2 First-order lower bound

It is the goal of this Section to bring, in analogy to Eq. (22), the many-particle operator $H_N$ in an approximate block-diagonal form, which allows us to obtain an asymptotically correct lower bound on the ground state energy $E_N$ in Corollary 2. First, we are going to rewrite the operator $H_N$ defined in Eq. (2) in the language of second quantization. For this purpose let $a_k^\dagger $ denote the operator that creates a particle in the mode $u_k$ , that is, for $\Phi _n\in L^2_{\mathrm {sym}} \left (\Lambda ^n\right )$ we define $a^\dagger _k\Phi _n\in L^2_{\mathrm {sym}} \left (\Lambda ^{n+1}\right )$ as

$$ \begin{align*} a^\dagger_k\Phi_n:=\frac{1}{\sqrt{n+1}}\Xi_{n+1}(u_k\otimes \Phi_n), \end{align*} $$

where $\Xi _{n+1}$ is the orthogonal projection onto $L^2_{\mathrm {sym}} \left (\Lambda ^{n+1}\right )\subseteq L^2 \left (\Lambda ^{n+1}\right )$ , and we write $a_k$ for its adjoint, which annihilates a particle in the mode $u_k(x):=e^{ik\cdot x}$ . With creation and annihilation operators at hand, we can write

(23) $$ \begin{align} H_N=\sum_{k\in (2\pi \mathbb{Z})^3}|k|^2 a_k^\dagger a_k+\frac{1}{6}\sum_{i j k,\ell m n\in (2\pi \mathbb{Z})^3}\left(V_N\right)_{ijk,\ell m n}a_i^\dagger a_j^\dagger a_k^\dagger a_{\ell}a_{m}a_{n}, \end{align} $$

where $\left (V_N\right )_{ijk,\ell m n}$ are the matrix elements of $V_N$ w.r.t. to the basis $u_i u_j u_k$ defined below Eq. (17). If not indicated otherwise, we will always assume that indices run in the set $(2\pi \mathbb {Z})^3$ , which we will usually neglect in our notation, and we write $k\neq 0$ in case the index runs in the set $(2\pi \mathbb {Z})^3\setminus \{0\}$ . Note that the operator on the right hand side of Eq. (23) is naturally defined on the full Fock space

$$ \begin{align*} \mathcal{F} \left(L^2 \left(\Lambda\right)\right):=\bigoplus_{n=0}^\infty L^2_{\mathrm{sym}} \left(\Lambda^n\right) , \end{align*} $$

while the left-hand side is only defined on $L^2_{\mathrm {sym}} \left (\Lambda ^N\right )\subseteq \mathcal {F} \left (L^2 \left (\Lambda \right )\right )$ , and therefore Eq. (23) has to be understood as being restricted to the subspace $L^2_{\mathrm {sym}} \left (\Lambda ^N\right )$ . Furthermore, we observe that $V_N$ is a translation-invariant multiplication operator, and therefore the matrix elements of $V_N$ satisfy $\left (V_N\right )_{ijk,\ell m n}=0$ in case $i+j+k\neq \ell +m+n$ and otherwise

(24) $$ \begin{align} \left(V_N\right)_{ijk,\ell m n}=\left(V_N\right)_{(i-\ell)(j-m)(k-n),0 0 0}=N^{-2}\widehat{V} \left(\frac{j-m}{\sqrt{N}},\frac{k-n}{\sqrt{N}}\right). \end{align} $$

Following the strategy proposed in [Reference Brooks8], we are going to introduce a many-particle counterpart to the three particle map T defined in Eq. (19), which is realized by the set of operators

(25) $$ \begin{align} c_k&:=a_k+\frac{1}{2}\sum_{i j,\ell m n}(T-1)_{ijk,\ell m n} \, a_i^\dagger a_j^\dagger a_{\ell}a_{m}a_{n}, \end{align} $$

(26) $$ \begin{align} \psi_{i j k}&:=\sum_{\ell m n}T_{ijk,\ell m n} \, a_{\ell}a_{m}a_{n}. \end{align} $$

Here denotes the matrix elements of T. The following Lemma 1 is the many-particle counterpart to Eq. (22), in the sense that it provides an (approximate) block-diagonal representation of the operator $H_N$ in terms of the new variables $c_k$ and $\psi _{ijk}$ .

Lemma 1. Let $\widetilde {V}_N$ be the operator defined in Eq. (21). Then we have

(27) $$ \begin{align} H_N=\sum_{k}|k|^2 & c_k^\dagger c_k + \frac{1}{6}\sum_{i j k,\ell m n}\left(\widetilde{V}_N\right)_{i j k,\ell m n}\psi_{i j k}^\dagger \psi_{\ell m n}-\mathcal{E}, \end{align} $$

where the residual term $\mathcal {E}$ is defined as

$$ \begin{align*} \mathcal{E}&:=\frac{1}{4}\sum_{i j k, \ell m n; i' j', \ell' m' m'}|k|^2 \overline{(T-1)_{i'j'k,\ell' m' n'}}(T-1)_{ijk,\ell m n}\, a_{\ell'}^\dagger a_{m'}^\dagger a_{n'}^\dagger \\& \quad \times \left(a_{i'}a_{j'}a_i^\dagger a_j^\dagger-\delta_{ii'}\delta_{jj'}-\delta_{ij'}\delta_{ji'}\right) a_{\ell}a_{m}a_{n}. \end{align*} $$

Proof. Using the permutation symmetry of T, we first identify $\sum _{k}|k|^2 (c_k-a_k)^\dagger (c_k-a_k)$ as

$$ \begin{align*} \frac{1}{4}& \sum_{i j k, \ell m n; i' j', \ell' m' m'}|k|^2 \overline{(T-1)_{i'j'k,\ell' m' n'}}(T-1)_{ijk,\ell m n}\, a_{\ell'}^\dagger a_{m'}^\dagger a_{n'}^\dagger a_{i'}a_{j'}a_i^\dagger a_j^\dagger a_{\ell}a_{m}a_{n}\\ & =\frac{1}{2} \sum_{i j k,\ell m n}\left\{(T-1)^\dagger (-\Delta_{x_3})(T-1)\right\}_{ijk,\ell m n} a_i^\dagger a_j^\dagger a_k^\dagger a_{\ell}a_{m}a_{n}+\mathcal{E}\\ & =\frac{1}{6} \sum_{i j k,\ell m n}\left\{(T-1)^\dagger (-\Delta_3)(T-1)\right\}_{ijk,\ell m n} a_i^\dagger a_j^\dagger a_k^\dagger a_{\ell}a_{m}a_{n}+\mathcal{E}, \end{align*} $$

where $\Delta _3$ is the Laplace operator on $L^2(\Lambda )^{\otimes 3}$ . Similarly

$$ \begin{align*} & \sum_{k}|k|^2 a_k^\dagger (c_k-a_k)+\mathrm{H.c.}=\frac{1}{6} \sum_{i j k,\ell m n}\Big\{(-\Delta_3)(T-1)+\mathrm{H.c.}\Big\}_{ijk,\ell m n} a_i^\dagger a_j^\dagger a_k^\dagger a_{\ell}a_{m}a_{n},\\& \sum_{i j k,\ell m n}\left(\widetilde{V}_N\right)_{i j k,\ell m n}\psi_{i j k}^\dagger \psi_{\ell m n}=\sum_{i j k,\ell m n}\left(T^\dagger \widetilde{V}_N T\right)_{i j k,\ell m n}a^\dagger_i a^\dagger_j a^\dagger_k a_\ell a_m a_n. \end{align*} $$

Since

$$ \begin{align*} (T - 1)^\dagger \Delta_3(T - 1) + \left\{\Delta_3(T - 1) + \mathrm{H.c.}\right\} = T^\dagger \Delta_3 T - \Delta_3 \end{align*} $$

we obtain

$$ \begin{align*} & \sum_{k}|k|^2 c_k^\dagger c_k + \frac{1}{6}\sum_{i j k,\ell m n}\left(\widetilde{V}_N\right)_{i j k,\ell m n}\psi_{i j k}^\dagger \psi_{\ell m n}\\ & \quad =\sum_{k}|k|^2 a_k^\dagger a_k + \frac{1}{6} \sum_{i j k,\ell m n}\left\{T^\dagger \left(-\Delta_3+\widetilde{V}_N\right)T + \Delta_3\right\}_{ijk,\ell m n} a_i^\dagger a_j^\dagger a_k^\dagger a_{\ell}a_{m}a_{n} + \mathcal{E}. \end{align*} $$

We observe that $T^\dagger \left (-\Delta _3+\widetilde {V}_N\right )T + \Delta _3=V_N$ by Eq. (22), which concludes the proof by the representation of $H_N$ in second quantization, see Eq. (23).

Making use of the sign $ (1-\pi _K) V_N (1-\pi _K) \geq 0$ , we immediately obtain that

$$ \begin{align*} \sum_{i j k,\ell m n}\left((1-\pi_K) V_N(1-\pi_K)\right)_{i j k,\ell m n}\psi_{ijk}^\dagger \psi_{\ell m n}\geq 0. \end{align*} $$

Therefore Lemma 1 allows us to bound $H_N$ from below by

(28) $$ \begin{align} \nonumber H_N & \geq \sum_{k}|k|^2 c_k^\dagger c_k+\frac{1}{6}\sum_{i j k,\ell m n}\left(\widetilde{V}_N-(1-\pi_K) V_N(1-\pi_K)\right)_{i j k,\ell m n}\psi_{ijk}^\dagger \psi_{\ell m n}-\mathcal{E}\\ &= \sum_{k}|k|^2 c_k^\dagger c_k+\frac{1}{6}\sum_{i j k,\ell m n}\left(\widetilde{V}_N-(1-\pi_K) V_N(1-\pi_K)\right)_{i j k,\ell m n}a_i^\dagger a_j^\dagger a_k^\dagger a_\ell a_m a_n-\mathcal{E}, \end{align} $$

where we have used the fact that $\psi _{ijk}=a_i a_j a_k$ in case one of the indices is zero, which is a direct consequence of the observation that $(T-1)_{ijk,\ell mn}=0$ in case one of the indices in $\{i,j,k\}$ is zero. Note that we can write

$$ \begin{align*} \widetilde{V}_N-(1-\pi_K) V_N(1-\pi_K)=A + B+B^* \end{align*} $$

with A and B defined as

$$ \begin{align*} A&:=\pi_K \left(V_N -V_N R V_N\right)\pi_K,\\ B&:= \left(1-\pi_K- Q^{\otimes 3} \right)\left(V_N -V_N R V_N \right)\pi_K. \end{align*} $$

Using the sets $\mathcal {L}^{(z)}:=\{(z,0,0),(0,z,0),(0,0,z)\}$ , let us first analyze the term involving A

$$ \begin{align*} & \frac{1}{6} \sum_{i j k,\ell m n}\left(A\right)_{i j k,\ell m n}a_i^\dagger a_j^\dagger a_k^\dagger a_\ell a_m a_n = \frac{1}{6} \sum_{(i j k),(\ell m n)\in \mathcal{L}_K}\left(V_N -V_N R V_N\right)_{i j k,\ell m n}a_i^\dagger a_j^\dagger a_k^\dagger a_\ell a_m a_n\\& \quad = \frac{1}{6}\left(V_N -V_N R V_N\right)_{000,000}(a_0^\dagger)^{3}a_0^3+ \sum_{0<|z|<K} \left\{\frac{1}{6} \sum_{(i j k),(\ell m n)\in \mathcal{L}^{(z)}} \left(V_N -V_N R V_N\right)_{i j k,\ell m n}\right\}a_0^{2\dagger}a_0^2 a_z^\dagger a_z. \end{align*} $$

Together with the definition of the coefficients

(29) $$ \begin{align} \lambda_{k,\ell}:=\frac{1}{18}\langle u_0 u_\ell u_{k-\ell},(V_N -V_N R V_N)(u_0 u_0 u_k+u_0 u_k u_0+u_k u_0 u_0)\rangle \end{align} $$

we can write

$$ \begin{align*} \frac{1}{6} \left(V_N -V_N R V_N\right)_{000,000}=\frac{1}{6}\langle u_0 u_0 u_0,(V_N -V_N R V_N)u_0 u_0 u_0\rangle=\lambda_{0,0}, \end{align*} $$

and utilizing the permutation symmetry of $V_N -V_N R V_N$ we furthermore obtain for $z\neq 0$

$$ \begin{align*} \frac{1}{6} \sum_{(i j k),(\ell m n)\in \mathcal{L}^{(z)}} \left(V_N -V_N R V_N\right)_{i j k,\ell m n} & =\frac{1}{6}\left\langle \sum_{(i j k)\in \mathcal{L}^{(z)}} u_iu_ju_k, \left(V_N -V_N R V_N\right) \sum_{(\ell m n)\in \mathcal{L}^{(z)}}u_\ell u_m u_n\right\rangle\\ & = \frac{1}{2}\left\langle u_0 u_0 u_z, \left(V_N -V_N R V_N\right) \sum_{(\ell m n)\in \mathcal{L}^{(z)}}u_\ell u_m u_n\right\rangle\\ & = 9\lambda_{z,0}. \end{align*} $$

Therefore,

(30) $$ \begin{align} \frac{1}{6} \sum_{i j k,\ell m n}\left(A\right)_{i j k,\ell m n}a_i^\dagger a_j^\dagger a_k^\dagger a_\ell a_m a_n = \lambda_{0,0}\, (a_0^\dagger)^{3}a_0^3+9 a_0^{2\dagger}a_0^2\sum_{0<|k|\leq K} \lambda_{k,0} a_k^\dagger a_k. \end{align} $$

To keep the notation light, we do not explicitly indicate the N dependence of $\lambda _{k,\ell }$ . Similarly

(31) $$ \begin{align} \frac{1}{6} \sum_{i j k,\ell m n}\left(B\right)_{i j k,\ell m n}a_i^\dagger a_j^\dagger a_k^\dagger a_\ell a_m a_n=3 a_0^\dagger a_0^3\sum_{\ell\neq 0}\lambda_{0,\ell}\, a_{\ell}^\dagger a_{-\ell}^\dagger+9 a_0^\dagger a_0^2\sum_{\ell,0<|k|\leq K}\lambda_{k,\ell} a_\ell^\dagger a_{k-\ell}^\dagger a_k. \end{align} $$

Putting together Eq. (28), Eq. (30) and Eq. (31) yields

(32) $$ \begin{align} H_N \geq \lambda_{0,0}(a_0^\dagger )^3 a_0^3 + \sum_{k} |k|^2 c_k^\dagger c_k + \mathbb{Q}_K + \mathcal{E}' - \mathcal{E}, \end{align} $$

where we define the operator $\mathbb {Q}_K$ and the error term $\mathcal {E}'$ as

(33) $$ \begin{align} \mathbb{Q}_K&:= 9 a_0^{2\dagger}a_0^2\sum_{0<|k|\leq K} \lambda_{k,0} a_k^\dagger a_k +3\left(a_0^\dagger a_0^3\sum_{0<|\ell|\leq K}\lambda_{0,\ell}\, a_{\ell}^\dagger a_{-\ell}^\dagger+\mathrm{H.c.}\right), \end{align} $$

(34) $$ \begin{align} \mathcal{E}' &:= \bigg(3 \sum_{|\ell|>K} \lambda_{0,\ell}\, a_{\ell}^\dagger a_{-\ell}^\dagger\, a_0^\dagger a_0^3 + 9 \sum_{\ell,0<|k|\leq K} \lambda_{k,\ell} a_\ell^\dagger a_{k-\ell}^\dagger a_k\, a_0^\dagger a_0^2 + \mathrm{H.c.}\bigg). \end{align} $$

Let us furthermore introduce the particle number operator

$$ \begin{align*} \mathcal{N}:=\sum_{k\neq 0}a_k^\dagger a_k, \end{align*} $$

which counts the number of excited particles, that is, the number of particles with momentum $k\neq 0$ . Since we have the operator identity $\sum _{k}a_k^\dagger a_k=N$ on the Hilbert space $L^2_{\mathrm {sym}}(\Lambda ^N)\subseteq \mathcal {F} \left (L^2 \left (\Lambda \right )\right )$ , we observe that $a_0^\dagger a_0=N-\mathcal {N}$ , see also Eq. (13), that is, the number of particles with momentum $k=0$ is given by the difference between the total number of particles N and the number of excited particles.

In order to control the terms arising in Eq. (32), it is imperative to understand the asymptotic behavior of the coefficients $\lambda _{k,\ell }$ and the matrix entries $T_{ijk,\ell m n}$ defined below Eq. (26). Since we want to focus our attention on the many-body analysis, we will postpone our study of the scattering coefficients to Section 7. For the convenience of the reader, we are going to state the relevant results, which are proven in Lemma 15 and Lemma 16 respectively,

(35) $$ \begin{align} & |\lambda_{k,\ell}| \leq \frac{C}{N^2}\left(1+\frac{|\ell|^2}{N }\right)^{-1}, \end{align} $$

(36)

(37) $$ \begin{align} & \left|6N^2\lambda_{0,0}-b_{\mathcal{M}}(V)\right| \leq \frac{1}{N}, \end{align} $$

(38) $$ \begin{align} & \left|6N^2\frac{\lambda_{k,\ell}}{6}-b_{\mathcal{M}}(V)\right| \leq \frac{C_{k,\ell }}{\sqrt{N}}. \end{align} $$

Furthermore, we need the following result, which is verified in Lemma A2, see Appendix A, and which allows us to compare the new operators $c_k$ with the annihilation operators $a_k$ ,

(39) $$ \begin{align} & \sum_{k}|k|^{2\sigma}(c_k-a_k)(c_k-a_k)^\dagger\lesssim \frac{1}{N}\mathcal{N}^2, \end{align} $$

(40) $$ \begin{align} & \sum_{k}|k|^{2\tau}a_k^\dagger\, \mathcal{N}^s a_k \lesssim \sum_{k}|k|^{2\tau} c_k^\dagger \mathcal{N}^s c_k+\frac{1}{N}\mathcal{N}^{s+2} +N^{\tau}\left(\mathcal{N}+1\right)^s \end{align} $$

for $0\leq \tau \leq 1$ , $0\leq \sigma <\frac {1}{2}$ and integers $s\geq 0$ . Utilizing Eq. (35)-(40), the following Lemma 2, Lemma 3 and Lemma 4, provide relevant bounds on the various terms appearing in Eq. (32), which will be instrumental in order to establish that the ground state energy $E_N$ of $H_N$ is, to leading order, bounded from below by $\frac {1}{6}b_{\mathcal {M}}(V) N$ , see Corollary 2. In our first Lemma 2 we provide a lower bound on

$$ \begin{align*} \lambda_{0,0}(a_0^\dagger)^3 a_0^3+\mathbb{Q}_K \end{align*} $$

for K large enough, which is an operator that is at most quadratic in the variables $a_k$ and $a_k^\dagger $ for $k\neq 0$ .

Lemma 2. Let $b_{\mathcal {M}}(V)$ be the modified scattering length defined in Eq. (4). Then there exists for all $\tau ,\alpha>0$ , a constant $K_0(\tau ,\alpha )$ and for all $K\geq K_0(\tau ,\alpha )$ a constant $C_K>0$ , such that

$$ \begin{align*} \lambda_{0,0}(a_0^\dagger)^3 a_0^3+\mathbb{Q}_K\geq \frac{1}{6}b_{\mathcal{M}}(V) N - \alpha \sum_{k}|k|^{2\tau}a_k^\dagger a_k-C_K\left(1+\frac{\mathcal{N}^2}{N}+\frac{\mathcal{N}}{\sqrt{N}}\right), \end{align*} $$

for $K\geq K_0(\tau ,\alpha )$ .

