ell _0} (x)$, for $|x| = r$)

$$\begin{aligned} \begin{aligned}&2N \lambda _\ell \int \chi _\ell (x) f_\ell ^2 (x) g_{\ell _0} (x) e^{-i p \cdot x} dx \\&\quad =\; \frac{4\pi N \lambda _\ell }{|p|}\int _{\mathfrak {a}/N}^\ell r f_\ell ^2(r) g_{\ell _0}(r) \sin (|p|r)dr \\&\quad = \frac{4\pi N \lambda _\ell }{|p|^2} \int _{\mathfrak {a}/N}^\ell \Big [ (f_\ell ^2(r) + 2r f_\ell (r)\, \partial _r f_\ell (r)) g_{\ell _0}(r) + r f_\ell ^2(r) \partial _r g_{\ell _0}(r)\Big ] \cos (|p|r) dr \\&\qquad \qquad - \frac{4 \pi \ell N \lambda _\ell }{|p|^2} g_{\ell _0}(\ell ) \cos (|p|\ell ). \end{aligned}\end{aligned}$$

From Lemma 2.1, we have $f_\ell (r), r |\partial _r f_\ell (r)| \le C$. From (5.11), we find (recalling that $g_{\ell _0} = 1 + {\check{\eta }} / N$) that $|\partial _r g_{\ell _0} (r)| \le C / (N\ell ^2)$. With the bound (2.4) (or (A.2)) for $\lambda _\ell$, we conclude that

$$\begin{aligned} \Big | 2N \lambda _\ell \int \chi _\ell (x) f_\ell ^2 (x) g_{\ell _0} (x) e^{-i p \cdot x} dx \Big | \le \frac{C}{\ell ^2 p^2}.\end{aligned}$$

The second bound in (5.13) can be proven analogously (on the r.h.s. $\ell$ is then replaced by $\ell _0$, which is chosen of order one).

Equations (5.15), (5.16) follow directly from (5.7). As for (5.17), we rewrite

$$\begin{aligned} \eta _0=\int {\check{\eta }}(x)\, dx=\int _{|x|<\ell } {\check{\eta }}(x)\, dx-N\int _{|x|>\ell }\omega _{\ell _0}(x)\, dx. \end{aligned}$$

(A.7)

Using (2.8),(A.3) and the fact that $g_{\ell _0} (x)=1+O(\mathfrak {a}/N\ell )$ for $|x|<\ell$, we obtain

$$\begin{aligned} \eta _0= & {} -N \int \omega _{\ell _0}(x)\, dx + \int _{|x|<\ell } {\check{\eta }}(x)\, dx+N\int _{|x|<\ell }\omega _{\ell _0}(x)\, dx\\= & {} -\frac{2}{5}\pi \mathfrak {a}\ell _0^2+\mathcal {O}\Big (\frac{\mathfrak {a}^2\ell _0}{N}\Big )+\mathcal {O}(\mathfrak {a}\ell ^2). \end{aligned}$$

To prove (5.18), we consider the Fourier coefficients $D_p$ defined in (5.14) and the corresponding function ${\check{D}} (x) = -\nabla \cdot \big [ (f_\ell ^2 (x) - 1) \nabla {\check{\eta }} (x) \big ]$. For any $p \in \Lambda ^*$, we have

$$\begin{aligned} |D_p| \le \int _{\mathfrak {a}/ N \le |c| \le \ell } | {\check{D}} (x)| dx. \end{aligned}$$

(A.8)

For $\text{\AA} / N \le |x| \le \ell$, we find

$$\begin{aligned} \begin{aligned} {\check{D}}(x)&= 2 f_\ell (x) (\partial _r f_\ell )(x) (\partial _r {\check{\eta }})(x) + (f^2_\ell (x)-1) \Delta {\check{\eta }}(x) \\ {}&= N g_{\ell _0}(x) (\lambda _\ell - \lambda _{\ell _0}) (f^2_\ell (x)-1) -2 (\partial _r {\check{\eta }})(x) \frac{(\partial _r f_\ell )(x)}{f_\ell (x)} \end{aligned} \end{aligned}$$

(A.9)

where in the second line we used the definition ${\check{\eta }} = N (g_{\ell _0} -1) = N (f_{\ell _0} / f_{\ell } - 1)$ and the scattering equation (2.1) for $f_\ell$ and $f_{\ell _0}$ to replace

$$\begin{aligned} \Delta {{\check{\eta }}} (x) = -2 N \frac{\nabla f_\ell (x)}{f_\ell (x)} \cdot \nabla g_{\ell _0} (x) + N (\lambda _\ell -\lambda _{\ell _0}) g_{\ell _0} (x). \end{aligned}$$

Using (A.4) to bound $|g_{\ell _0} (x) | \le C$, (2.7) to show $|f_\ell ^2 (x) - 1| \le C \text{\AA} / (N |x|)$ and using (A.5), (A.6) to control the second term on the r.h.s. of (A.9), we find

$$\begin{aligned} \big | {\check{D}} (x) \big | \le \frac{C \mathfrak {a}}{|x|} (\lambda _\ell - \lambda _{\ell _0}) \le \frac{C \mathfrak {a}^2}{N |x| \ell ^3} \end{aligned}$$

(A.10)

for all $\mathfrak {a}/ N \le |x| \le \ell$. Inserting (A.10) in (A.8), we arrive at

$$\begin{aligned} |D_p| \le C / (N \ell ). \end{aligned}$$

(A.11)