Proof. First of all we observe that we can write

$$ \begin{align*} (a_0^\dagger)^3 a_0^3 & =N^3-3N^2(\mathcal{N}+3)+N(3\mathcal{N}^2+6\mathcal{N}+2)-\mathcal{N}^3-3\mathcal{N}^2 -2\mathcal{N}\\ &\geq N^3-3N^2 \mathcal{N}-N^2 D, \end{align*} $$

for a suitable $D>0$ . Defining

$$ \begin{align*} \mathcal{N}_{>}:=\mathcal{N}-\sum_{0<|k|\leq K}a_k^\dagger a_k, \end{align*} $$

we therefore obtain in combination with Eq. (37) and Eq. (38)

$$ \begin{align*} & \lambda_{0,0}(a_0^\dagger)^3 a_0^3+\mathbb{Q}_K-\frac{1}{6}b_{\mathcal{M}}(V) N\\&\quad \geq \frac{1}{6}b_{\mathcal{M}}(V)\left\{9\frac{(a_0^\dagger)^2 a_0^2}{N^2}\sum_{0<|k|\leq K}a_k^\dagger a_k+3\left(\frac{a_0^\dagger a_0^3}{N^2}\sum_{0<|k|\leq K}a_k^\dagger a_{-k}^\dagger +\mathrm{H.c.}\right)-3\mathcal{N}-DN^{-\frac{1}{2}}\mathcal{N}\right\}-D\\&\quad =\frac{1}{6}b_{\mathcal{M}}(V)\bigg\{\left(9\frac{(a_0^\dagger)^2 a_0^2}{N^2}-3-DN^{-\frac{1}{2}}\right)\sum_{0<|k|\leq K}a_k^\dagger a_k\\&\qquad +3\left(\frac{a_0^\dagger a_0^3}{N^2}\sum_{0<|k|\leq K}a_k^\dagger a_{-k}^\dagger +\mathrm{H.c.}\right)-3\mathcal{N}_{>}\bigg\}-D, \end{align*} $$

for a suitable constant $D>0$ . Since

$$ \begin{align*} 3\frac{a_0^\dagger a_0^3}{N^2}\sum_{0<|k|\leq K}a_k^\dagger a_{-k}^\dagger +\mathrm{H.c.}\leq 3\sum_{0<|k|\leq K} \big(2a_k^\dagger a_k+1\big)\leq 6\sum_{0<|k|\leq K}a_k^\dagger a_k+3\left(\frac{K}{\pi}+1\right)^3, \end{align*} $$

and since we have

$$ \begin{align*} \left(9\frac{(a_0^\dagger)^2 a_0^2}{N^2}-9-DN^{-\frac{1}{2}}\right)\sum_{0<|k|\leq K}a_k^\dagger a_k\geq -9\frac{\mathcal{N}^2}{N}-D\frac{\mathcal{N}}{\sqrt{N}}, \end{align*} $$

we obtain

$$ \begin{align*} \lambda_{0,0}(a_0^\dagger)^3 a_0^3+\mathbb{Q}_K\geq \frac{1}{6}b_{\mathcal{M}}(V) N-\frac{1}{2}b_{\mathcal{M}}(V)\mathcal{N}_>-C_K\left(1+\frac{\mathcal{N}^2}{N}+\frac{\mathcal{N}}{\sqrt{N}}\right) \end{align*} $$

for a suitable, K-dependent, constant $C_K$ . Finally, we choose $K_0(\tau ,\alpha )$ large enough such that $\frac {1}{2}b_{\mathcal {M}}(V)K_0(\tau ,\alpha )^{-2\tau }\leq \alpha $ , and therefore we have for all $K\geq K_0(\tau ,\alpha )$

$$ \begin{align*} \frac{1}{2}b_{\mathcal{M}}(V)\mathcal{N}_>\leq \frac{1}{2}b_{\mathcal{M}}(V)K^{-2\tau}\sum_{k}|k|^{2\tau}a_k^\dagger a_k\leq \alpha\sum_{k}|k|^{2\tau}a_k^\dagger a_k.\\[-45pt] \end{align*} $$

In the Lemma 3, we will provide estimates on the residual term $\mathcal {E}$ defined in Lemma 1, which will allow us to compare the size of $\mathcal {E}$ with the kinetic energy $ \sum _{k}|k|^2 c_k^\dagger c_k$ in the variables $c_k$ . Before we come to Lemma 3, we need the following Corollary 1, which is a consequence of Eq. (40) and allows us to estimate monomials in the operators $a_k$ and $a_k^\dagger $ by the kinetic energy in the variables $c_k$ and powers of the particle number operator $\mathcal {N}$ .

Corollary 1. Let $\mathcal {K}_{\tau ,t}:=\sum _{i=1}^t (-\Delta _{x_i})^\tau $ . Given $0\leq \tau \leq 1$ , and integers $s,t\geq 1$ and $\alpha ,\beta \geq 0$ , there exist $\delta>0$ and $C>0$ , such that for $\epsilon>0$

$$ \begin{align*} &\pm \left(\sum_{i_1\dots i_s,j_1\dots j_t} G_{i_1\dots i_s,j_1\dots j_t}a_{j_t}^\dagger\dots a_{j_1}^\dagger X\ a_{i_1}\dots a_{i_s}+\mathrm{H.c.}\right)\leq C\left\|\mathcal{K}_{\tau,t}^{-\frac{1}{2}}G\mathcal{K}_{\tau,s}^{-\frac{1}{2}}\right\| \big\| \mathcal{N}^{-\frac{\beta}{2}}X\mathcal{N}^{-\frac{\alpha}{2}}\big\| \\ & \ \ \ \ \ \times \left\{\sum_{k}|k|^2 c_k^\dagger \left(\epsilon \mathcal{N}^{s+\alpha-1}+\epsilon^{-1}\mathcal{N}^{t+\beta-1}\right) c_k + \left(\mathcal{N}+N^\tau \right) \left(\epsilon \mathcal{N}^{s+\alpha-1}+\epsilon^{-1}\mathcal{N}^{t+\beta-1}\right)\right\}, \end{align*} $$

where $G:\left (\mathrm {ran}Q\right )^{\otimes s}\longrightarrow \left (\mathrm {ran}Q\right )^{\otimes t}$ and $X:\mathcal {F}\left (L^2(\Lambda )\right )\longrightarrow \mathcal {F}\left (L^2(\Lambda )\right )$ . In case $s=0$

$$ \begin{align*} &\pm \left(\sum_{j_1\dots j_t} G_{j_1\dots j_t}a_{j_t}^\dagger\dots a_{j_1}^\dagger X +\mathrm{H.c.}\right) \leq C\left\|\mathcal{K}_{\tau,t}^{-\frac{1}{2}}G\right\| \big\| \mathcal{N}^{-\frac{\beta}{2}}X\mathcal{N}^{-\frac{\alpha}{2}}\big\| \\ & \ \ \ \times \left\{\epsilon \mathcal{N}^\alpha+\epsilon^{-1} \sum_{k}|k|^2 c_k^\dagger \mathcal{N}^{t+\beta-1} c_k +\epsilon^{-1} \left(\mathcal{N}+N^\tau \right) \mathcal{N}^{t+\beta-1}\right\}. \end{align*} $$

Proof. Let us define for $s,t\geq 1$ the operator-valued vector and operator-valued matrix

(41) $$ \begin{align} \left(\Phi_{\tau,s}\right)_{k_1\dots k_s}&:=\left(|k_1|^{2\tau}+\dots +|k_s|^{2\tau}\right)^{\frac{1}{2}}a_{k_1}\dots a_{k_s},\\\nonumber \Upsilon_{i_1\dots i_s,j_1\dots j_t}&:=\left(\mathcal{K}_{\tau,t}^{-\frac{1}{2}}G\, \mathcal{K}_{\tau,s}^{-\frac{1}{2}}\right)_{i_1\dots i_s,j_1\dots j_t} \mathcal{N}^{-\frac{\beta}{2}}X\mathcal{N}^{-\frac{\alpha}{2}}, \end{align} $$

which allow us to represent

$$ \begin{align*} \sum_{i_1\dots i_s,j_1\dots j_t} G_{i_1\dots i_s,j_1\dots j_t}a_{j_t}^\dagger\dots a_{j_1}^\dagger X\ a_{i_1}\dots a_{i_s}=\Phi_{\tau,t}^\dagger \mathcal{N}^{\frac{\beta}{2}}\Upsilon\mathcal{N}^{\frac{\alpha}{2}} \Phi_{\tau,s}. \end{align*} $$

Using the fact that $\|\Upsilon \|\leq \left \|\mathcal {K}_{\tau ,t}^{-\frac {1}{2}}G\, \mathcal {K}_{\tau ,s}^{-\frac {1}{2}}\right \|\big \| \mathcal {N}^{-\frac {\beta }{2}}X\mathcal {N}^{-\frac {\alpha }{2}}\big \|$ , we obtain by Cauchy-Schwarz

$$ \begin{align*} & \ \ \ \pm \left(\Phi_t^\dagger \Upsilon \Phi_s+\mathrm{H.c.}\right)\leq \left\|\mathcal{K}_{\tau,t}^{-\frac{1}{2}}G\, \mathcal{K}_{\tau,s}^{-\frac{1}{2}}\right\|\big\| \mathcal{N}^{-\frac{\beta}{2}}X\mathcal{N}^{-\frac{\alpha}{2}}\big\|\left(\epsilon \Phi_{\tau,t}^\dagger \mathcal{N}^\beta \Phi_{\tau,t}+\epsilon^{-1}\Phi_{\tau,s}^\dagger \mathcal{N}^\alpha \Phi_{\tau,s}\right)\\ &=\left\|\mathcal{K}_{\tau,t}^{-\frac{1}{2}}G\, \mathcal{K}_{\tau,s}^{-\frac{1}{2}}\right\|\big\| \mathcal{N}^{-\frac{\beta}{2}}X\mathcal{N}^{-\frac{\alpha}{2}}\big\|\left(\epsilon s\sum_{k}|k|^{2\tau}a_k^\dagger\, \mathcal{N}^{s+\alpha -1} a_k+\epsilon^{-1}\, t\sum_{k}|k|^{2\tau}a_k^\dagger \, \mathcal{N}^{t+\beta -1} a_k\right). \end{align*} $$

Since $\mathcal {N}\leq N$ we furthermore have by Eq. (40)

$$ \begin{align*} \sum_{k}|k|^{2\tau}a_k^\dagger\, \mathcal{N}^{s+\alpha-1} a_k \lesssim \sum_{k}|k|^2 c_k^\dagger \mathcal{N}^{s+\alpha -1} c_k +(\mathcal{N}+1)^{s+\alpha-1}\left(N^{\tau}+\mathcal{N}\right).\\[-45pt] \end{align*} $$

With Corollary 1 at hand, we can verify the subsequent Lemma 3.

Lemma 3. For $K\geq 0$ , there exists a constant $C_K>0$ , such that

$$ \begin{align*} \pm \mathcal{E}\leq C_K \sum_{k}|k|^2 c_k^\dagger \left(\frac{\mathcal{N}}{N}+N^{-\frac{1}{2}}\right) c_k+C_K\left(\frac{\mathcal{N}}{N}+N^{-\frac{1}{3}}\right) (\mathcal{N}+1). \end{align*} $$

Proof. Let us denote with $\mathcal {I}\subseteq (2\pi \mathbb {Z})^{3\times 3}$ the index set

$$ \begin{align*} \mathcal{I}:=\{(0,0,0)\}\cup \bigcup_{0<|\ell|\leq K}\{(\ell,0,0),(0,\ell,0),(0,0,\ell)\} \end{align*} $$

Then we define for $I=(I_1,I_2,I_3),I'=(I^{\prime }_1,I^{\prime }_2,I^{\prime }_3)\in \mathcal {I}$ the operator $K^{(I,I')}$ acting on $L^2(\Lambda )$ and the operator $G^{(I,I')}$ acting on $L^2(\Lambda )^{\otimes 2}$ as

$$ \begin{align*} K^{(I,I')}_{i,i'}&:=\frac{1}{2}\sum_{j k }|k|^2\overline{(T-1)_{i'jk,I' }}\left((T-1)_{ijk,I}+(T-1)_{jik,I}\right),\\ G^{(I,I')}_{ij,i' j'}&:=\frac{1}{4}\sum_{k}|k|^2 \overline{(T-1)_{i'j'k,I' }}(T-1)_{ijk,I}, \end{align*} $$

as well as $\mathcal {K}_{\tau ,2}:=(-\Delta _{x_1})^{\tau }+(-\Delta _{x_2})^{\tau }$ acting on $L^2(\Lambda )^{\otimes 2}$ . Then we can write $\mathcal {E}$ as

(42) $$ \begin{align} \mathcal{E}=\sum_{I,I'\in \mathcal{I}}\left(\sum_{i,i'}K^{(I,I')}_{i,i'} a^\dagger_i \left(a_{I_1}^\dagger a_{I_2}^\dagger a_{I_3}^\dagger a_{I^{\prime}_1} a_{I^{\prime}_2} a_{I^{\prime}_3}\right)a_{i'}+\sum_{ij,i'j'}G^{(I,I')}_{ij,i'j'}a^\dagger_{i}a_j^\dagger \left(a_{I_1}^\dagger a_{I_2}^\dagger a_{I_3}^\dagger a_{I^{\prime}_1} a_{I^{\prime}_2} a_{I^{\prime}_3}\right) a_{i'}a_{j'}\right). \end{align} $$

By the weighted Schur test, the operator norm of $\mathcal {K}_{\tau ,2}^{-\frac {1}{2}} G^{(I,I')}\mathcal {K}_{\tau ,2}^{-\frac {1}{2}}$ is bounded by

$$ \begin{align*} \|\mathcal{K}_{\tau,2}^{-\frac{1}{2}} G^{(I,I')}\mathcal{K}_{\tau,2}^{-\frac{1}{2}}\|\leq \sqrt{\alpha^{(I,I')}\alpha^{(I',I)}}, \end{align*} $$

where we define $\alpha ^{(I,I')}:=\sup _{i'j'}\sum _{ij}\frac {|G^{(I,I')}_{ij,i' j'}|}{|i|^{2\tau }+|j|^{2\tau }}$ . Let us furthermore introduce $s:=I_1+I_2+I_3$ and $s':=I^{\prime }_1+I^{\prime }_2+I^{\prime }_3$ . Making use of Eq. (36), we obtain for the concrete choice $\tau :=\frac {2}{3}$

$$ \begin{align*} \alpha^{(I,I')} & \leq \sup_{i'j'} \sum_{ijk} \frac{|k|^2 |(T - 1)_{i'j'k,I' }|\, |(T - 1)_{ijk,I}|}{|i|^{2\tau} + |j|^{2\tau}} \lesssim N^{-4} \sup_{i'j'} \sum_{ijk\neq 0}\frac{ \delta_{i'+j'+k=s'} \delta_{i+j+k=s}}{(|i|^{2\tau} + |j|^{2\tau})(|i|^2 + |j|^2 + |k|^2)} \\ & \leq N^{-4} \sum_{i\neq 0}\frac{1}{|i|^{2+2\tau}}\lesssim N^{-4}. \end{align*} $$

Consequently $\|\mathcal {K}_{\tau ,2}^{-\frac {1}{2}} G^{(I,I')}\mathcal {K}_{\tau ,2}^{-\frac {1}{2}}\|\lesssim N^{-4}$ . Furthermore, the operator

$$ \begin{align*} X^{(I,I')}:=a_{I_1}^\dagger a_{I_2}^\dagger a_{I_3}^\dagger a_{I^{\prime}_1} a_{I^{\prime}_2} a_{I^{\prime}_3} \end{align*} $$

satisfies $\|X^{(I,I')}\|\leq N^3$ . Therefore we obtain by Corollary 1

$$ \begin{align*} \sum_{ij,i'j'}G^{(I,I')}_{ij,i'j'}a^\dagger_{i}a_j^\dagger \left(a_{I_1}^\dagger a_{I_2}^\dagger a_{I_3}^\dagger a_{I^{\prime}_1} a_{I^{\prime}_2} a_{I^{\prime}_3}\right) a_{i'}a_{j'}\lesssim \sum_{k}|k|^2 c_k^\dagger \frac{\mathcal{N}}{N}c_k + (\mathcal{N}+N^\tau)\frac{\mathcal{N}}{N}. \end{align*} $$

Again by Lemma 15 we have $\|K^{(I,I')}\|\lesssim N^{-\frac {7}{2}}$ , which concludes the proof by Corollary 1, together with the observation that the set $\mathcal {I}$ in the definition of $\mathcal {E}$ in Eq. (42) is finite.

The next Lemma 4 will give us sufficient bounds on the error term $\mathcal {E}'$ defined in Eq. (34), which will be responsible for the appearance of an order $O_{N\rightarrow \infty } \left (\sqrt {N}\right )$ error in the main results of this Section Theorem 3 and Corollary 2.

Lemma 4. There exist constants $C,C_K>0$ such that for $K\leq \sqrt {N}$ , where K is as in the definition of $\pi _K$ below Eq. (17), and $\epsilon>0$

(43) $$ \begin{align} \pm \left(\sum_{|\ell|>K}\lambda_{0,\ell}\, a_{\ell}^\dagger a_{-\ell}^\dagger\, a_0^\dagger a_0^3 + \mathrm{H.c.}\right)&\leq \epsilon \sum_\ell |\ell|^2 c_\ell^\dagger c_\ell + \epsilon\, \mathcal{N} + C\frac{\mathcal{N}}{\sqrt{N}} + \frac{C}{\epsilon} \left(\sqrt{N} + \frac{\mathcal{N}}{\sqrt{K + 1}}\right), \end{align} $$

(44) $$ \begin{align} \pm \left(\sum_{0<|k|\leq K,\ell}\lambda_{k,\ell} a_\ell^\dagger a_{k-\ell}^\dagger a_k\, a_0^\dagger a_0^2+\mathrm{H.c.}\right)&\lesssim \epsilon \sum_\ell |\ell|^2 c_\ell^\dagger c_\ell+\epsilon \frac{\mathcal{N}^2}{N}+C_K\frac{\mathcal{N}}{N}+\frac{C_K}{\epsilon}\left(\frac{\mathcal{N}}{\sqrt{N}}+\frac{\mathcal{N}^2}{N}\right). \end{align} $$

Furthermore, we have

(45) $$ \begin{align} \pm \left(\sum_{|\ell|>K}\lambda_{0,\ell}\, a_{\ell}^\dagger a_{-\ell}^\dagger\, a_0^\dagger a_0^3\frac{\mathcal{N}}{N} + \mathrm{H.c.}\right) \leq N^{-\frac{1}{2}}\left(\sum_\ell |\ell|^2 c_\ell^\dagger c_\ell+\mathcal{N}\right)+N^{-\frac{3}{2}}\mathcal{N}^2\left(\mathcal{N}+\sqrt{N}\right). \end{align} $$

Proof. Given $m\in \{0,1\}$ , let us define the operator $X:=a_0^\dagger a_0^3\frac {\mathcal {N}^m}{N^m}$ and the coefficients

$$ \begin{align*} \Lambda^{(n)}_{\ell,k}:=\overline{(T-1)_{(n-k)(k-\ell)\ell,n 0 0}}+\overline{(T-1)_{(n-k)(k-\ell)\ell,0 n 0}}+\overline{(T-1)_{(n-k)(k-\ell)\ell,0 0 n}}, \end{align*} $$

and observe that by Eq. (36) there exists a constant $C>0$ such that

(46) $$ \begin{align} |\Lambda^{(n)}_{\ell,k}|\leq \frac{C}{N^2(|\ell|^2+|k|^2)}\left(1+\frac{|\ell|^2+|k|^2}{N}\right)^{-1}, \end{align} $$

where we have assumed w.l.o.g. that $|n|\leq \sqrt {N}$ , since $\Lambda ^{(n)}_{\ell ,k}=0$ in case $|n|>K$ and $K\leq \sqrt {N}$ . In order to verify Eq. (43), respectively Eq. (45), let us write

(47) $$ \begin{align} \nonumber & \sum_{|\ell|>K}\lambda_{0,\ell}\, a_{\ell}^\dagger a_{-\ell}^\dagger a_0^\dagger a_0^3\frac{\mathcal{N}^m}{N^m}=\sum_{|\ell|>K}\lambda_{0,\ell}\, c_{\ell}^\dagger a_{-\ell}^\dagger X-\sum_{|\ell|>K}\lambda_{0,\ell}\, (c_\ell-a_{\ell})^\dagger a_{-\ell}^\dagger X\\&\quad = \sum_{|\ell|>K}\lambda_{0,\ell}\, c_{\ell}^\dagger a_{-\ell}^\dagger X-\sum_{|\ell|>K}\lambda_{0,\ell}\, a_{-\ell}^\dagger (c_\ell-a_{\ell})^\dagger X-\sum_{|\ell|>K}\sum_{n\neq 0} \lambda_{0,\ell}\, \Lambda^{(n)}_{\ell,0} a_0^{2\dagger }a_n^\dagger a_n X. \end{align} $$

Regarding the first term in Eq. (47), note that we have for $\epsilon>0$ the estimate

$$ \begin{align*} \pm \sum_{|\ell|>K}\lambda_{0,\ell}\, c_{\ell}^\dagger a_{-\ell}^\dagger X\pm \mathrm{H.c.}\leq \epsilon\sum_{\ell}|\ell|^2c_\ell^\dagger c_\ell+\frac{1}{\epsilon} X^\dagger \left(\sum_{|\ell|>K}\frac{|\lambda_{0,\ell}|^2}{\ell^2}a_\ell^\dagger a_\ell+\sum_{|\ell|>K}\frac{|\lambda_{0,\ell}|^2}{\ell^2} \right)X. \end{align*} $$

Using $|\lambda _{k,\ell }|\lesssim N^{-2}(1+\frac {|\ell |^2}{N})^{-1}$ , see Eq. (35), we have $\sum _{|\ell |>K}\frac {|\lambda _{0,\ell }|^2}{\ell ^2}\lesssim N^{-\frac {7}{2}}$ and $\frac {|\lambda _{0,\ell }|^2}{\ell ^2}\leq \frac {1 }{N^4 (K^2+1)}$ for $|\ell |>K$ , and therefore we obtain for such K

$$ \begin{align*} \pm \sum_{|\ell|>K}\lambda_{0,\ell}\, c_{\ell}^\dagger a_{-\ell}^\dagger X \pm \mathrm{H.c.}&\lesssim \epsilon\sum_{\ell}|\ell|^2c_\ell^\dagger c_\ell + \frac{1}{\epsilon N^4 }X^\dagger \left(\frac{\mathcal{N} }{ K^2+1} + \sqrt{N}\right)X\\& \lesssim \epsilon\sum_{\ell}|\ell|^2c_\ell^\dagger c_\ell + \frac{1}{\epsilon}\frac{\mathcal{N}^{2m}}{N^{2m}}\left(\frac{\mathcal{N}}{K^2+1} + \frac{\sqrt{N}}{\epsilon}\right), \end{align*} $$

where we have used $X^\dagger X\lesssim N^2 \frac {\mathcal {N}^{2m}}{N^{2m}}$ and $[X,\mathcal {N}]=0$ . Regarding the second term in Eq. (47), let us use $\sum _{|\ell |>K}|\ell |^{-\frac {1}{2}}|N^4\lambda _{0,\ell }|^2 a_{\ell }^\dagger a_{\ell }\lesssim \frac {\mathcal {N}}{\sqrt {K+1}}$ as well as the fact that

$$ \begin{align*} \sum_{\ell}\sqrt{|\ell|}(c_\ell-a_\ell) (c_\ell-a_\ell)^\dagger\lesssim \mathcal{N} \end{align*} $$

by Eq. (39), to estimate for $\kappa>0$

$$ \begin{align*} \pm \sum_{|\ell|>K}\lambda_{0,\ell}\, a_{-\ell}^\dagger (c_\ell - a_{\ell})^\dagger X \pm \mathrm{H.c.}\lesssim \frac{1}{\kappa \sqrt{K+1}}\mathcal{N}+\frac{\kappa }{N^4}X^\dagger \mathcal{N} X\leq \frac{1}{\kappa \sqrt{K+1}}\mathcal{N}+\kappa \frac{\mathcal{N}^{2m}}{N^{2m}}\mathcal{N}. \end{align*} $$

Regarding the final term in Eq. (47) we have that $\sum _{\ell }|\Lambda ^{(n)}_{\ell ,0}|\lesssim N^{-\frac {3}{2}}$ by Eq. (46), and hence

$$ \begin{align*} \pm \sum_{|\ell|>K}\sum_{n\neq 0} \lambda_{0,\ell}\, \Lambda^{(n)}_{\ell,0} a_0^{2\dagger }a_n^\dagger a_n X \pm \mathrm{H.c.}\lesssim N^{-\frac{1}{2}}\mathcal{N}. \end{align*} $$

For $m=0$ , the choice $\kappa :=\epsilon $ yields Eq. (43) and for $m=1$ the choice $\kappa :=\sqrt {N}$ and $\epsilon :=\frac {1}{\sqrt {N}}$ yields Eq. (45).

Regarding the proof of Eq. (44), let us define the operators $d_{k,\ell }:=\lambda _{k,\ell } N^{\frac {3}{2}} a_{k-\ell }^\dagger a_k$ and write $a^\dagger _\ell d_{k,\ell }=c_\ell ^\dagger d_{k,\ell }+d_{k,\ell } (c_\ell -a_\ell )^\dagger +[(c_\ell -a_\ell )^\dagger ,d_{k,\ell }]$ . We compute

$$ \begin{align*} \sum_\ell &[(c_\ell-a_\ell)^\dagger,d_{k,\ell}]=\frac{1}{2} N^{\frac{3}{2}}\Big(a^{3\dagger}_{0} \sum_{i j \ell}\frac{1}{3}\Lambda^{(0)}_{\ell,-i}\lambda_{k,\ell}[ a_i a_{-i-\ell} ,a_{k-\ell}^\dagger a_k]\\ & \ \ \ \ +a^{2\dagger}_{0}\sum_{|n|\leq K}\sum_{i \ell}\Lambda^{(n)}_{\ell,n-i}\lambda_{k,\ell}[ a^\dagger_{n} a_i a_{n-i-\ell} ,a_{k-\ell}^\dagger a_k]\Big)\frac{a_0^\dagger a_0^2}{N^{\frac{3}{2}}}\\ &=\frac{a_0^{3\dagger}}{N^{\frac{3}{2}}}\mu^{(1)}_k a_{k}a_{-k}\frac{a_0^\dagger a_0^2}{N^{\frac{3}{2}}}+\frac{a_0^{2\dagger}}{N}\sum_{|n|\leq K}\mu^{(2)}_{k,n} a^{\dagger}_n a_{k}a_{n-k}\frac{a_0^\dagger a_0^2}{N^{\frac{3}{2}}}-\frac{a_0^{2\dagger}}{N}\sum_{i,\ell }\mu^{(3)}_{k, i,\ell}a^\dagger_{k-\ell}a_i a_{k-i-\ell}\frac{a_0^\dagger a_0^2}{N^{\frac{3}{2}}}, \end{align*} $$

where we define the coefficients

$$ \begin{align*} \mu^{(1)}_k&:=N^{3}\sum_{\ell }\frac{1}{3}\Lambda^{(0)}_{\ell, k}\lambda_{k,\ell},\\\mu^{(2)}_{k,n}&:=N^{\frac{5}{2}}\sum_{\ell}\Lambda^{(n)}_{\ell, k}\lambda_{k,\ell},\\\mu^{(3)}_{k,i,\ell }&:=N^{\frac{5}{2}}\Lambda^{(k)}_{\ell,k-i}\lambda_{k,\ell}. \end{align*} $$

Using again Eq. (46) and $|\lambda _{k,\ell }|\lesssim N^{-2}(1+\frac {|\ell |^2}{N})^{-1}$ , we immediately obtain $|\mu ^{(1)}_k|\lesssim \frac {1}{\sqrt {N}}$ , $|\mu ^{(2)}_{k,n}|\lesssim \frac {1}{N}$ , $|\mu ^{(3)}_{k,i,\ell }|\lesssim N^{-\frac {3}{2}}$ and $\sum _{i}|\mu ^{(3)}_{k,i,\ell }|\lesssim \frac {1}{N}$ , and therefore by Cauchy-Schwarz

$$ \begin{align*} \sum_\ell \left([(c_\ell-a_\ell)^\dagger,d_{k,\ell}]\frac{a_0^\dagger a_0^2}{N^{\frac{3}{2}}}+\mathrm{H.c.}\right)\lesssim \frac{\mathcal{N}}{\sqrt{N}}. \end{align*} $$

Consequently,

$$ \begin{align*} & \ \ \ \ \pm \left(\sum_{0<|k|\leq K,\ell}\lambda_{k,\ell} a_\ell^\dagger a_{k-\ell}^\dagger a_k\, a_0^\dagger a_0^2+\mathrm{H.c.}\right)= \left(\sum_{\ell}a_\ell^\dagger d_{k,\ell} \frac{a_0^\dagger a_0^2}{N^{\frac{3}{2}}}+\mathrm{H.c.}\right)\\ & \lesssim \sum_{0<|k|\leq K}\bigg(\frac{\epsilon}{K^3}\sum_{\ell}|\ell|^2c_\ell^\dagger c_\ell+\frac{K^3}{\epsilon}\sum_{\ell}\frac{1}{|\ell|^2} d_{k,\ell}^\dagger d_{k,\ell}+\frac{\epsilon}{K^3}\sum_{\ell}d_{k,\ell} d_{k,\ell}^\dagger\\ & \ \ \ \ +\frac{K^3}{\epsilon}\sum_{\ell}\frac{a_0^{3\dagger} a_0}{N^2}(c_\ell-a_\ell) (c_\ell-a_\ell)^\dagger \frac{a_0^\dagger a_0^3}{N^2}+\frac{\mathcal{N}}{\sqrt{N}}\bigg)\\ & \lesssim \epsilon\sum_{\ell}|\ell|^2 c_\ell^\dagger c_\ell+\frac{K^3}{\epsilon}\sum_{0<|k|\leq K,\ell}\frac{1}{|\ell|^2} d_{k,\ell}^\dagger d_{k,\ell}+\frac{\epsilon}{K^3} \sum_{0<|k|\leq K,\ell}d_{k,\ell} d_{k,\ell}^\dagger\\ & \ \ \ \ +\frac{K^3}{\epsilon}\sum_{0<|k|\leq K,\ell}\frac{a_0^{3\dagger} a_0}{N^2}(c_\ell-a_\ell) (c_\ell-a_\ell)^\dagger \frac{a_0^\dagger a_0^3}{N^2}+K^3\frac{\mathcal{N}}{\sqrt{N}}. \end{align*} $$

Similar to the proof of Eq. (43), we observe that $\frac {1}{K^3}\sum _{0<|k|\leq K,\ell }d_{k,\ell } d_{k,\ell }^\dagger \lesssim \frac {\mathcal {N}}{N}\mathcal {N}$ and, using Eq. (39),

$$ \begin{align*} & \sum_{0<|k|\leq K,\ell}\frac{a_0^{3\dagger} a_0}{N^2}(c_\ell-a_\ell) (c_\ell-a_\ell)^\dagger \frac{a_0^\dagger a_0^3}{N^2}\lesssim K^3 \frac{a_0^{3\dagger} a_0}{N^2}\frac{\mathcal{N}^2}{N} \frac{a_0^\dagger a_0^3}{N^2}\leq K^3 \frac{\mathcal{N}^2}{N},\\& \sum_{0<|k|\leq K,\ell}\frac{1}{|\ell|^2} d_{k,\ell}^\dagger d_{k,\ell}\lesssim K^3 \frac{\mathcal{N}}{\sqrt{N}}+K^3 N^3\sum_{\ell}\frac{1}{|\ell|^2}|\lambda_{k,\ell}|^2 a_k^\dagger a_{k-\ell}^\dagger a_{k-\ell} a_k\lesssim K^3 \frac{\mathcal{N}}{\sqrt{N}}+K^3\frac{\mathcal{N}^2}{N}.\\[-47pt] \end{align*} $$

Having Lemma 2, Lemma 3 and Lemma 4 at hand, we can use the lower bound in Eq. (32) in order to derive the following Theorem 3, which provides strong lower bounds on the quantity . Note, however, that Theorem 3 is only applicable for states $\Psi $ which satisfy the strong condition that $\Psi $ is in the spectral subspace $\mathcal {N}\leq \epsilon N$ , that is,

(48)

where the orthogonal projection is defined by means of functional calculus. In the following we will refer to the property in Eq. (48) as (BEC) in the spectral sense . Furthermore, we refer to a Hilbert space element $\Psi $ as a state, in case it satisfies $\|\Psi \|=1$ .

Theorem 3. There exist constants $\delta ,C,N_0>0$ and $\epsilon>0$ , such that

$$ \begin{align*} \langle \Psi,H_N\Psi\rangle\geq \frac{1}{6} b_{\mathcal{M}}(V)N+\delta \langle \Psi,\sum_{k}|k|^2 c_k^\dagger c_k \Psi \rangle+\delta\langle \Psi,\mathcal{N} \Psi\rangle-C\sqrt{N} \end{align*} $$

for any state $\Psi $ satisfying and $N\geq N_0$ , where $\mathcal {N}:=\sum _{k\neq 0}a_k^\dagger a_k$ .

Proof. By Eq. (32) together with the estimates in Lemma 2, Lemma 3 and Lemma 4 we have for $\alpha ,\tau ,\epsilon '>0$ and $K\geq K_0(\alpha ,\tau )$

(49) $$ \begin{align} H_N& \geq \frac{1}{6} b_{\mathcal{M}}(V)N + (1-\epsilon')\sum_k |k|^2 c_k^\dagger c_k -C_{K,\epsilon'} \sum_k |k|^2 c_k^\dagger \left(\frac{\mathcal{N}}{N}+N^{-\frac{1}{2}}\right) c_k\\\nonumber & \quad - \alpha\sum_k |k|^{2\tau}a_k^\dagger a_k - \left(\epsilon'+C_{K,\epsilon'}\frac{\mathcal{N}}{N}+C_{K,\epsilon'}N^{-\frac{1}{3}}+\frac{C}{\epsilon' \sqrt{K+1}}\right) \mathcal{N}-C_{K,\epsilon'}\sqrt{N}, \end{align} $$

where $C,C_{K,\epsilon '}$ and $K_0(\alpha ,\tau )$ are suitable constants. In the following let $\Psi $ be a state satisfying and define $\Psi _k:=c_k \Psi $ . By the definition of $c_k$ it is clear that , and therefore

(50)

In a similar fashion we have

(51) $$ \begin{align} \langle \Psi,\mathcal{N}^2\Psi\rangle\leq \epsilon N \langle \Psi,\mathcal{N} \Psi\rangle. \end{align} $$

Furthermore, note that for a suitable constant $D_1>0$

(52) $$ \begin{align} \sum_k |k|^{2\tau}a_k^\dagger a_k\leq D_1 \sum_k |k|^2 c_k^\dagger c_k+D_1\frac{\mathcal{N}^2}{N}+D_1N^{\frac{1}{2}} \end{align} $$

by Eq. (40) for $\tau <\frac {1}{2}$ , and by Eq. (39) we have

$$ \begin{align*} \mathcal{N}\left(1-R\frac{\mathcal{N}}{N}\right)\leq R \sum_k |k|^2 c_k^\dagger c_k+R \end{align*} $$

for a suitable constant $R>0$ . Using with $\epsilon $ small enough such that $R\epsilon <1$ , we therefore obtain

(53) $$ \begin{align} \langle \Psi,\mathcal{N}\Psi\rangle\leq D_2\left\langle \Psi, \sum_k |k|^2 c_k^\dagger c_k\Psi\right\rangle+D_2, \end{align} $$

for a suitable constant $D_2>0$ . Combining Eq. (49)-(53), therefore yields for suitable constants D and $D_{K,\epsilon ',\alpha }$ , and $0<\epsilon <\frac {1}{R}$ ,

$$ \begin{align*} \langle \Psi, H_N\Psi\rangle & \geq \left[1-D\left(\epsilon'+\alpha+\frac{D}{\epsilon' \sqrt{K+1}}\right)-D_{K,\epsilon',\alpha}\left(\epsilon+N^{-\frac{1}{3}}\right)\right] \langle \Psi, \sum_k |k|^2 c_k^\dagger c_k\Psi\rangle\\& \quad +\frac{1}{6} b_{\mathcal{M}}(V)N -D_{K,\epsilon',\alpha}\sqrt{N}. \end{align*} $$

We can now make our choice of parameters concrete. First we take choose $\tau $ such that $0<\tau < \frac {1}{2}$ and $\alpha ,\epsilon '>0$ small enough, such that $D\left (\epsilon '+\alpha \right )<\frac {1}{2}$ , and then we take $K\geq K_0(\alpha ,\tau )$ large enough, such that

$$ \begin{align*} D\left(\epsilon'+\alpha+\frac{D}{\epsilon' \sqrt{K+1}}\right)\leq \frac{1}{2}. \end{align*} $$

Finally, we take $0<\epsilon <\frac {1}{R}$ small enough and N large enough, such that

$$ \begin{align*} D_{K,\epsilon',\alpha}\left(\epsilon+N^{-\frac{1}{3}}\right)\leq \frac{1}{2}.\\[-41pt] \end{align*} $$

It has been verified in [Reference Nam, Ricaud and Triay23], for the more general setting of particles being confined by an additional external potential, that any approximate ground state $\Psi _N$ of the operator $H_N$ satisfies complete Bose-Einstein condensation

(54) $$ \begin{align} \langle \Psi_N,\mathcal{N}\Psi_N\rangle=o_{N\rightarrow \infty}(N). \end{align} $$

Adapting the localization procedure presented in [Reference Lieb and Solovej19, Theorem A.1] in the form stated in [Reference Lewin, Nam, Serfaty and Solovej16, Proposition 6.1] for the following Lemma 5 allows us to lift Bose-Einstein condensation in the sense of Eq. (54), to Bose-Einstein condensation in the spectral sense, which is a crucial assumption of the previous Theorem 3.