From the scattering equation (5.15), we can estimate

$$\begin{aligned} |\eta _p| \le \frac{C}{|p|^2} \Big ( |D_p| + |({{\widehat{V}}}_\ell *{\widehat{g}}_{\ell _0})(p) |+|N\lambda _{\ell _0}( \widehat{\chi _{\ell _0} f_\ell ^2} *{\widehat{g}}_{\ell _0})(p)|\Big ). \end{aligned}$$

(A.12)

Combining (A.11) with the first bounds in (5.12), (5.13), we immediately conclude that $|\eta _p| \le C / p^2$. To prove the remaining bounds in (5.18), we write

$$\begin{aligned} D_p = \frac{4\pi }{|p|} \int _{\mathfrak {a}/N}^\ell r \, {\check{D}}(r) \sin (|p|r) dr = \frac{4\pi }{|p|^2} \int _{\mathfrak {a}/N}^\ell \Big [ {\check{D}}(r) + r \, \partial _r {\check{D}}(r) \Big ] \cos (|p|r) dr.\nonumber \\ \end{aligned}$$

(A.13)

From (A.9) we get

$$\begin{aligned} \begin{aligned} \partial _r {\check{D}}(r) =\;&\partial _r{\check{\eta }}(r) (\lambda _\ell - \lambda _{\ell _0})(f_\ell ^2(r)-1) + N g_{\ell _0}(r) (\lambda _\ell - \lambda _{\ell _0}) 2 f_\ell (r) \partial _r f_\ell (r) \\&-2 \partial _r \Big ((\partial _r {\check{\eta }})(r) \frac{(\partial _r f_\ell )(r)}{f_\ell (r)}\Big ). \end{aligned} \end{aligned}$$

(A.14)

Using the bounds $|\partial _r{\check{\eta }}(r) |\le C r^{-2}$, $|\partial _r f_\ell (r)|\le (C N r^2)^{-1}$, the boundness of $f_\ell$ and $g_{\ell _0}$ and (2.4) we easily see that the first line of (A.14) is bounded by $\mathcal {O}\big ( (N \ell ^3 r^2)^{-1} \big )$. As for the second line of (A.14), we find, using (A.5) and (A.6),

$$\begin{aligned} \begin{aligned}&- \partial _r \Big ((\partial _r {\check{\eta }})(r) \frac{(\partial _r f_\ell )(r)}{f_\ell (r)}\Big ) \\&\quad = N g_{\ell _0}(r) \Bigg [ \bigg ( \frac{\sqrt{\lambda _\ell }}{\tan \big (\sqrt{\lambda _\ell }(r -\mathfrak {a}/N)\big )}- \frac{\sqrt{\lambda _{\ell _0}}}{\tan \big (\sqrt{\lambda _{\ell _0}}(r -\mathfrak {a}/N)\big )} \bigg )^2 \bigg ( \frac{\sqrt{\lambda _\ell }}{\tan \big (\sqrt{\lambda _\ell }(r -\mathfrak {a}/N)\big )} - \frac{1}{r} \bigg ) \\&\qquad \qquad + \bigg ( \frac{\lambda _\ell }{\sin ^2\big (\sqrt{\lambda _\ell }(r -\mathfrak {a}/N)\big )} - \frac{\lambda _{\ell _0}}{\sin ^2\big (\sqrt{\lambda _{\ell _0}}(r -\mathfrak {a}/N)\big )}\bigg ) \bigg ( \frac{\sqrt{\lambda _\ell }}{\tan \big (\sqrt{\lambda _\ell }(r -\mathfrak {a}/N)\big )} - \frac{1}{r} \bigg ) \bigg ) \\&\qquad \qquad - \bigg ( \frac{\sqrt{\lambda _\ell }}{\tan \big (\sqrt{\lambda _\ell }(r -\mathfrak {a}/N)\big )}- \frac{\sqrt{\lambda _{\ell _0}}}{\tan \big (\sqrt{\lambda _{\ell _0}}(r -\mathfrak {a}/N)\big )} \bigg )\bigg ( \frac{1}{r^2}- \frac{\lambda _\ell }{\sin ^2\big (\sqrt{\lambda _\ell }(r -\mathfrak {a}/N)\big )} \bigg )\Bigg ]. \end{aligned}\end{aligned}$$

Expanding $1/ \tan (s)= 1/s + s/3 + \mathcal {O}(s^3)$ and $1/\sin ^2 (s) = 1/ s^{2} +1/ 3 + \mathcal {O}(s^2)$, we obtain

$$\begin{aligned} \Big | \partial _r \Big ((\partial _r {\check{\eta }})(r) \frac{(\partial _r f_\ell )(r)}{f_\ell (r)}\Big ) \Big | \le \frac{C}{r^2} (\lambda _\ell -\lambda _{\ell _0}) \le \frac{C}{N \ell ^3 r^2}. \end{aligned}$$

Thus, $| r \partial _r {\check{D}} (r)| \le C / (N \ell ^3 r) \le C / \ell ^3$ for all $\text{\AA} / N \le |x| \le \ell$. Combined with (A.10) and (A.13), we conclude that

$$\begin{aligned} |D_p| \le \frac{C}{p^2 \ell ^2}. \end{aligned}$$

Inserting this estimate in (A.12), together with the second bounds in (5.12), (5.13), we obtain $|\eta _p| \le C/ (\ell ^2 |p|^4)$, which finishes the proof of (5.18). Equation (5.19) is a simple consequence of (5.18). $\square$