Lemma 5. Let $\Psi $ satisfy with $\delta \leq N$ . Then there exists a constant $C>0$ , such that there exists for all $1\leq M\leq N$ states $\Phi $ satisfying and

(55) $$ \begin{align} \langle \Phi, H_N \Phi\rangle\leq E_N+ C \left(1-\frac{2\langle \Psi,\mathcal{N}\Psi}\rangle{M}\right)^{-1} \left(\frac{\sqrt{N}}{M}+\frac{N}{M^2}+\delta\right). \end{align} $$

Furthermore, there exists a state $\widetilde {\Phi }$ such that and

(56) $$ \begin{align} \langle \Psi, \mathcal{N}\Psi\rangle \leq \langle \Phi,\mathcal{N} \Phi \rangle+\frac{C N}{\langle \widetilde{\Phi},H_N \widetilde{\Phi}}-E_N\rangle \left(\frac{\sqrt{N}}{M}+\frac{N}{M^2}+\delta \right). \end{align} $$

Proof. In the following let $f,g:\mathbb {R}\rightarrow [0,1]$ be smooth functions satisfying $f^2+g^2=1$ and $f(x)=1$ for $x\leq \frac {1}{2}$ , as well as $f(x)=0$ for $x\geq 1$ , and let us define $ m:= \|f \left (\frac {\mathcal {N}}{M}\right )\Psi \|^2$ and

$$ \begin{align*} \Phi&:=\frac{1}{\sqrt{m}} f \left(\frac{\mathcal{N}}{M}\right)\Psi, \\ \widetilde \Phi&:=\frac{1}{\sqrt{1-m}} g \left(\frac{\mathcal{N}}{M}\right)\Psi. \end{align*} $$

Note that $\|\Phi \|=\|\widetilde \Phi \|=1$ , $0\leq m \leq 1$ and clearly we have and . Making use of the algebraic identity

$$ \begin{align*} H_N=f \left(\frac{\mathcal{N}}{M}\right) H_N f \left(\frac{\mathcal{N}}{M}\right)+g \left(\frac{\mathcal{N}}{M}\right) H_N g \left(\frac{\mathcal{N}}{M}\right)+\mathcal{E} \end{align*} $$

with the residual term

$$ \begin{align*} \mathcal{E}:=\frac{1}{2}\left[f \left(\frac{\mathcal{N}}{M}\right),\left[H_N, f \left(\frac{\mathcal{N}}{M}\right)\right]\right]+\frac{1}{2}\left[g \left(\frac{\mathcal{N}}{M}\right),\left[H_N, g \left(\frac{\mathcal{N}}{M}\right)\right]\right], \end{align*} $$

we obtain

$$ \begin{align*} m \langle \Phi,H_N \Phi\rangle+(1-m)\langle \widetilde \Phi,H_N \widetilde \Phi\rangle= E_N+\delta-\langle \Psi,\mathcal{E}\Psi\rangle. \end{align*} $$

In order to estimate , let $\pi ^0$ denote the projection onto the constant function in $L^2(\Lambda )$ and $\pi ^1:=1-\pi ^0$ . Then we can rewrite $\mathcal {E}$ as

$$ \begin{align*} \mathcal{E}=\frac{1}{4M^2}\sum_{I,J\in \{0,1\}^3}\sum_{ijk,\ell m n}(\pi^{I_1}\pi^{I_2}\pi^{I_3} V_N \pi^{J_1}\pi^{J_2}\pi^{J_3})_{ijk, \ell m n }a_k^\dagger a_j^\dagger a_i^\dagger X_{I,J} a_\ell a_m a_n, \end{align*} $$

with

$$ \begin{align*} X_{I,J}:=M^2 \left[f \left(\frac{\mathcal{N}+\, \#_I}{M}\right)-f \left(\frac{\mathcal{N}+\, \#_J}{M}\right)\right]^2+M^2 \left[g \left(\frac{\mathcal{N}+\, \#_I}{M}\right)-g \left(\frac{\mathcal{N}+\, \#_J}{M}\right)\right]^2 \end{align*} $$

and $\#_I$ counting the number of indices in I that are equal to $1$ . Using $0\leq X_{I,J}\leq X$ , where

we obtain by the Cauchy-Schwarz inequality

(57) $$ \begin{align} \pm \mathcal{E}\lesssim \frac{1}{4M^2}\sum_{I\in \{0,1\}^3}\sum_{ijk,\ell m n}(\pi^{I_1}\pi^{I_2}\pi^{I_3} V_N \pi^{I_1}\pi^{I_2}\pi^{I_3})_{ijk, \ell m n }a_k^\dagger a_j^\dagger a_i^\dagger X a_\ell a_m a_n. \end{align} $$

In the following we want to show that for any $I\in \{0,1\}^3$ , the $\Psi $ -expectation value of the corresponding term appearing in the sum on the right side of Eq. (57) is of the order $\sqrt {N}M+N$ . For $I=(0,0,0)$ we have

$$ \begin{align*} (V_N)_{000,000}(a_0^\dagger)^3 X a_0^3 \lesssim N^{-2}\|X\|(a_0^\dagger)^3 a_0^3\leq \left(\|\nabla f\|^2+\|\nabla g\|^2\right)N. \end{align*} $$

Similarly,

$$ \begin{align*} \sum_{k\neq 0}(V_N)_{001,001}(a_0^\dagger)^2 a_k^\dagger X a_k a_0^2\lesssim M\leq N \end{align*} $$

in the case $I=(0,0,1)$ . Regarding the case $I=(0,1,1)$ , let us first observe that we have the upper bound

(58) $$ \begin{align} \sum_{jk,mn\neq 0}(V_N)_{0jk,0mn}a_0^\dagger a_k^\dagger a_j^\dagger X a_m a_n a_0\leq C_N \sum_{k}|k|^2 a_k^\dagger\left(\sum_{j\neq 0}a_0^\dagger a_j^\dagger X a_j a_0\right)a_k \end{align} $$

with the constant $C_N$ being defined as

$$ \begin{align*} C_N:=\sup_{m, n \neq0}\left\{ \sum_{j,k\neq 0}\frac{\left|(V_N)_{0jk,0mn}\right|}{|k|^2}\right\}=\frac{1}{N^2}\left\{\sup_{p,q\neq 0}\sum_{t:p+t\neq 0}\frac{\big|V \big(N^{-\frac{1}{2}}t\big)\big|}{|p+t|^2}\right\}. \end{align*} $$

Due to our regularity assumptions on V we have $\big |V \big (N^{-\frac {1}{2}}t\big )\big |\lesssim \frac {1}{1+N^{-1}|t|^2}$ and therefore

(59) $$ \begin{align} \nonumber C_N & \lesssim N^{-2}\sum_{t:p+t\neq 0}\frac{1}{|p+t|^2(1+N^{-1}|t|^2)}\lesssim N^{-2}\int_{\mathbb{R}^3}\frac{\mathrm{d}x}{|p+x|^2(1+N^{-1}|x|^2)}\\ & \leq N^{-2}\int_{\mathbb{R}^3}\frac{\mathrm{d}x}{|x|^2(1+N^{-1}|x|^2)} = N^{-2}4\pi \int_0^\infty \frac{\mathrm{d}r}{1+N^{-1}r^2}=2\pi^2N^{-\frac{3}{2}}, \end{align} $$

where we have used the Hardy-Littlewood inequality in the first estimate of Eq. (59). Since

$$ \begin{align*} \sum_{j\neq 0}a_0^\dagger a_j^\dagger X a_j a_0\lesssim MN \end{align*} $$

we obtain by Eq. (58)

$$ \begin{align*} & \left\langle \Psi, \sum_{jk,mn\neq 0}(V_N)_{0jk,0mn}a_0^\dagger a_k^\dagger a_j^\dagger X a_m a_n a_0 \, \Psi\right\rangle\lesssim N^{-\frac{1}{2}}M\left\langle \Psi, \sum_{k}|k|^2 a_k^\dagger a_k \, \Psi\right\rangle\\ & \ \ \ \ \ \ \ \leq N^{-\frac{1}{2}}M\left\langle \Psi, H_N \, \Psi\right\rangle\leq N^{-\frac{1}{2}}M\left(E_N+N\right)\lesssim N^{\frac{1}{2}}M, \end{align*} $$

where we have used the assumption $\delta \leq N$ and the upper bound on $E_N$ derived in Theorem 4. The only distinguished case left is $I=(1,1,1)$ . We start its analysis by defining

$$ \begin{align*} \mathbb{V}_{\alpha,\beta}: =\frac{1}{4M^2}\underset{\# I=\alpha, \# J=\beta}{\sum_{I,J\in \{0,1\}^3:}}\sum_{ijk,\ell m n}(\pi^{I_1}\pi^{I_2}\pi^{I_3} V_N \pi^{J_1}\pi^{J_2}\pi^{J_3})_{ijk, \ell m n }a_k^\dagger a_j^\dagger a_i^\dagger X_{I,J} a_\ell a_m a_n, \end{align*} $$

which allows us to estimate, using the Cauchy-Schwarz inequality,

$$ \begin{align*} \mathbb{V}_{3,3}\leq H_N - \left(\mathbb{V}_{2,3} + \mathbb{V}_{3,2} + \mathbb{V}_{1,3} + \mathbb{V}_{3,1} + \mathbb{V}_{0,3} + \mathbb{V}_{3,0}\right)\leq H_N + \frac{1}{2}\mathbb{V}_{3,3} + 6\left(\mathbb{V}_{0,0} + \mathbb{V}_{1,1} + \mathbb{V}_{2,2}\right). \end{align*} $$

From the previous cases we know that

$$ \begin{align*} \langle \Psi,(\mathbb{V}_{0,0} + \mathbb{V}_{1,1} + \mathbb{V}_{2,2})\Psi\rangle&\lesssim N^{\frac{1}{2}}M+N,\\\langle \Psi,H_N \Psi\rangle&\lesssim N, \end{align*} $$

and therefore . Summarizing what we have so far yields the inequality

$$ \begin{align*} m\langle \Phi,H_N \Phi\rangle+(1-m)\langle \widetilde \Phi,H_N \widetilde \Phi\rangle\leq E_N+\delta+C \left(\frac{\sqrt{N}}{M}+\frac{N}{M^2} \right). \end{align*} $$

Using and the simple observation that immediately yields Eq. (55), and using we obtain for a suitable constant $C>0$

$$ \begin{align*} 1-m\leq \frac{C }{\langle \widetilde{\Phi},H_N \widetilde{\Phi}}-E_N\rangle \left(\frac{\sqrt{N}}{M}+\frac{N}{M^2}+\delta \right). \end{align*} $$

In order to derive Eq. (56), we note that $\mathcal {N}=f \left (\frac {\mathcal {N}}{M}\right )\mathcal {N} f \left (\frac {\mathcal {N}}{M}\right )+g \left (\frac {\mathcal {N}}{M}\right ) \mathcal {N} g \left (\frac {\mathcal {N}}{M}\right )$ and therefore

$$ \begin{align*} \langle \Psi,\mathcal{E}\Psi\rangle=m\langle \Phi,\mathcal{N}\Phi\rangle+(1-m)\langle \widetilde \Phi,\mathcal{N}\widetilde \Phi\rangle\leq \langle \Phi,\mathcal{N}\Phi\rangle+(1-m)N.\\[-37pt] \end{align*} $$

Before we come to the lower bound on the ground state energy $E_N$ in the main result of this Section Corollary 2, let us first state the corresponding upper bound in the subsequent Theorem 4, which has essentially been verified in [Reference Nam, Ricaud and Triay23]. To be precise, it has been shown in [Reference Nam, Ricaud and Triay23] that $E_N\leq \frac {1}{6}b_{\mathcal {M}}(V)N+CN^{\frac {2}{3}}$ , and, as is explained in [Reference Nam, Ricaud and Triay25], the method in [Reference Nam, Ricaud and Triay23] can be improved to yield $E_N\leq \frac {1}{6}b_{\mathcal {M}}(V)N+CN^{\frac {1}{2}}$ as well. However, since the computations in the proof of Theorem 4 are relevant for the proof of the upper bound in Theorem 1, and for the sake of completeness, we are nevertheless going to verify Theorem 4 in detail in the subsequent Section 3.

Using Bose-Einstein condensation in the spectral sense, Theorem 3 allows us to derive an asymptotically correct lower bound on the ground state energy in Corollary 2 with an error of the order $\sqrt {N}$ , see Eq. (62). In this context we call $\Psi _N$ an approximate ground state, in case $\|\Psi _N\|=1$ and there exists a constant $C>0$ such that

(60) $$ \begin{align} \langle \Psi_N,H_N\Psi_N\rangle\leq E_N+C. \end{align} $$

Note that the assumption in Eq. (60) is more restrictive compared to the one employed in [Reference Nam, Ricaud and Triay23], where the authors call $\Psi _N$ an approximate ground state in case $\|\Psi _N\|=1$ and

(61) $$ \begin{align} \lim_{N\rightarrow \infty}\frac{1}{N}\langle \Psi_N,H_N\Psi_N\rangle= \frac{1}{6}b_{\mathcal{M}}(V). \end{align} $$

The fact that Eq. (60) implies Eq. (61) follows immediately from the leading-order asymptotics in Eq. (3), which has been verified in [Reference Nam, Ricaud and Triay23], together with the trivial lower bound .

Corollary 2. The ground state $\Psi _N^{\mathrm {GS}}$ of the operator $H_N$ satisfies for a suitable $C>0$

$$ \begin{align*} \langle \Psi_N^{\mathrm{GS}},\mathcal{N}\Psi_N^{\mathrm{GS}}\rangle\leq C\sqrt{N}, \end{align*} $$

and we have the lower bound

(62) $$ \begin{align} E_N\geq \frac{1}{6}b_{\mathcal{M}}(V)N-C\sqrt{N}. \end{align} $$

Furthermore there exists a constant $C>0$ and states $\Phi _N$ , such that $\Phi _N$ is an approximate ground state of $H_N$ satisfying (BEC) in the spectral sense with rate $\frac {1}{\sqrt {N}}$ , that is,

and we have the estimate on the kinetic energy $\left \langle \Phi _N, \sum _{k}|k|^2 c_k^\dagger c_k \Phi _N \right \rangle \leq C\sqrt {N}$ .

Proof. From the results in [Reference Nam, Ricaud and Triay23] we know that the ground state $\Psi ^{\mathrm {GS}}_N$ of $H_N$ satisfies

$$ \begin{align*} \langle \Psi^{\mathrm{GS}}_N,\mathcal{N}\Psi^{\mathrm{GS}}_N\rangle=o_{N\rightarrow \infty}(N). \end{align*} $$

Consequently, we know by Lemma 5 that there exist states $\xi _N$ satisfying

(63) $$ \begin{align} \langle \xi_N,H_N \xi_N\rangle\leq E_N+C\left(\frac{1}{\epsilon \sqrt{N}}+\frac{1}{\epsilon^2 N}\right) \end{align} $$

and , where we choose $\epsilon>0$ as in Theorem 3. By Theorem 3 and Theorem 4 we therefore obtain for a suitable constant $C>0$

(64) $$ \begin{align} \nonumber & \frac{1}{6} b_{\mathcal{M}}(V)N+\delta \left\langle \xi_N , \sum_{k}|k|^2 c_k^\dagger c_k \xi_N \right\rangle+\delta\langle \xi_N,\mathcal{N} \xi_N\rangle-C\sqrt{N} \leq \langle \xi_N,H_N \xi_N\rangle\\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \leq E_N +C\leq \frac{1}{6} b_{\mathcal{M}}(V)N+C \sqrt{N}. \end{align} $$

This immediately implies $E_N\geq \frac {1}{6} b_{\mathcal {M}}(V)N-C\sqrt {N}$ for a suitable constant $C>0$ and

(65) $$ \begin{align} \langle \xi_N,\mathcal{N}\xi_N\rangle=O_{N\rightarrow \infty} \left(\sqrt{N}\right) , \end{align} $$

as well as $\left \langle \xi _N , \sum _{k}|k|^2 c_k^\dagger c_k \xi _N \right \rangle =O_{N\rightarrow \infty } \left (\sqrt {N}\right ) $ . Furthermore, there exist by Lemma 5 states $\widetilde {\xi }_N$ satisfying and

(66) $$ \begin{align} \langle \Psi^{\mathrm{GS}}_N, \mathcal{N}\Psi^{\mathrm{GS}}_N\rangle \lesssim \langle \xi_N,\mathcal{N} \xi_N \rangle+\frac{ \sqrt{N}}{\langle \widetilde {\xi}_N,H_N \widetilde {\xi}_N}-E_N\rangle\lesssim \sqrt{N}+\frac{ \sqrt{N}}{\langle \widetilde {\xi}_N,H_N \widetilde {\xi}_N\rangle -E_N} . \end{align} $$

In the following we show by a contradiction argument, similar to the one employed in the proof of [Reference Boccato, Brennecke, Cenatiempo and Schlein4, Theorem 1.1], that

(67) $$ \begin{align} \liminf_N \left\{\langle \widetilde \xi_N,H_N \widetilde \xi_N\rangle-E_N\right\}=\infty. \end{align} $$

For this purpose let us assume that Eq. (67) is violated, that is, we assume that there exists a subsequence $N_j$ and a constant $C>0$ such that . Let us complete this subsequence to a proper sequence by defining $\xi ^{\prime }_N:=\widetilde \xi _N$ in case $N=N_j$ for some j and $\xi ^{\prime }_N:=\Psi ^{\mathrm {GS}}_N$ otherwise. Clearly $\xi ^{\prime }_N$ is a sequence of approximate ground states, see Eq. (60), and as such $\xi ^{\prime }_N$ satisfies complete (BEC) by the results in [Reference Nam, Ricaud and Triay23], that is, . This is, however, a contradiction to

$$ \begin{align*} \langle \xi^{\prime}_{N_j},\mathcal{N}\xi^{\prime}_{N_j}\rangle=\langle \widetilde \xi_{N_j},\mathcal{N}\widetilde \xi_{N_j}\rangle\geq \frac{\epsilon}{2}N, \end{align*} $$

which concludes the proof of Eq. (67). Combining Eq. (66) and Eq. (67) yields

$$ \begin{align*} \langle \Psi^{\mathrm{GS}}_N,\mathcal{N}\Psi^{\mathrm{GS}}_N\rangle\leq C\sqrt{N}, \end{align*} $$

for a suitable constant $C>0$ . Applying again Lemma 5 for the state $\Psi ^{\mathrm {GS}}_N$ and $M:=K \sqrt {N}$ , we obtain states $\Phi _N$ satisfying and

$$ \begin{align*} \langle \Phi_N, H_N \Phi_N\rangle\leq E_N+\frac{C}{1-\frac{2}{K\sqrt{N}}\langle \Psi^{\mathrm{GS}}_N,\mathcal{N}\Psi^{\mathrm{GS}}_N}\rangle\leq E_N+\frac{C}{1-\frac{2C}{K}}, \end{align*} $$

for a large enough $C>0$ . Consequently $\Phi _N$ is a sequence of approximate ground states for $K>2C$ . Finally we notice that the states $\Phi _N$ satisfy the chain of inequalities in Eq. (64) as well, and therefore

$$ \begin{align*} \left\langle \Phi_N , \sum_{k}|k|^2 c_k^\dagger c_k \Phi_N \right\rangle=O_{N\rightarrow \infty} \left(\sqrt{N}\right).\\[-47pt] \end{align*} $$

3 First-order upper bound

It is the goal of this section to introduce a trial state $\Gamma $ , which simultaneously annihilates the variables $c_k$ for $k\neq 0$ and $\psi _{\ell m n}$ in case $(\ell ,m,n)\neq 0$ , at least in an approximate sense, which allows us to verify the upper bound on the ground state energy $E_N$ in Theorem 4. For the rest of this Section we specify the parameter K introduced above the definition of $\pi _K$ in Eq. (17) as $K:=0$ , which especially means that with $\eta _{ijk}:=(T-1)_{i j k, 0 0 0}$ we have

(68) $$ \begin{align} c_k &=a_k+\frac{1}{2}\sum_{i j}\eta_{ijk} \, a_i^\dagger a_j^\dagger a_{0}^3, \end{align} $$

(69) $$ \begin{align} \psi_{i j k} &=a_ia_ja_k+\eta_{ijk} \, a_0^3. \end{align} $$

In order to find a suitable state $\Gamma $ , let $\eta _{ijk}:=(T-1)_{i j k, 0 0 0}$ and let us follow the strategy in [Reference Nam, Ricaud and Triay23], respectively in the case of Bose gases with two-particle interactions see, for example, [Reference Boccato, Brennecke, Cenatiempo and Schlein3, Reference Brennecke, Caporaletti and Schlein7, Reference Hainzl, Schlein and Triay13], by defining the generator

(70) $$ \begin{align} \mathcal{G}:=\frac{1}{6}\sum_{i j k}\eta_{ijk}\, a_i^\dagger a_j^\dagger a_k^\dagger a_0^3 \end{align} $$

of a unitary group $U_s:=e^{s\mathcal {G}^\dagger -s\mathcal {G}}$ and $U:=U_1$ . The generator $\mathcal {G}$ is chosen, such that

$$ \begin{align*} c_k=a_k+[a_k,\mathcal{G}], \end{align*} $$

and in particular, as we show this section, the unitary U has the property $U^{-1}c_k U\approx a_k$ and $U^{-1}\psi _{ijk}U\approx a_i a_j a_k$ in a suitable sense. Denoting with

(71) $$ \begin{align} \Gamma_0(x_1,\dots ,x_N):=1 \end{align} $$

the constant function in $L^2_{\mathrm {sym}} \left (\Lambda ^N\right )$ , that is, $a_k \Gamma _0=0$ for $k\neq 0$ , we observe that $\Gamma :=U \Gamma _0$ is a suitable trial state for the (approximate) annihilation of $c_k$ , given $k\neq 0$ , and $\psi _{\ell m n}$ , given $(\ell ,m,n)\neq 0$ , where the action of the unitary U introduces a three-particle correlation structure on the completely uncorrelated wavefunction $\Gamma _0$ . We note at this point, that the action of the unitary operator U only creates an $O(1)$ amount of particles, in the sense that

(72) $$ \begin{align} U_{-s} \, \mathcal{N}^m U_s\leq e^{C_m |s|} (\mathcal{N}+1)^m, \end{align} $$

as is proven in Appendix A, see Lemma A1.

For the purpose of verifying that $U^{-1}\psi _{ijk}U$ is approximately identical to $a_i a_j a_k$ , we first apply Duhamel’s formula, which yields

(73) $$ \begin{align} U^{-1} a_i a_j a_k U=a_i a_j a_k - \int_0^1 U_{-s} \big[a_i a_j a_k ,\mathcal{G}\big]U_s \, \mathrm{d}s+\int_0^1 U_{-s} \big[a_i a_j a_k ,\mathcal{G}^\dagger \big]U_s \, \mathrm{d}s. \end{align} $$

Furthermore, note that we can write

$$ \begin{align*} \big[a_i a_j a_k ,\mathcal{G}\big]=\eta_{ijk}a_0^3+(\delta_1 \psi)_{ijk}+(\delta_2 \psi)_{ijk} \end{align*} $$

using the definition

Therefore we can identify the transformed operators $U^{-1}\psi _{ijk}U$ as

(74) $$ \begin{align} \nonumber U^{-1}\psi_{ijk} U & = a_i a_j a_k + \int_0^1 U_{-s} \left\{ \big[a_i a_j a_k ,\mathcal{G}^\dagger \big] - (\delta_1 \psi)_{ijk} - (\delta_2 \psi)_{ijk} - \eta_{ijk}a_0^3 \right\} U_s \mathrm{d}s + \eta_{ijk} U^{-1} a_0^3 U\\\nonumber &= a_i a_j a_k + \int_0^1 U_{-s} \big[a_i a_j a_k ,\mathcal{G}^\dagger \big] U_s \mathrm{d}s - \int_0^1 U_{-s} (\delta_1 \psi)_{ijk} U_s \mathrm{d}s - \int_0^1 U_{-s} (\delta_2 \psi)_{ijk} U_s \mathrm{d}s\\& \quad +\int_0^1\int_s^1 U_{-t} \eta_{ijk} \big[a_0^3 ,\mathcal{G}^\dagger \big]U_t\mathrm{d}t\mathrm{d}s, \end{align} $$

where we have used Duhamel’s formula to express $U^{-1}\eta _{ijk}a_0^3U-U_{-s}\eta _{ijk}a_0^3U_s$ . The following Lemma 6 demonstrates that we can treat the quantities $\delta _1 \psi $ and $\delta _2 \psi $ in Eq. (74) as error terms. In order to formulate Lemma 6, recall the set $\mathcal {L}_0:=\{(0,0,0)\}$ from Eq. (16) and let us define

$$ \begin{align*} A := (2\pi \mathbb{Z})^9\setminus \mathcal{L}_0 =(2\pi \mathbb{Z})^9\setminus \{(0,0,0)\}, \end{align*} $$

and the potential energy $\mathcal {E}_{\mathcal {P}}$ of an operator-valued three particle vector $\Theta _{i_1 i_2 i_3}$ as

(75) $$ \begin{align} \mathcal{E}_{\mathcal{P}}(\Theta):=\sum_{(i_1 i_2 i_3),(i_1' i_2' i_3')\in A}(V_N)_{i_1 i_2 i_3,i_1' i_2' i_3'}\Theta_{i_1 i_2 i_3}^\dagger \Theta_{i_1' i_2' i_3'}. \end{align} $$

To keep the notation light, we will occasionally write $\mathcal {E}_{\mathcal {P}}(\Theta _{i_1 i_2 i_3})$ for $\mathcal {E}_{\mathcal {P}}(\Theta )$ with dummy indices $i_1 i_2 i_3$ .

Lemma 6. There exists a constant $C>0$ , such that

(76) $$ \begin{align} \mathcal{E}_{\mathcal{P}}(\delta_1 \psi) & \leq CN^{\frac{1}{2}}(\mathcal{N}+1), \end{align} $$

(77) $$ \begin{align} \mathcal{E}_{\mathcal{P}}(\delta_2 \psi) & \leq C (\mathcal{N}+1)^4. \end{align} $$

Furthermore, $\mathcal {E}_P \left (\big [a_{i_1} a_{i_2} a_{i_3} ,\mathcal {G}^\dagger \big ]\right )\leq C N^{-\frac {3}{2}}(\mathcal {N}+1)^5$ .

Proof. Let us define $A_j$ as the set of all s such that $(-s, j-s, 0)\in A$ and

$$ \begin{align*} \alpha_j:=9\sum_{s,t\in A_j}\eta_{-s (s-j) j}\overline{\eta_{-t (t-j) j}}(V_N)_{-s (j-s) 0,-t (j-t) 0}. \end{align*} $$

Making use of the fact that $V_N\geq 0$ , we obtain by the Cauchy-Schwarz inequality

$$ \begin{align*} \sum_{(i_1 i_2 i_3),(i^{\prime}_1 i^{\prime}_2 i^{\prime}_3)\in A} (V_N)_{i_1 i_2 i_3,i^{\prime}_1 i^{\prime}_2 i^{\prime}_3}(\delta_1 \psi)_{i_1 i_2 i_3}^\dagger (\delta_1 \psi)_{i^{\prime}_1 i^{\prime}_2 i^{\prime}_3}\leq (a_0^\dagger)^4 a_0^4\sum_{j}\alpha_j\, a_{j}a_j^\dagger\leq N^4 \left(\sum_j \alpha_j\right) (\mathcal{N} + 1). \end{align*} $$

By Lemma 15 and the fact that $|(V_N)_{-s (j-s) 0,-t (j-t) 0}|\lesssim \frac {1}{N^2}$ , we have $N^4\sum _j \alpha _j\lesssim C N^{\frac {1}{2}}$ , which concludes the proof of Eq. (76). In order to verify Eq. (77), let us first define

$$ \begin{align*} (\widetilde{\delta}_{2,\sigma} \psi)_{i_1 i_2 i_3}:=\sum_{j k} \eta_{i_{\sigma_1}j k}\, a_k^\dagger a_{j}^\dagger a_{i_{\sigma_2}}a_{i_{\sigma_3}}a_0^3. \end{align*} $$

Then we obtain, using the sign $V_N\geq 0$ and a Cauchy-Schwarz estimate,

$$ \begin{align*} \mathcal{E}_{\mathcal{P}}(\delta_2 \psi) \lesssim \sum_{\sigma\in S_3} \mathcal{E}_{\mathcal{P}} \left(\widetilde{\delta}_{2,\sigma} \psi\right) + \mathcal{E}_{\mathcal{P}} \left(\delta_2 \psi - \frac{1}{4} \sum_{\sigma\in S_3} \widetilde{\delta}_{2,\sigma} \psi\right) = 6\mathcal{E}_{\mathcal{P}} \left(\widetilde{\delta}_{2,\mathrm{id}} \psi\right) + \mathcal{E}_{\mathcal{P}} \left(\delta_2 \psi - \frac{1}{4}\sum_{\sigma\in S_3}\widetilde{\delta}_{2,\sigma} \psi\right). \end{align*} $$

Proceeding as in the proof of Eq. (76), we obtain

$$ \begin{align*} \mathcal{E}_{\mathcal{P}} \left(\delta_2 \psi - \frac{1}{4}\sum_{\sigma\in S_3}\widetilde{\delta}_{2,\sigma} \psi\right)\lesssim N^{-\frac{1}{2}}(\mathcal{N}+1)^2. \end{align*} $$

Regarding the term $\mathcal {E}_{\mathcal {P}} \left (\widetilde {\delta }_{2,\mathrm {id}} \psi \right )$ , let us define

$$ \begin{align*} (G)_{i_1.. i_4, j_1..j_4}:=(V_N)_{i_1 i_2 (-j_3-j_4),j_1 j_2 (-i_3-i_4)}\eta_{i_3 i_4 (-i_3-i_4)}\overline{\eta_{j_3 j_4 (-j_3-j_4)}} \end{align*} $$

for $(i_1,i_2,-j_3-j_4)\in A$ and $(j_1,j_2,-i_3-i_4)\in A$ , and $(G)_{i_1.. i_4, j_1..j_4}:=0$ otherwise. Then

(78) $$ \begin{align} \nonumber \mathcal{E}_{\mathcal{P}} \left(\widetilde{\delta}_{2,\mathrm{id}} \psi\right)&= \sum_{i_1.. i_4, j_1..j_4}(G)_{i_1.. i_4, j_1..j_4}(a_0^\dagger)^3 a_{i_1}^\dagger a_{i_2}^\dagger a_{j_3}a_{j_4}a_{i_3}^\dagger a_{i_4}^\dagger a_{j_1}a_{j_2} a_0^3\\\nonumber & = \sum_{i_1.. i_4, j_1..j_4} (G)_{i_1.. i_4, j_1..j_4}(a_0^\dagger)^3 a_{i_1}^\dagger a_{i_2}^\dagger a_{i_3}^\dagger a_{i_4}^\dagger a_{j_3}a_{j_4} a_{j_1}a_{j_2} a_0^3 + 2\\& \quad \times \sum_{i_1.. i_3, j_1..j_3,k} (G')_{i_1.. i_3 , j_1..j_3 }(a_0^\dagger)^3 a_{i_1}^\dagger a_{i_2}^\dagger a_{i_3}^\dagger a_{j_3} a_{j_1}a_{j_2} a_0^3 \nonumber \\& \quad + \sum_{i_1 i_2, j_1 j_2} (G")_{i_1 i_2 , j_1 j_2}(a_0^\dagger)^3 a_{i_1}^\dagger a_{i_2}^\dagger a_{j_1}a_{j_2} a_0^3, \end{align} $$

with $(G')_{i_1.. i_3, j_1..j_3}:=\sum _k G_{i_1.. i_3 k, j_1..j_3 k}$ and $(G")_{i_1 i_2, j_1 j_2}:=\sum _{k_1 k_2} G_{i_1 i_2 k_1 k_2, j_1 j_2 k_1 k_2}$ . In the following let us study the term involving $G"$ , which is responsible for the largest contribution. Since $(V_N)_{i_1 i_2 (-k_1-k_2),j_1 j_2 (-k_1-k_2)}=(V_N)_{i_1 i_2 0,j_1 j_2 0}$ , see Eq. (24), we obtain

$$ \begin{align*} & \sum_{i_1 i_2, j_1 j_2} (G")_{i_1 i_2 , j_1 j_2}(a_0^\dagger)^3 a_{i_1}^\dagger a_{i_2}^\dagger a_{j_1}a_{j_2} a_0^3 = \left(\sum_{k_1 k_2}|\eta_{k_1 k_2 (-k_1-k_2)}|^2 \right) \sum_{i_1 i_2,j_1 j_2} (V_N)_{i_1 i_2 0,j_1 j_2 0}(a_0^\dagger)^3 a_{i_1}^\dagger a_{i_2}^\dagger a_{j_1}a_{j_2} a_0^3\\& \ \ \ \ \ \lesssim N^{-3}\sum_{i_1 i_2,j_1 j_2} (V_N)_{i_1 i_2 0,j_1 j_2 0}(a_0^\dagger)^3 a_{i_1}^\dagger a_{i_2}^\dagger a_{j_1}a_{j_2} a_0^3 \lesssim N^{-3} (a_0^\dagger)^3 (\mathcal{N}+1)^2 a_0^3\leq (\mathcal{N}+1)^2, \end{align*} $$

where we have used

$$ \begin{align*} & \sum_{i_1 i_2,j_1 j_2} (V_N)_{i_1 i_2 0,j_1 j_2 0} a_{i_1}^\dagger a_{i_2}^\dagger a_{j_1}a_{j_2}\lesssim (\mathcal{N}+1)^2,\\& \sum_{k_1 k_2}|\eta_{k_1 k_2 (-k_1-k_2)}|^2\lesssim N^{-3}, \end{align*} $$

see Lemma 15. Proceeding similarly for the other terms in Eq. (78), concludes the proof of Eq. (77). Regarding the bound on $\mathcal {E}_P \left (\big [a_{i_1} a_{i_2} a_{i_3} ,\mathcal {G}^\dagger \big ]\right )$ , let us identify

where $\mathbb {A}:=\frac {1}{6}\sum _{ijk}\eta _{ijk}a_i a_j a_k$ . Due to the sign $V_N\geq 0$ and the permutations symmetry of $V_N$ , as well as to the fact that there are $6$ permutations of the set $\{1,2,3\}$ , we can bound the operator $ \mathcal {E}_P \left (\big [a_{i_1} a_{i_2} a_{i_3} ,\mathcal {G}^\dagger \big ]\right )$ from above by

In the following we focus on , the other terms can be treated in a similar fashion. By Lemma 15 we have $\mathbb {A}^\dagger \mathbb {A}\lesssim N^{-3}(\mathcal {N}+1)^3$ , and therefore

where $A^0$ contains all pairs $(jk)$ such that $(0jk)\in A$ .

As a consequence of Lemma 6, we obtain that the trial state $\Gamma $ defined below Eq. (71) has a potential energy $\mathcal {E}_{\mathcal {P}}(\psi )$ of the order $O_{N\rightarrow \infty } \left (\sqrt {N}\right )$ , see the following Corollary 3.

Corollary 3. There exists a constant $C>0$ , such that $ \left \langle \Gamma , \mathcal {E}_{\mathcal {P}}(\psi +\delta _1 \psi )\Gamma \right \rangle \leq C$ and

$$ \begin{align*} \left\langle \Gamma, \mathcal{E}_{\mathcal{P}}(\psi)\Gamma\right\rangle\leq C\sqrt{N}. \end{align*} $$

Proof. Recall that we can express the transformed quantity $U^{-1}\psi _{ijk} U$ by Eq. (74) as

$$ \begin{align*} U^{-1}\psi_{ijk} U & = a_i a_j a_k + \int_0^1 U_{-s} \big[a_i a_j a_k ,\mathcal{G}^\dagger \big] U_s \mathrm{d}s - \int_0^1 U_{-s} (\delta_1 \psi)_{ijk} U_s \mathrm{d}s - \int_0^1 U_{-s} (\delta_2 \psi)_{ijk} U_s \mathrm{d}s\\& \quad +\int_0^1\int_s^1 U_{-t} \eta_{ijk} \big[a_0^3 ,\mathcal{G}^\dagger \big]U_t\mathrm{d}t\mathrm{d}s, \end{align*} $$

where we have used Duhamel’s formula to express $U^{-1}\eta _{ijk}a_0^3U-U_{-s}\eta _{ijk}a_0^3U_s$ . Using the sign $V_N\geq 0$ and Lemma 6, we estimate using the Cauchy-Schwarz inequality

(79) $$ \begin{align} \nonumber \mathcal{E}_{\mathcal{P}} \left(\int_0^1 U_{-s} (\delta_1 \psi)_{ijk} U_s \mathrm{d}s\right) &\leq \int_0^1 U_{-s}\mathcal{E}_{\mathcal{P}}(\delta_1 \psi)U_s\, \mathrm{d}s \leq CN^{\frac{1}{2}} \int_0^1 U_{-s}(\mathcal{N} + 1)^4 U_s \mathrm{d}s \leq C'N^{\frac{1}{2}}(\mathcal{N} + 1),\\\mathcal{E}_{\mathcal{P}} \left(\int_0^1 U_{-s} (\delta_2 \psi)_{ijk} U_s \mathrm{d}s\right) &\leq \int_0^1 U_{-s}\mathcal{E}_{\mathcal{P}}(\delta_2 \psi)U_s\, \mathrm{d}s \leq CN^{\frac{1}{2}} \int_0^1 U_{-s}(\mathcal{N} + 1)^4 U_s \mathrm{d}s \leq C'(\mathcal{N} + 1)^4. \end{align} $$

for suitable $C,C'$ , where we utilize Eq. (72) in order to estimate $U_{-s}(\mathcal {N}+1)^4 U_s$ . Similarly

(80) $$ \begin{align} \mathcal{E}_{\mathcal{P}} \left(\int_0^1 U_{-s} \big[a_i a_j a_k ,\mathcal{G}^\dagger \big] U_s \mathrm{d}s\right)\leq C'N^{-\frac{3}{2}}(\mathcal{N} + 1)^5 \end{align} $$

follows from Lemma 6. Regarding the term in the last line of Eq. (74), we note that

$$ \begin{align*} \big[a_0^3 ,\mathcal{G}^\dagger \big]^\dagger \big[a_0^3 ,\mathcal{G}^\dagger \big]\lesssim N(\mathcal{N}+1)^3 \end{align*} $$

follows from an analogous argument as we have seen in the proof of Lemma 6 and

$$ \begin{align*} \sum_{(i j k),(\ell m n)\in A} \left( V_N\right)_{i j k,\ell m n} \overline{\eta_{ijk}}\eta_{\ell m n}\lesssim N^{-2} \end{align*} $$

by Eq. (36). Therefore

(81) $$ \begin{align} & \mathcal{E}_{\mathcal{P}}\left(\int_0^1\int_s^1 U_{-t} \eta_{ijk} \big[a_0^3 ,\mathcal{G}^\dagger \big]U_t\mathrm{d}t\mathrm{d}s\right)\leq \frac{1}{2}\int_0^1\int_s^1 U_{-t}\mathcal{E}_{\mathcal{P}} \left(\eta_{ijk} \big[a_0^3 ,\mathcal{G}^\dagger \big]\right) U_t\, \mathrm{d}t\mathrm{d}s\lesssim N^{-1}(\mathcal{N}+1)^3, \end{align} $$

where we have used Eq. (72) again. Using $a_i a_j a_k \Gamma _0=0$ in case $(ijk)\in A$ , we obtain by Eq. (74) together with Eq. (79), Eq. (80) and Eq. (81) for a suitable constant C

$$ \begin{align*} & \left\langle \Gamma, \mathcal{E}_{\mathcal{P}} \left(\psi\right)\Gamma\right\rangle=\left\langle \Gamma_0, \mathcal{E}_{\mathcal{P}} \left(U^{-1}\psi U\right)\Gamma_0\right\rangle \leq CN^{\frac{1}{2}}\left\langle \Gamma_0, (\mathcal{N}+1)^5 \Gamma_0\right\rangle=CN^{\frac{1}{2}}. \end{align*} $$

Analogously we obtain $ \left \langle \Gamma , \mathcal {E}_{\mathcal {P}}(\psi +\delta _1 \psi )\Gamma \right \rangle \leq C$ .

Regarding the variable $c_k=a_k+[a_k,\mathcal {G}]$ from Eq. (68), let us apply Duahmel’s formula

(82) $$ \begin{align} U^{-1}c_k U & =a_k-\int_0^1 U_{-s}[a_k,\mathcal{G}]U_s \, \mathrm{d}s+U^{-1}[a_k,\mathcal{G}]U=a_k+\int_0^1 \int_s^1 U_{-t} \left[[a_k,\mathcal{G}],\mathcal{G}^\dagger \right]U_t \, \mathrm{d}t\mathrm{d}s, \end{align} $$

where we have used $\left [a_k,\mathcal {G}^\dagger \right ]=0$ for $k\neq 0$ and $[[a_k,\mathcal {G}],\mathcal {G}]=0$ , which follows from the observation that $\eta _{ijk}=0$ in case one of the indices in $\{i,j,k\}$ is zero. The following Lemma 7 provides useful estimates on the quantity $\left [[a_k,\mathcal {G}],\mathcal {G}^\dagger \right ]$ . In order to formulate Lemma 7 let us define the kinetic energy of an operator-valued one-particle vector $\Theta _k$ , written as $\mathcal {E}_{\mathcal {K}}(\Theta )$ or $\mathcal {E}_{\mathcal {K}}(\Theta _k)$ with k being a dummy index, as

(83) $$ \begin{align} \mathcal{E}_{\mathcal{K}}(\Theta):=\sum_k |k|^2 \Theta_k^\dagger \Theta_k. \end{align} $$

Lemma 7. For $m\geq 0$ there exists a constant $C_m>0$ , such that

$$ \begin{align*} \mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m \left[[a_k,\mathcal{G}],\mathcal{G}^\dagger \right]\right)\leq C_m N^{-1}(\mathcal{N}+1)^{5+2m}. \end{align*} $$

Proof. Let us write the double commutator as $\left [[a_k,\mathcal {G}],\mathcal {G}^\dagger \right ]= (\delta _1 c)_k+(\delta _2 c)_k+(\delta _3 c)_k$ , where

$$ \begin{align*} (\delta_1 c)_k &:=(a_0^\dagger)^3 a_0^3 \sum_{ij}|\eta_{ijk}|^2 a_k,\\ (\delta_2 c)_k&:=(a_0^\dagger)^3 a_0^3 \sum_{ij,j' k'}\overline{\eta_{i j' k'}}\eta_{ijk} a_j^\dagger a_{j'}a_{k'},\\ (\delta_3 c)_k&:=\left[a_0^3, (a_0^\dagger)^3\right] \left(\sum_{ij}\eta_{ijk}a_j^\dagger a_k^\dagger\right) \left(\sum_{i'j'k'}\overline{\eta_{i'j'k'}}a_{i'} a_{j'}a_{k'}\right). \end{align*} $$

By Eq. (36) it is clear that

$$ \begin{align*} \sum_{ij}|\eta_{ijk}|^2\lesssim \frac{1}{N^4}\sum_{t}\frac{1}{(|k|^2+|t|^2)^2}\lesssim \frac{1}{N^4 |k|}, \end{align*} $$

and therefore

$$ \begin{align*} \mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m \delta_1 c\right)=\sum_k |k|^2 (\delta_1 c)_k^\dagger \mathcal{N}^{2m} (\delta_1 c)_k\lesssim \frac{1}{N^8}\sum_{k\neq 0}a_k^\dagger \left((a_0^\dagger)^3 a_0^3\right)^2 \mathcal{N}^{2m} a_k\leq \frac{1}{N^2}\mathcal{N}^{2m+1}. \end{align*} $$

Using $J_{p_1 p_2 p_3,p_1' p_2' p_3'}:=\sum _{q q' k}|k|^2\overline {\eta _{q' p_2' p_3'}\eta _{q p_1' k}}\eta _{q' p_1 k}\eta _{q p_2 p_3}$ and $\widetilde {J}_{p_2 p_3,p_2' p_3'}:=\sum _{p_1}J_{p_1 p_2 p_3,p_1 p_2' p_3'}$

$$ \begin{align*} \mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^{m} \delta_2 c\right) & =\left((a_0^\dagger)^3 a_0^3\right)^2\sum_{j p' n' , p j' k'}J_{j p' n' , p j' k'} \, a_j^\dagger a_{p'}^\dagger a_{n'}^\dagger (\mathcal{N}+2)^{2m} a_p a_{j'}a_{k'}\\ & \ \ \ \ \ + \left((a_0^\dagger)^3 a_0^3\right)^2\sum_{ p' n' , j' k'}\widetilde{J}_{p' n' , j' k'} \, a_{p'}^\dagger a_{n'}^\dagger (\mathcal{N}+1)^{2m} a_{j'}a_{k'}. \end{align*} $$

Utilizing the operator $X_{jk,j'k'}:=\sum _q |k|\eta _{q j k}\overline {\eta _{q j' k'}}$ acting on $L^2(\Lambda )^{\otimes 2}$ and the permutation operator $(S\Psi )(x_1,x_2,x_3):=\Psi (x_2,x_1,x_3)$ acting on $L^2(\Lambda )^{\otimes 3}$ , we can write

$$ \begin{align*} J=(1\otimes X^\dagger )S (1\otimes X), \end{align*} $$

and $\widetilde {J}=X^\dagger X$ . Consequently

$$ \begin{align*} \|J\|\leq \|S\|\, \|1\otimes X\|^2=\|X\|^2=\|\widetilde{J}\|. \end{align*} $$

By Eq. (36) we have $\|\widetilde {J}\|\leq C N^{-\frac {15}{2}}$ for a suitable constant C. Consequently

$$ \begin{align*} \mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^{m} \delta_2 c\right)\leq CN^{-\frac{15}{2}}\left((a_0^\dagger)^3 a_0^3\right)^2(\mathcal{N}+2)^{3+2m}\leq C N^{-\frac{3}{2}}(\mathcal{N}+2)^{3+2m}. \end{align*} $$

Similarly one can show that $\mathcal {E}_{\mathcal {K}} \left (\mathcal {N}^{m} \delta _3 c\right )\lesssim N^{-1}(\mathcal {N}+1)^{5+2m}$ , and therefore

$$ \begin{align*} & \mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m \left[[a_k,\mathcal{G}],\mathcal{G}^\dagger \right]\right)=\mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m \delta_1 + \mathcal{N}^m \delta_2 + \mathcal{N}^m \delta_3\right)\\&\quad \leq 3\mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m \delta_1\right)+3\mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m \delta_2\right)+3\mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m \delta_3\right)\lesssim C_m N^{-1}(\mathcal{N}+1)^{5+2m}.\\[-41pt] \end{align*} $$

As a consequence of Lemma 7, we obtain that the trial state $\Gamma $ defined below Eq. (71) has a kinetic energy $\mathcal {E}_{\mathcal {K}}(c)$ of the order $O_{N\rightarrow \infty } \left (1\right )$ in the subsequent Corollary 4. Since in the residual term $\mathcal {E}$ defined in Lemma 1 the term $\mathcal {E}_{\mathcal {K}} \left (\sqrt {\mathcal {N}}c\right )\leq \frac {1}{2}\mathcal {E}_{\mathcal {K}} \left (c\right )+\frac {1}{2}\mathcal {E}_{\mathcal {K}} \left (\mathcal {N} c\right )$ appears, it will be convenient to estimate the expectation value in the state $\Gamma $ of $\mathcal {E}_{\mathcal {K}} \left (\mathcal {N}^m c\right )$ for $m\geq 1$ as well.

Proof. By Eq. (72) we have

$$ \begin{align*} U^{-1}\mathcal{N}^{2m}U=(U^{-1}\mathcal{N}^{m}U)^\dagger U^{-1}\mathcal{N}^{m}U\lesssim (\mathcal{N}^{m}+1)^2 \end{align*} $$

and hence

$$ \begin{align*} U^{-1}\mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m c\right)U=\mathcal{E}_{\mathcal{K}} \left(U^{-1}\mathcal{N}^m U\, U^{-1} c\, U\right)\lesssim \mathcal{E}_{\mathcal{K}} \left((\mathcal{N}^m+1) U^{-1} c\, U\right), \end{align*} $$

where we have used that for operators $f_k$ and $A,B$ satisfying $A^\dagger A\leq C B^\dagger B$ we have

$$ \begin{align*} \mathcal{E}_{\mathcal{K}} \left(A f_k\right)\leq C\mathcal{E}_{\mathcal{K}} \left(B f_k\right). \end{align*} $$

Proceeding as in the proof of Corollary 3, we obtain by Eq. (82) and Lemma 7

$$ \begin{align*} & \mathcal{E}_{\mathcal{K}} \left((\mathcal{N}^m+1) U^{-1} c\, U\right)\lesssim \mathcal{E}_{\mathcal{K}} \left((\mathcal{N}^m+1) a\right)+\int_0^1 \mathcal{E}_{\mathcal{K}} \left((\mathcal{N}^m+1) U_{-t} \left[[a_k,\mathcal{G}],\mathcal{G}^\dagger \right] U_t\right)\mathrm{d}t\\ & \ \ \ \ \ =\mathcal{E}_{\mathcal{K}} \left((\mathcal{N}^m+1) a\right)+\int_0^1 U_{-t}\mathcal{E}_{\mathcal{K}} \left(U_t(\mathcal{N}^m+1)U_{t} \left[[a_k,\mathcal{G}],\mathcal{G}^\dagger \right] \right)U_t\mathrm{d}t\\ & \ \ \ \ \ \lesssim \mathcal{E}_{\mathcal{K}} \left((\mathcal{N}^m+1) a\right)+\int_0^1 U_{-t}\mathcal{E}_{\mathcal{K}} \left((\mathcal{N}^m+1) \left[[a_k,\mathcal{G}],\mathcal{G}^\dagger \right] \right)U_t\mathrm{d}t\\ & \ \ \ \ \ \lesssim \mathcal{E}_{\mathcal{K}} \left((\mathcal{N}^m+1) a\right)+ N^{-1}\int_0^1 U_{-t}(\mathcal{N}+1)^{5+2m} U_t\mathrm{d}t\\ & \ \ \ \ \ \lesssim \mathcal{E}_{\mathcal{K}} \left((\mathcal{N}^m+1) a\right)+ N^{-1}(\mathcal{N}+1)^{5+2m}. \end{align*} $$

where we have made use of Eq. (36) again. Using $a_k\Gamma _0=0$ for $k\neq 0$ , therefore yields

$$ \begin{align*} & \left\langle \Gamma, \mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m c\right)\Gamma\right\rangle=\left\langle \Gamma_0, U^{-1}\mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m c\right)U \Gamma_0\right\rangle\\&\quad \lesssim \left\langle \Gamma_0, \mathcal{E}_{\mathcal{K}} \left((\mathcal{N}^m+1) a\right) \Gamma_0\right\rangle +N^{-1} \left\langle \Gamma_0, (\mathcal{N}+1)^{5+2m} \Gamma_0\right\rangle =\frac{1}{N}.\\[-41pt] \end{align*} $$

Having Corollary 3 and Corollary 4 at hand, we are in a position to verify the upper bound on the ground state energy $E_N$ in Theorem 4.

Proof of Theorem 4.

Let $A:=(2\pi \mathbb {Z})^9 \setminus \{(0,0,0)\}$ , and let $\Gamma $ be the state defined below Eq. (71). Using Eq. (27) and Eq. (21), and the fact that $\left (\widetilde {V}_N\right )_{i j k,\ell m n}=\left (V_N\right )_{i j k,\ell m n}$ for index triples $(ijk),(\ell m n)\in A$ , we obtain

$$ \begin{align*} H_N & =\sum_{k}|k|^2 c_k^\dagger c_k +\lambda_{0,0} (a_0^\dagger)^3 a_0^3 + \frac{1}{6}\sum_{(i j k),(\ell m n)\in A}\left(V_N\right)_{i j k,\ell m n}\psi_{i j k}^\dagger \psi_{\ell m n}\\ & \ \ \ \ \ \ +\left(3 a_0^\dagger a_0^3\sum_{\ell\neq 0}\lambda_{0,\ell}\, a_{\ell}^\dagger a_{-\ell}^\dagger + \mathrm{H.c.}\right)-\mathcal{E}\\ & =\lambda_{0,0} (a_0^\dagger)^3 a_0^3 +\mathcal{E}_{\mathcal{K}} \left(c\right) + \mathcal{E}_{\mathcal{P}}(\psi)+\left(3 a_0^\dagger a_0^3\sum_{\ell\neq 0}\lambda_{0,\ell}\, a_{\ell}^\dagger a_{-\ell}^\dagger + \mathrm{H.c.}\right)-\mathcal{E}. \end{align*} $$

By a symmetry argument it is clear that . Applying Corollary 3 as well as Corollary 4, with $m=0$ , yields for suitable constants $C>0$

$$ \begin{align*} & \left\langle \Gamma, \mathcal{E}_{\mathcal{P}}(\psi)\Gamma\right\rangle\leq C\sqrt{N}, \ \ \ \ \ \ \ \ \ \ \left\langle \Gamma, \mathcal{E}_{\mathcal{K}} \left(c\right)\Gamma\right\rangle\leq \frac{C}{N}. \end{align*} $$

Furthermore, observe that $N^3\lambda _{0,0}\leq \frac {1}{6}b_{\mathcal {M}}(V)N +C'$ by Eq. (37) for a suitable $C'$ . In order to estimate the final term , note that we have by Eq. (72) for $m\in \mathbb {N}$

(84) $$ \begin{align} \langle \Gamma,\mathcal{N}^m \Gamma\rangle=\langle \Gamma_0, U^{-1}\mathcal{N}^m U \Gamma_0\rangle\lesssim \langle \Gamma_0, (\mathcal{N}+1)^m \Gamma_0\rangle=1. \end{align} $$

Using Lemma 3 together with the estimate from Corollary 4 for $m=0$ and $m=1$ , we therefore obtain .

4 Refined correlation structure

Utilizing the set of operators defined in Eq. (25) and Eq. (26), we were able to identify the ground state energy $E_N$ up to errors of the magnitude $O_{N\rightarrow \infty }(\sqrt {N})$ in the previous Sections 2 and 3. It is the purpose of this Section to obtain a higher resolution of the energy, which especially captures the subleading term proportional to $\sqrt {N}$ in the asymptotic expansion of $E_N$ , using a more refined correlation structure compared to the one introduced in Subsection 1.1. On a technical level, the new correlation structure is implemented by the new set of operators $d_k$ and $\xi _{ijk}$ defined below in Eq. (89) and Eq. (90), which constitute a refined version of the operators $c_k$ and $\psi _{ijk}$ respectively. Writing the operator $H_N$ in terms of $d_k$ and $\xi _{ijk}$ will then allow us to verify the lower bound from Theorem 1 in Subsection 4.2 and the corresponding upper bound in Section 5.

The approach presented in Sections 2 and 3 fails to capture the correct term of order $\sqrt {N}$ for two reasons: (I) The following expression appearing in Eq. (32)

(85) $$ \begin{align} 3 \sum_{|\ell|>K} N^2\lambda_{0,\ell}\, a_{\ell}^\dagger a_{-\ell}^\dagger\, \frac{a_0^\dagger a_0^3}{N^2} \end{align} $$

is expected to lower the ground state energy by an amount proportional to $\sqrt {N}$ , to be precise naive perturbation theory suggests that the term in Eq. (85) is giving rise to an energy correction proportional to

(86) $$ \begin{align} \sum_{|\ell|>K}\frac{|N^2 \lambda_{0,\ell}|^2}{|\ell|^2}=O(\sqrt{N}), \end{align} $$

see Eq. (35), and therefore consistent with our estimate in Lemma 4. (II) In the pursue of an upper bound on $E_N$ we expressed the unitary conjugated variables $U^{-1} \psi _{ijk} U$ as a sum of $a_i a_j a_k$ and various error terms according to Eq. (74) as

(87) $$ \begin{align} \nonumber U^{-1}\psi_{ijk} U & = a_i a_j a_k + \int_0^1 U_{-s} \big[a_i a_j a_k ,\mathcal{G}^\dagger \big] U_s \mathrm{d}s - \int_0^1 U_{-s} (\delta_1 \psi)_{ijk} U_s \mathrm{d}s - \int_0^1 U_{-s} (\delta_2 \psi)_{ijk} U_s \mathrm{d}s\\& \quad +\int_0^1\int_s^1 U_{-t} \eta_{ijk} \big[a_0^3 ,\mathcal{G}^\dagger \big]U_t\mathrm{d}t\mathrm{d}s. \end{align} $$

While most of the terms appearing in Eq. (87) give a contribution of the order $o_{N\rightarrow } \left (\sqrt N \right )$ , the term $\delta _1 \psi $ is expected to increase the ground state energy by an amount proportional to $\sqrt {N}$ , which is consistent with our estimate in Eq. (76). In order to extract the energy shift due to the expression in Eq. (85), we follow the strategy in Subsection 1.1 and introduce an additional two-particle correlation structure via a map acting on the two-particle space

$$ \begin{align*} T_2:L^2(\Lambda^2)\longrightarrow L^2(\Lambda^2) \end{align*} $$

in Eq. (88), which will give rise to the negative energy correction $-\mu (V)\sqrt {N}$ from Theorem 1. To be precise, we define the map $T_2$ via its matrix elements as

(88) $$ \begin{align} (T_2-1)_{\ell (-\ell),00}:=(T_2-1)_{00, \ell (-\ell)}:=3N\frac{\lambda_{0,\ell}}{|\ell|^2}, \end{align} $$

for $|\ell |>K$ and $(T_2-1)_{jk,mn}:=0$ otherwise, where $\lambda _{k,\ell }$ is defined below Eq. (32). Regarding the energy shift associated with $\delta _1 \psi $ , it is a natural idea to include this term in the definition of our new operators $\xi _{ijk}$ , giving rise to the positive energy correction $\gamma (V)\sqrt {N}$ from Theorem 1. However, a computation in Eq. (93) demonstrates that the presence of $\delta _1 \psi $ produces new four-particle correlation terms of the form

$$ \begin{align*} \sum_{uijk} \theta_{uijk} a_u^\dagger a_i^\dagger a_j^\dagger a_k^\dagger \frac{a_0^4}{N^2} +\mathrm{H.c.}, \end{align*} $$

with coefficients $\theta _{uijk}$ proportional to $N^2\sum _{mn}(V_N)_{ijk,0mn}\eta _{mnu}$ , which behave like

for momenta of the order $\sqrt {N}$ , and decay fast for higher momenta. Therefore, the four-particle correlation terms are expected to lower the ground state energy, similar to Eq. (86), by an amount of the order

Again we extract the correlation energy by introducing a map, acting this time on the four-particle space

$$ \begin{align*} T_4:L^2(\Lambda^4) \longrightarrow L^2(\Lambda^4) \end{align*} $$

in Eq. (95), which gives rise to the negative energy correction $-\sigma (V)\sqrt {N}$ from Theorem 1.

In the following let $T:L^2(\Lambda ^3)\longrightarrow L^2(\Lambda ^3)$ be the map constructed in Eq. (19), and for now let us think of $T_2:L^2(\Lambda ^2)\longrightarrow L^2(\Lambda ^2)$ and $T_4:L^2(\Lambda ^4)\longrightarrow L^2(\Lambda ^4)$ as generic bounded permutation-symmetric operators modeling the two-particle and four-particle correlation structure respectively. Following the approach in Section 2, we are implementing many-particle counterparts to the transformations T, $T_2$ , and $T_4$ as

(89) $$ \begin{align} d_k&:=a_k+\sum_{j,m n}(T_2-1)_{jk,m n} \, a_j^\dagger a_{m}a_{n}+\frac{1}{2}\sum_{i j,\ell m n}(T-1)_{ijk,\ell m n} \, a_i^\dagger a_j^\dagger a_{\ell}a_{m}a_{n}\\\nonumber &\quad +\frac{1}{6}\sum_{u i j,v \ell m n}(T_4-1)_{u ijk,v \ell m n} \, a_u^\dagger a_i^\dagger a_j^\dagger a_v a_{\ell}a_{m}a_{n}, \end{align} $$

(90) $$ \begin{align} \xi_{i j k}&:=\sum_{\ell m n}(T)_{ijk,\ell m n} \, a_{\ell}a_{m}a_{n}+(\delta_1 \psi)_{ijk}+\sum_{u,v\ell m n}(T_4-1)_{uijk,v\ell m n} \, a_u^\dagger a_v a_{\ell}a_{m}a_{n}. \end{align} $$

Note that $T_2$ is not included in the definition of $\xi _{ijk}$ , as it would only give contributions of the order $O_{N\rightarrow \infty }(1)$ . Using the Laplace operator $\Delta _s$ acting on the space $L^2(\Lambda )^{\otimes s}$ and the coefficients

(91)

let us furthermore define the operators $X_2: =T_2^\dagger (-\Delta _2)T_2+\Delta _2$ and

(92) $$ \begin{align} X_4&:= \bigg(\left\{(-\Delta_4 +4( \widetilde{V}_N\otimes 1))(T_4-1)+ 4( \widetilde{V}_N\otimes 1)\chi\right\}+\mathrm{H.c.}\bigg) \\\nonumber & \quad + (T_4-1)^\dagger (-\Delta_4)(T_4-1)+\left((T\otimes 1 - 1)^\dagger 4( \widetilde{V}_N\otimes 1)(T_4-1+\chi)+\mathrm{H.c.}\right)\\\nonumber & \quad +(T_4-1+\chi)^\dagger 4( \widetilde{V}_N\otimes 1)(T_4-1+\chi). \end{align} $$

A straightforward computation, similar to the one in Eq. (27), reveals that up to excess terms involving $X_2$ , $X_4$ and an error term $\widetilde {\mathcal {E}}$ , we can write the operator $H_N$ as a sum of squares in the variables $d_k$ and $\xi _{ijk}$ according to

(93) $$ \begin{align} \nonumber & \sum_{k}|k|^2 d_k^\dagger d_k + \frac{1}{6}\sum_{i j k,\ell m n}\left(\widetilde{V}_N\right)_{i j k,\ell m n}\xi_{i j k}^\dagger \xi_{\ell m n}\\& \quad =H_N+\frac{1}{2}\sum_{jk,mn} (X_2)_{jk,mn} \, a_j^\dagger a_k^\dagger a_m a_n+\frac{1}{24}\sum_{uijk,v \ell m n}(X_4)_{uijk,v \ell m n} \, a_u^\dagger a_i^\dagger a_j^\dagger a_k^\dagger a_v a_\ell a_m a_n+ \widetilde{\mathcal{E}}, \end{align} $$

where the error $\widetilde {\mathcal {E}}$ contains all the non-fully contracted products appearing in the squares

(94) $$ \begin{align} & \sum_{k}|k|^2 (d_k-a_k)^\dagger (d_k - a_k) , \frac{1}{6}\sum_{i j k,\ell m n}\left(\widetilde{V}_N\right)_{i j k,\ell m n}(\xi_{i j k}-\psi_{ijk})^\dagger (\xi_{\ell m n} -\psi_{ijk}). \end{align} $$

In this context we define the fully contracted part of a product of monomials

$$ \begin{align*} \left(a_{i_1}^\dagger \dots a_{i_r}^\dagger a_{j_1}\dots a_{j_t}\right)\left(a_{i^{\prime}_1}^\dagger \dots a_{i^{\prime}_{r'}}^\dagger a_{j^{\prime}_1}\dots a_{j^{\prime}_{t'}}\right) \end{align*} $$

as $C_{j_1\dots j_{t},i^{\prime }_1\dots i^{\prime }_{r'}} a_{i_1}^\dagger \dots a_{i_r}^\dagger a_{j^{\prime }_1}\dots a_{j^{\prime }_{t'}}$ with $C_{j_1\dots j_{t},i^{\prime }_1\dots i^{\prime }_{r'}}$ being the expectation of $a_{j_1}\dots a_{j_t}a_{i^{\prime }_1}^\dagger \dots a_{i_{r'}}^\dagger $ in the vacuum. For a term by term definition of $\widetilde {\mathcal {E}}$ see Eq. (111) in Subsection 4.1.

In the following we want to choose $T_4$ , such that the term $4( \widetilde {V}_N\otimes 1)\chi $ is cancelled in the expression $\{..\}$ from Eq. (92), at least after symmetrization and projection onto the range of $Q^{\otimes 4}$ , that is, we define

(95) $$ \begin{align} T_4:=1-R_4 \Pi_{\mathrm{sym}}Q^{\otimes 4}4( \widetilde{V}_N\otimes 1)\chi=1-R_4 \Pi_{\mathrm{sym}}Q^{\otimes 4}4( V_N\otimes 1)\chi, \end{align} $$

where $\Pi _{\mathrm {sym}}$ is the orthogonal projection onto the subspace $L^2_{\mathrm {sym}}(\Lambda ^4)\subseteq L^2(\Lambda ^4)$ and $R_4$ is the pseudoinverse of

(96) $$ \begin{align} Q^{\otimes 4}\left(-\Delta_4 +4( \widetilde{V}_N\otimes 1)\right)Q^{\otimes 4}=Q^{\otimes 4}\left(-\Delta_4 +4( V_N\otimes 1)\right)Q^{\otimes 4}. \end{align} $$

In order to obtain an improved representation of the operator $X_4$ defined in Eq. (92), let us introduce the constants

(97) $$ \begin{align} \sigma_N&:=\frac{N^4}{6}\langle (\widetilde{V}_N\otimes 1)\chi u_0^{\otimes 4},(1-T_4) u_0^{\otimes 4}\rangle,\\\nonumber &= \frac{N^4}{24} \left\langle (T_4-1) u_0^{\otimes 4},\left(-\Delta_4 +4 V_N\otimes 1\right)(T_4-1) u_0^{\otimes 4}\right\rangle, \end{align} $$

(98) $$ \begin{align} \gamma_N&:=\frac{N^4}{6}\langle ( \widetilde{V}_N \otimes 1)\chi u_0^{\otimes 4},\chi u_0^{\otimes 4}\rangle=\frac{N^4}{6}\langle ( V_N \otimes 1)\chi u_0^{\otimes 4},\chi u_0^{\otimes 4}\rangle, \end{align} $$

which allow us to write $\gamma _N-\sigma _N=\frac {N^4}{24}(X_4)_{0000,0000}$ . Furthermore, we define the three-particle state $\Theta $ as

(99) $$ \begin{align} (\Theta)_{ijk}&:=4\left(\Pi_{\mathrm{sym}}\frac{X_4}{24} \Pi_{\mathrm{sym}}\right)_{0ijk,000},\\ \nonumber (\Theta)_{0jk}&:=6\left(\Pi_{\mathrm{sym}}\frac{X_4}{24} \Pi_{\mathrm{sym}}\right)_{00jk,000} \end{align} $$

for $\{i,j,k\}$ all different from zero and $ (\Theta )_{ijk}:=0$ otherwise. According to the definition of $T_4$ we have $Q^{\otimes 4}\Pi _{\mathrm {sym}}X_4\Pi _{\mathrm {sym}} P^{\otimes 4}=0$ , and therefore

(100) $$ \begin{align} \nonumber &\frac{1}{24} \sum_{uijk,v \ell m n} (X_4)_{uijk,v \ell m n} \, a_u^\dagger a_i^\dagger a_j^\dagger a_k^\dagger a_v a_\ell a_m a_n\\& \quad = \frac{1}{24} \sum_{uijk,v \ell m n} (\Pi_{\mathrm{sym}}X_4 \Pi_{\mathrm{sym}})_{uijk,v \ell m n} \, a_u^\dagger a_i^\dagger a_j^\dagger a_k^\dagger a_v a_\ell a_m a_n \nonumber\\& \quad =(a_0^\dagger )^4 a_0^4\, N^{-4}(\gamma_N-\sigma_N) +\left(\sum_{ijk, \ell m n}(\Theta)_{ijk} \, a_i^\dagger a_j^\dagger a_k^\dagger\, a_0^\dagger a_0^4 + \mathrm{H.c.}\right). \end{align} $$

In order to understand the size of the term in Eq. (100) better, we are going to rewrite it in terms of the variables $\psi _{ijk}$ defined in Eq. (26), respectively the variables

(101) $$ \begin{align} \widetilde{\psi}_{i j k} &=a_ia_ja_k+\eta_{ijk} \, a_0^3 \end{align} $$

defined in Eq. (26) for the concrete choice $K:=0$ , see Eq. (69), with the corresponding operator $T_{K=0}:=1+RV_N\pi _0$ , see Eq. (17). Note that

$$ \begin{align*} (T_{K=0}^{-1 })^\dagger \Theta = \Theta +2(\sigma_N-\gamma_N) u_0^{\otimes 3} , \end{align*} $$

and therefore

$$ \begin{align*} & \frac{1}{24} \sum_{uijk,v \ell m n} (X_4)_{uijk,v \ell m n} \, a_u^\dagger a_i^\dagger a_j^\dagger a_k^\dagger a_v a_\ell a_m a_n = N^{-4}(a_0^\dagger )^4 a_0^4\, ( \sigma_N-\gamma_N) \\& \quad + \left(\sum_{ijk, \ell m n}(\Theta)_{ijk} \, \widetilde{\psi}_{ijk}^\dagger \, a_0^\dagger a_0^4 + \mathrm{H.c.}\right). \end{align*} $$

In order to address the correlation term in Eq. (85), we use the concrete choice for $T_2$ from Eq. (88). With

(102) $$ \begin{align} \mu_N:=\frac{N^2}{2} (X_2)_{00,00}, \end{align} $$

this choice for a transformation $T_2$ yields

$$ \begin{align*} \frac{1}{2}\sum_{jk,mn} (X_2)_{jk,mn} \, a_j^\dagger a_k^\dagger a_m a_n= N^{-2}(a_0^\dagger )^2 a_0^2\, \mu_N +\bigg(3 N a_0^2\sum_{|\ell|>K}\lambda_{0,\ell}\, a_{\ell}^\dagger a_{-\ell}^\dagger + \mathrm{H.c.}\bigg). \end{align*} $$

Summarizing what we have so far allows us to write the operator $H_N$ in terms of the new variables $d_k$ and $\xi _{ijk}$ as

(103) $$ \begin{align} \nonumber H_N&=\sum_{k}|k|^2 d_k^\dagger d_k + \frac{1}{6}\sum_{i j k,\ell m n}\left(\widetilde{V}_N\right)_{i j k,\ell m n}\xi_{i j k}^\dagger \xi_{\ell m n}+ N^{-4}(a_0^\dagger )^4 a_0^4\, (\gamma_N-\sigma_N)-N^{-2}(a_0^\dagger )^2 a_0^2\, \mu_N \\& \quad -\bigg(3 N a_0^2\sum_{|\ell|>K}\lambda_{0,\ell}\, a_{\ell}^\dagger a_{-\ell}^\dagger + \mathrm{H.c.}\bigg) - \left(\sum_{ijk, \ell m n}(\Theta)_{ijk} \, \widetilde{\psi}_{ijk}^\dagger \, a_0^\dagger a_0^4 + \mathrm{H.c.}\right) - \widetilde{\mathcal{E}}. \end{align} $$

Defining the error term

(104) $$ \begin{align} \mathcal{E}_*:= 3 \left(a_0^\dagger a_0 - N\right) a_0^2 \sum_{|\ell|>K} \lambda_{0,\ell}\, a_{\ell}^\dagger a_{-\ell}^\dagger + 9 a_0^\dagger a_0^2 \sum_{\ell,0<|k|\leq K} \lambda_{k,\ell} a_\ell^\dagger a_{k-\ell}^\dagger a_k, \end{align} $$

we obtain as a consequence of Eq. (103) the following Corollary 5.

Corollary 5. Let $d_k$ and $\xi _{ijk}$ be as in Eq. (89) and Eq. (90), with $T_2$ defined in Eq. (88) and $T_4$ defined in Eq. (95), $\gamma _N,\sigma _N$ , and $\mu _N$ as in Eq. (98), Eq. (97) and Eq. (102), and let $\mathcal {E}_*$ be as in Eq. (104), $\Theta $ as in Eq. (99) and $\widetilde {\psi }_{ijk}$ as in Eq. (101). Furthermore, recall the definition of $\lambda _{0,0}$ in Eq. (29) and $\mathbb {Q}_K$ in Eq. (33). Then

(105) $$ \begin{align} & H_N\geq (a_0^\dagger )^3 a_0^3 \lambda_{0,0}+ N^{-4}(a_0^\dagger )^4 a_0^4\, (\gamma_N - \sigma_N) -N^{-2}(a_0^\dagger )^2 a_0^2\, \mu_N +\sum_{k}|k|^2 d_k^\dagger d_k +\mathbb{Q}_K\\ \nonumber & \ \ \ \ \ \ \ \ \ - \left(\sum_{ijk, \ell m n}(\Theta)_{ijk} \, \widetilde{\psi}_{ijk}^\dagger \, a_0^\dagger a_0^4 + \mathrm{H.c.}\right) + \left(\mathcal{E}_*+\mathcal{E}_*^\dagger \right)- \widetilde{\mathcal{E}}. \end{align} $$

Making use of the notation $\mathcal {E}_{\mathcal {K}}(d)$ from Eq. (83) and $\mathcal {E}_{\mathcal {P}}(\xi )$ from Eq. (75), we obtain in the case $K=0$ the identity

(106) $$ \begin{align} \nonumber & H_N =\lambda_{0,0}(a_0^\dagger)^3 a_0^3 + (\gamma_N - \sigma_N) N^{-4}(a_0^\dagger)^4 a_0^4 - \mu_N N^{-2}(a_0^\dagger)^2 a_0^2 +\mathcal{E}_{\mathcal{K}}(d)+ \mathcal{E}_{\mathcal{P}}(\xi) \\ & \ \ \ \ \ \ \ \ \ \ - \left(\sum_{ijk, \ell m n}(\Theta)_{ijk} \, \widetilde{\psi}_{ijk}^\dagger \, a_0^\dagger a_0^4 + \mathrm{H.c.}\right)+ \left(\mathcal{E}_*+\mathcal{E}_*^\dagger \right) - \widetilde{\mathcal{E}}. \end{align} $$

Proof. Using Eq. (103) and the definition of $\widetilde {V}_N$ in Eq. (21), as well as the identities in Eq. (30) and Eq. (31), we obtain

$$ \begin{align*} & H_N=\lambda_{0,0}(a_0^\dagger)^3 a_0^3 + (\gamma_N - \sigma_N) N^{-4}(a_0^\dagger)^4 a_0^4 - \mu_N N^{-2}(a_0^\dagger)^2 a_0^2 +\sum_{k}|k|^2 d_k^\dagger d_k+\mathbb{Q}_K\\ & \ \ \ \ \ \ \ \ \ \ + \frac{1}{6}\sum_{i j k,\ell m n}\left((1-\pi_K)V_N(1-\pi_K)\right)_{i j k,\ell m n}\xi_{i j k}^\dagger \xi_{\ell m n}\\ & \ \ \ \ \ \ \ \ \ \ -\left(\sum_{ijk, \ell m n}(\Theta)_{ijk} \, \widetilde{\psi}_{ijk}^\dagger \, a_0^\dagger a_0^4 + \mathrm{H.c.}\right) + \left(\mathcal{E}_*+\mathcal{E}_*^\dagger \right) - \widetilde{\mathcal{E}}. \end{align*} $$

Since $\mathbb {Q}_0=0$ and

$$ \begin{align*} & \frac{1}{6}\sum_{i j k,\ell m n}\left((1-\pi_K)V_N(1-\pi_K)\right)_{i j k,\ell m n}\xi_{i j k}^\dagger \xi_{\ell m n}\geq 0,\\ & \frac{1}{6}\sum_{i j k,\ell m n}\left((1-\pi_0)V_N(1-\pi_0)\right)_{i j k,\ell m n}\xi_{i j k}^\dagger \xi_{\ell m n}=\mathcal{E}_{\mathcal{P}}(\xi), \end{align*} $$

we immediately obtain Eq. (105), respectively Eq. (106).

4.1 Analysis of the error terms

In the following we are providing an explicit representation of the error term $\widetilde {\mathcal {E}}$ introduced in Eq. (93), which we subsequently use in Lemma 8 in order to control $\widetilde {\mathcal {E}}$ . For this purpose, we are going to utilize the following estimates on the matrix elements of $T,T_2$ and $T_4$

(107)

(108)

(109)

which are verified in Lemma 15 and Lemma 18 respectively. Furthermore, it is useful to introduce the two-particle state $(\varphi _2^0)_{jk}:=N(T_2-1)_{jk,00}$ , the three-particle states $(\varphi ^0_3)_{ijk}:=\frac {N^{\frac {3}{2}}}{2}(T-1)_{ijk,0 0 0}$ and for $m\in (2\pi \mathbb {Z})^3\setminus \{0\}$

$$ \begin{align*} (\varphi_3^m)_{ijk}:=\frac{N^{\frac{3}{2}}}{2}(T-1)_{ijk,m 0 0}+\frac{N^{\frac{3}{2}}}{2}(T-1)_{ijk,0 m 0}+\frac{N^{\frac{3}{2}}}{2}(T-1)_{ijk,0 0 m}, \end{align*} $$

and the four particle state $(\varphi ^0_4)_{uijk}:=\frac {N^2}{6} (T_4-1)_{uijk,0000}$ as well as

$$ \begin{align*} (\varphi_4)_{uijk}:=N^2 (T_4-1+\chi)_{uijk,0000}. \end{align*} $$

Additionally, let us introduce for $\varphi \in L^2(\Lambda ^s)$ and $\psi \in L^2(\Lambda ^t)$ with $s,t\geq 0$ , and $\ell \leq \min \{s,t\}$ , the operator

(110) $$ \begin{align} G_{\ell}(\varphi,\psi):=\mathrm{Tr}_{1\rightarrow \ell} \left[(-\Delta)_{x_1} \varphi \psi^\dagger \right] \end{align} $$

acting on $L^2(\Lambda ^{t-\ell })\longrightarrow L^2(\Lambda ^{s-\ell })$ . In coordinates, the operator is given by

$$ \begin{align*} \Big(G_{\ell}(\varphi,\psi)\Big)_{i_1\dots i_{s-\ell},j_1\dots j_{t-\ell}} := \sum_{k_1\dots k_\ell} |k_1|^2 \varphi_{k_1\dots k_\ell i_1\dots i_{s-\ell}}\overline{\psi}_{k_1\dots k_\ell j_1\dots j_{t-\ell}}. \end{align*} $$

Finally let $\widetilde {G}:=\mathrm {Tr}_{1\rightarrow 3} \left [\widetilde {V}_N\otimes 1 \varphi _4 \varphi _4^\dagger \right ]$ . With this at hand we can write

(111) $$ \begin{align} \nonumber \widetilde{\mathcal{E}} & = \sum_{(s,t,\ell,m,n)\in \mathcal{S}}C_{s,t,\ell}\underset{j_1\dots j_{t-\ell}}{\sum_{i_1\dots i_{s-\ell}}} \Big(G_{\ell}(\varphi_s^m,\varphi_t^n)\Big)_{i_1\dots i_{s-\ell},j_1\dots j_{t-\ell}}a_{i_{s-\ell}}^\dagger \dots a_{i_1}^\dagger \frac{a_m^\dagger (a_0^\dagger)^{s-1} a_0^{t-1}a_n}{N^{ \frac{s+t}{2} }} a_{j_1}\dots a_{j_{t-\ell}}\\& \quad +\sum_{i,j}\left(\widetilde{G}\right)_{i,j}a_i^\dagger \frac{(a_0^\dagger)^4 a_0^4}{N^4} a_j, \end{align} $$

where $C_{s,t,\ell }:=\frac {(s-1)! (t-1)!}{(\ell -1)! (s-\ell )! (t-\ell )!}$ and the set of allowed configurations $(s,t,\ell ,m,n)\in \mathcal {S}$ is defined by the rules $\ell \leq \min \{s,t\}$ and $\ell <\max \{s,t\}$ , where $2\leq s,t\leq 4$ and $|n|,|m|\leq K$ with $m=0$ , respectively $n=0$ , in case $s\neq 3$ , respectively $t\neq 3$ . Note that the criterion $\ell <\max \{s,t\}$ makes sure that we only include non-fully contracted parts of the first product in Eq. (94) and $\widetilde {G}$ is the kernel associated with the non-fully contracted part of the second product in Eq. (94). Furthermore, $C_{s,t,\ell }$ counts the various different ways to have an $\ell -1$ fold contraction between $s-1$ many annihilation operators and $t-1$ many creation operators, that is, $C_{s,t,\ell }$ counts the number of partially defined injective functions

$$ \begin{align*} f:\{1,\dots ,s-1\}\longrightarrow \{1,\dots ,t-1\} \end{align*} $$

with $\# \mathrm {dom}f=\ell -1$ . Let us illustrate the derivation of Eq. (94), looking at the term

$$ \begin{align*} \sum_{k}|k|^2 (d_k-a_k)^\dagger (d_k-a_k), \end{align*} $$

from Eq. (94). According to the definition of $d_k$ in Eq. (89), the term $d_k-a_k$ decomposes into three terms, and in the following we focus only on the last one involving $T_4-1$

(112) $$ \begin{align} \nonumber & \!\sum_{k} |k|^2 \left( \frac{1}{6} \sum_{u i j,v \ell m n}\!(T_4 - 1)_{u ijk,v \ell m n} \, a_u^\dagger a_i^\dagger a_j^\dagger a_v a_{\ell}a_{m}a_{n}\! \right)^\dagger \left( \frac{1}{6} \sum_{u i j,v \ell m n}\!(T_4 - 1)_{u ijk,v \ell m n} \, a_u^\dagger a_i^\dagger a_j^\dagger a_v a_{\ell}a_{m}a_{n} \! \right)\\\nonumber & \quad = N^{-4}\sum_{k,uij,u'i'j'}|k|^2\overline{(\varphi^4_0)_{uijk}}(\varphi^4_0)_{u'i'j'k} (a_0^\dagger)^4 a_u a_i a_j a_{u'}^\dagger a_{i'}^\dagger a_{j'}^\dagger a_0^4\\\nonumber & \quad = N^{-4}\sum_{k,uij,u'i'j'}|k|^2\overline{(\varphi^4_0)_{uijk}}(\varphi^4_0)_{u'i'j'k} a_{u'}^\dagger a_{i'}^\dagger a_{j'}^\dagger (a_0^\dagger)^4 a_0^4 a_u a_i a_j \\\nonumber & \qquad +9N^{-4}\sum_{jk,ui,u'i'}|k|^2\overline{(\varphi^4_0)_{uijk}}(\varphi^4_0)_{u'i'j k} a_{u'}^\dagger a_{i'}^\dagger (a_0^\dagger)^4 a_0^4 a_u a_i\\\nonumber & \qquad + 18 N^{-4}\sum_{ijk,u,u'}|k|^2\overline{(\varphi^4_0)_{uijk}}(\varphi^4_0)_{u'i'j'k} a_{u'}^\dagger (a_0^\dagger)^4 a_0^4 a_u \\\nonumber & \qquad +6N^{-4}\sum_{uijk}|k|^2\overline{(\varphi^4_0)_{uijk}}(\varphi^4_0)_{u'i'j'k} (a_0^\dagger)^4 a_0^4\\\nonumber & \quad = \sum_{\ell=1}^3 C_{4,4,\ell}\underset{j_1\dots j_{4-\ell}}{\sum_{i_1\dots i_{4-\ell}}} \Big(G_{\ell}(\varphi_4^0,\varphi_4^0)\Big)_{i_1\dots i_{4-\ell},j_1\dots j_{4-\ell}}a_{i_{4-\ell}}^\dagger \dots a_{i_1}^\dagger \frac{ (a_0^\dagger)^{4} a_0^{4}}{N^{4 }} a_{j_1}\dots a_{j_{4-\ell}}\\& \qquad +6N^{-4}\sum_{uijk}|k|^2\overline{(\varphi^4_0)_{uijk}}(\varphi^4_0)_{u'i'j'k} (a_0^\dagger)^4 a_0^4. \end{align} $$

Notably, the final term in Eq. (112) is a fully contracted contribution and therefore part of the operator $X_4$ defined in Eq. (92) and not of the error term $ \widetilde {\mathcal {E}} $ .

Estimating the various terms appearing in Eq. (111) individually allows us to prove the following Lemma 8.

Lemma 8. There exists a constant $C>0$ and a function $\epsilon :[0,\infty )\longrightarrow (0,C)$ satisfying ${\lim _{K\rightarrow \infty }\epsilon (K)=0}$ , such that we have for K as in the definition of $\pi _K$ below Eq. (17)

$$ \begin{align*} \pm \widetilde{\mathcal{E}}\leq C N^{-\frac{1}{4}} \sum_k |k|^2 c_k^\dagger \left(\frac{\mathcal{N}}{\sqrt{N}}+1\right)^2 c_k+CN^{-\frac{1}{4}} \left(\frac{\mathcal{N}}{\sqrt{N}}+1\right)^2 \left(\mathcal{N}+\sqrt{N}\right)+\epsilon(K) \mathcal{N}. \end{align*} $$

Proof. In the following let $\tau \leq \frac {1}{2}$ . Using the fact that $\|\frac {a_m^\dagger (a_0^\dagger )^{s-1} a_0^{t-1}a_n}{N^{ \frac {s+t}{2} }}\|\leq 1$ , there exists by Corollary 1 a constant $C>0$ such that for $\delta>0$ and $s,t\geq \ell +1$

(113) $$ \begin{align} \nonumber & \pm\left(\underset{j_1\dots j_{t-\ell}}{\sum_{i_1\dots i_{s-\ell}}} \Big(G_{\ell,\sigma,\tau}(\varphi_s^m,\varphi_t^n)\Big)_{i_1\dots i_{s-\ell},j_1\dots j_{t-\ell}}(\Phi_{\sigma,s})_{i_1\dots i_{s-\ell}}^\dagger \frac{a_m^\dagger (a_0^\dagger)^{s-1} a_0^{t-1}a_n}{N^{ \frac{s+t}{2} }}(\Phi_{\tau,t})_{j_1\dots j_{t-\ell}}+\mathrm{H.c.}\right)\\\nonumber & \quad \leq C \left\|\mathcal{K}_{\tau,s-\ell}^{-\frac{1}{2}} G_{\ell}(\varphi_s^m,\varphi_t^n)\mathcal{K}_{\tau,t-\ell}^{-\frac{1}{2}}\right\| \Bigg( \sum_k |k|^{2 }c_k^\dagger\left( \delta \mathcal{N}^{s-\ell-1}+\delta^{-1}\mathcal{N}^{t-\ell-1}\right)c_k \\& \qquad + \left(\mathcal{N}+\sqrt{N}\right) \left(\delta \mathcal{N}^{s-\ell-1}+\delta^{-1}\mathcal{N}^{t-\ell-1}\right) \Bigg). \end{align} $$

For $\tau =\sigma =0$ , we have the improved bound on the left-hand side of Eq. (113)

(114) $$ \begin{align} C \left\|G_{\ell}(\varphi_s^m,\varphi_t^n)\right\|\left(\delta \mathcal{N}^{s-\ell}+\delta^{-1}\mathcal{N}^{t-\ell}\right). \end{align} $$

In the case that either s or t is equal to $\ell $ , for example, $t=\ell $ we obtain by Corollary 1

(115) $$ \begin{align} \nonumber & \pm\left(\sum_{i_1\dots i_{s-\ell}} \Big(G_{\ell,\sigma,\tau}(\varphi_s^m,\varphi_t^n)\Big)_{i_1\dots i_{s-\ell}}(\Phi_{\sigma,s})_{i_1\dots i_{s-\ell}}^\dagger \frac{a_m^\dagger (a_0^\dagger)^{s-1} a_0^{t-1}a_n}{N^{ \frac{s+t}{2} }}+\mathrm{H.c.}\right)\\& \quad \leq C \left\|\mathcal{K}_{\tau,s-\ell}^{-\frac{1}{2}} G_{\ell}(\varphi_s^m,\varphi_t^n)\right\| \Bigg( \delta^{-1}+\delta \sum_k |k|^{2 }c_k^\dagger \mathcal{N}^{s-\ell-1}c_k + \delta \left(\mathcal{N}+\sqrt{N}\right) \mathcal{N}^{s-\ell-1} \Bigg). \end{align} $$

In order to obtain good estimates on the operator norms $\left \|\mathcal {K}_{\sigma ,s-\ell }^{-\frac {1}{2}} G_{\ell }(\varphi _s^m,\varphi _t^n)\mathcal {K}_{\tau ,t-\ell }^{-\frac {1}{2}}\right \|$ , observe that we obtain by a Cauchy-Schwarz argument

$$ \begin{align*} \left\|\mathcal{K}_{\sigma,s-\ell}^{-\frac{1}{2}} G_{\ell}(\varphi_s^m,\varphi_t^n)\mathcal{K}_{\tau,t-\ell}^{-\frac{1}{2}}\right\|\leq \sqrt{\left\|\mathcal{K}_{\sigma,s-\ell}^{-\frac{1}{2}} G_{\ell}(\varphi_s^m,\varphi_s^m)\mathcal{K}_{\sigma,s-\ell}^{-\frac{1}{2}}\right\|\left\|\mathcal{K}_{\tau,t-\ell}^{-\frac{1}{2}} G_{\ell}(\varphi_t^n,\varphi_t^n)\mathcal{K}_{\tau,t-\ell}^{-\frac{1}{2}}\right\|}, \end{align*} $$

that is, it is enough to control the norm of the symmetric ones. In the following we choose $\sigma =\tau =\frac {1}{2}$ , except for the case $s=t=2$ where we choose $\sigma =\tau =0$ . By Eq. (107) and Eq. (109) we obtain for a suitable $C>0$

$$ \begin{align*} & \left\|\mathcal{K}_{\frac{1}{2},3}^{-\frac{1}{2}} G_{1}(\varphi_4^0,\varphi_4^0)\mathcal{K}_{\frac{1}{2},3}^{-\frac{1}{2}}\right\|\leq C N^{-\frac{3}{2}}, & \left\|\mathcal{K}_{\frac{1}{2},2}^{-\frac{1}{2}} G_{1}(\varphi_3^m,\varphi_3^m)\mathcal{K}_{\frac{1}{2},2}^{-\frac{1}{2}}\right\|\leq C N^{-1},\\ & \left\|\mathcal{K}_{\frac{1}{2},2}^{-\frac{1}{2}} G_{2}(\varphi_4^0,\varphi_4^0)\mathcal{K}_{\frac{1}{2},2}^{-\frac{1}{2}}\right\|\leq C N^{-\frac{3}{2}}, & \left\|\mathcal{K}_{\frac{1}{2},1}^{-\frac{1}{2}} G_{2}(\varphi_3^m,\varphi_3^m)\mathcal{K}_{\frac{1}{2},1}^{-\frac{1}{2}}\right\|\leq C N^{-\frac{1}{2}},\\ & \left\|\mathcal{K}_{\frac{1}{2},1}^{-\frac{1}{2}} G_{3}(\varphi_4^0,\varphi_4^0)\mathcal{K}_{\frac{1}{2},1}^{-\frac{1}{2}}\right\|\leq C N^{-1}, & \left\|G_{3}(\varphi_3^m,\varphi_3^m)\right\|\leq C N. \end{align*} $$

Furthermore by Eq. (108), $\left \| G_{1}(\varphi _2^0,\varphi _2^0)\right \|\leq \epsilon $ in case K is large enough and $\left \| G_{2}(\varphi _2^0,\varphi _2^0)\right \|\leq C \sqrt {N}$ , as well as $\left \| \widetilde {G}\right \|\leq C N^{-1}$ by Eq. (109). Choosing $\delta :=N^{\frac {t-s}{4}}$ , and combining the estimates on the operator norms with Eq. (113), respectively Eq. (114), and Eq. (115) concludes the proof by Eq. (111).

Following the ideas in the proof of Lemma 8, we can furthermore compare the operator $\sum _{k}|k|^{2\tau }a_k^\dagger a_k$ with the corresponding operator $\sum _{k}|k|^{2\tau }d_k^\dagger d_k$ in the variables $d_k$ defined in Eq. (89). This is the content of the subsequent Lemma 9.

Lemma 9. Let $0\leq \tau < \frac {1}{4}$ . Then there exists $K_0,C>0$ such that for $K\geq K_0$ , with K as in the definition of $\pi _K$ below Eq. (17),

$$ \begin{align*} \sum_{k}|k|^{2\tau}a_k^\dagger a_k\leq C\sum_{k}|k|^{2}d_k^\dagger d_k+CN^{-\frac{3}{2}}\mathcal{N}^3+CN^{\tau}. \end{align*} $$

Proof. By Eq. (40), there exists a constant $C>0$ such that

$$ \begin{align*} \sum_{k}|k|^{2\tau}a_k^\dagger a_k\leq C\sum_{k}|k|^{2\tau}c_k^\dagger c_k+CN^{-1}\mathcal{N}^2+CN^{\tau}. \end{align*} $$

Furthermore, we have by Cauchy-Schwarz the estimate

$$ \begin{align*} \sum_{k}|k|^{2\tau}c_k^\dagger c_k\leq 2\sum_{k}|k|^{2\tau}d_k^\dagger d_k+2\sum_{k}|k|^{2\tau}(d_k-c_k)^\dagger (d_k-c_k). \end{align*} $$

Similar to the definition of $G_\ell $ in Eq. (110) let us introduce

$$ \begin{align*} G^{\prime}_{\ell}&:=\mathrm{Tr}_{1\rightarrow \ell} \left[(-\Delta)_{x_1} \varphi^0_2 (\varphi^0_2)^\dagger \right],\\ G^{\prime\prime}_{\ell}&:=\mathrm{Tr}_{1\rightarrow \ell} \left[(-\Delta)_{x_1} \varphi_4^0 (\varphi_4^0)^\dagger \right]. \end{align*} $$

A similar computation as in Eq. (111) together with a Cauchy-Schwarz estimate yields

$$ \begin{align*} & \sum_{k}|k|^{2\tau}(d_k-c_k)^\dagger (d_k-c_k) \leq CG^{\prime}_2 \frac{ (a_0^\dagger)^{2} a_0^{2}}{N^{ 2 }} +C\sum_{i,j}(G^{\prime}_1)_{i,j}a_i^\dagger \frac{ (a_0^\dagger)^{2} a_0^{2}}{N^{ 2 }} a_j\\ & \ \ \ \ \ \ +C\sum_{\ell=1}^4\underset{j_1\dots j_{4-\ell}}{\sum_{i_1\dots i_{4-\ell}}} \Big(G^{\prime\prime}_{\ell}\Big)_{i_1\dots i_{4-\ell},j_1\dots j_{4-\ell}}a_{i_{4-\ell}}^\dagger \dots a_{i_1}^\dagger \frac{ (a_0^\dagger)^{4} a_0^{4}}{N^{ 4 }} a_{j_1}\dots a_{j_{4-\ell}}, \end{align*} $$

for a suitable constant $C>0$ . Utilizing the estimates in Eq. (108) and Eq. (109) we obtain that $|G^{\prime }_2|\lesssim 1$ , $\|G^{\prime }_2\|\lesssim \frac {1}{K^2}$ , $|G^{\prime \prime }_4|\lesssim N^{\tau -\frac {1}{2}}\leq 1$ and $\|G^{\prime \prime }_\ell \|\lesssim N^{\tau -2}\leq N^{-\frac {3}{2}}$ for $\ell \leq 3$ . Consequently there exists a $C>0$ such that

$$ \begin{align*} \sum_{k}|k|^{2\tau}(d_k-c_k)^\dagger (d_k-c_k) \leq C+\frac{C}{K^2}\mathcal{N}+CN^{-\frac{3}{2}}(\mathcal{N}+1)^3. \end{align*} $$

Using $\mathcal {N}\leq \sum _{k}|k|^{2\tau }a_k^\dagger a_k$ and $\sum _{k}|k|^{2\tau }d_k^\dagger d_k\leq \sum _{k}|k|^{2}d_k^\dagger d_k$ we therefore obtain

$$ \begin{align*} \sum_{k}|k|^{2\tau}a_k^\dagger a_k & \leq \frac{C}{K^2}\sum_{k}|k|^{2\tau}a_k^\dagger a_k+C\sum_{k}|k|^{2}d_k^\dagger d_k+CN^{-\frac{3}{2}}\mathcal{N}^3+CN^{\tau}+CN^{-1}\mathcal{N}^2\\ & \leq \frac{C}{K^2}\sum_{k}|k|^{2\tau}a_k^\dagger a_k+C\sum_{k}|k|^{2}d_k^\dagger d_k+2CN^{-\frac{3}{2}}\mathcal{N}^3+2CN^{\tau}. \end{align*} $$

Choosing K large enough such that $\frac {C}{K^2}<1$ concludes the proof.

Before we come to the proof of the lower bound in Theorem 1 in the following Subsection 4.2, we are going to derive sufficient estimates on

$$ \begin{align*} \sum_{ijk, \ell m n}(\Theta)_{ijk} \, \widetilde{\psi}_{ijk}^\dagger \, a_0^\dagger a_0^4 + \mathrm{H.c.} \end{align*} $$

in the following Lemma 10. In order to verify Lemma 10, we require the estimate

(116) $$ \begin{align} \sum_k |k|^{2} \tilde{c}_k^\dagger \tilde{c}_k\leq C\left(\sum_k |k|^2 c_k^\dagger c_k+\mathcal{N}+1\right) \end{align} $$

for $ \tilde {c}_k:=a_k +\frac {1}{2}\sum _{ij} (T-1)_{ijk,000}a_i^\dagger a_j^\dagger a_0^3$ , which is verified in Appendix A, see Lemma A3.

Lemma 10. Let $0\leq \gamma <\frac {1}{4}$ . Then there exists a $C>0$ such that

$$ \begin{align*} \pm \left(\sum_{ijk, \ell m n}(\Theta)_{ijk} \, \widetilde{\psi}_{ijk}^\dagger \, a_0^\dagger a_0^4 + \mathrm{H.c.}\right)\leq N^{-\frac{1}{4}}\sum_{k}|k|^2 c_k^\dagger c_k+N^{-\frac{1}{4}} \mathcal{N}+CN^{\frac{1}{4}}. \end{align*} $$

Proof. Let us define for $ijk\neq 0$

(117) $$ \begin{align} \zeta_{ijk}&:= \frac{1}{24} \Big( \Pi_{\mathrm{sym}} 4( \widetilde{V}_N\otimes 1)(T_4 - 1 + \chi )\Big)_{0 i j k,0 0 0 0} =\frac{1}{24} \Big( \Pi_{\mathrm{sym}} 4( V_N\otimes 1)(T_4 - 1 + \chi )\Big)_{0 i j k,0 0 0 0} ,\\ \nonumber \zeta_k&:= \frac{1}{24}\Big( \Pi_{\mathrm{sym}} 4( T^\dagger \widetilde{V}_N\otimes 1)(T_4-1+\chi )\Big)_{0 0 k (-k),0 0 0 0}, \end{align} $$

where $\chi $ is defined in Eq. (91). Then we have the decomposition

(118) $$ \begin{align} \nonumber & \left(\sum_{ijk, \ell m n}(\Theta)_{ijk} \, \widetilde{\psi}_{ijk}^\dagger \, a_0^\dagger a_0^4 + \mathrm{H.c.}\right) \\& \quad = 4\left(\sum_{ijk\neq 0}\overline{\zeta_{ijk}} (a_0^\dagger )^4 a_0 \, \widetilde{\psi}_{ijk} +\mathrm{H.c.}\right)+6 \left(\sum_{k\neq 0}\overline{\zeta_{k}} (a_0^\dagger )^4 a_0^2\, a_k a_{-k} +\mathrm{H.c.}\right). \end{align} $$

Note that $\zeta _k= \frac {1}{24}\Big ( \Pi _{\mathrm {sym}} 4( V_N\otimes 1)(T_4-1+\chi )\Big )_{0 0 k (-k),0 0 0 0}$ for $|k|>K$ . Using the regularity of V and the bounds derived in Eq. (107) and Eq. (109), we observe that we have $|N^3 \zeta _k|\lesssim N^{-\frac {1}{2}}\left (1+\frac {|k|^2}{N}\right )^{-1}$ , and therefore

$$ \begin{align*} \pm \left( \sum_{k\neq 0} \overline{ \zeta_{k}} (a_0^\dagger )^4 a_0^2 a_k a_{-k} + \mathrm{H.c.} \right) \lesssim \epsilon \sum_{k\neq 0} |k|^{2\tau} a_k^\dagger a_k + \epsilon^{-1} N^{\frac{1}{2}-\tau} \lesssim \epsilon \sum_{k\neq 0} |k|^{2\tau} c_k^\dagger c_k + \epsilon \mathcal{N} + \epsilon N^{\tau} + \epsilon^{-1}N^{\frac{1}{2}-\tau}, \end{align*} $$

where we have used Eq. (40). Choosing $\epsilon $ of the order $N^{-\frac {1}{4}}$ and $\tau =\frac {1}{2}$ concludes the analysis of the second term in Eq. (118). Regarding the first term, we use the definition of $\tilde {c}_k$ below Eq. (116), in order to identify $\sum _{ijk}\overline {\zeta _{ijk}} (a_0^\dagger )^4 \widetilde {\psi }_{ijk}a_0$ as

(119) $$ \begin{align} \nonumber & \sum_{ijk}\overline{\zeta_{ijk}}(a_0^\dagger )^4 a_i a_j \tilde{c}_k a_0 - \sum_{ijk}\overline{\zeta_{ijk}}(a_0^\dagger )^4 a_i a_j (\tilde{c}_k-a_k) a_0+ \sum_{ijk}\zeta_{ijk}(a_0^\dagger )^4 a_0^4 (T-1)_{ijk,000}\\\nonumber & \quad = \sum_{ijk}\overline{\zeta_{ijk}}(a_0^\dagger )^4 a_i a_j \tilde{c}_k a_0 - \frac{1}{2}\sum_{ijk,i' j'}\overline{\zeta_{ijk}}(T-1)_{i'j'k,000}a_{i'}^\dagger a_{j'}^\dagger a_0^{4\dagger}a_0^4 a_i a_j\\& \qquad -2\sum_{ijk,i' }\overline{\zeta_{ijk}}(T-1)_{i'j k,000}a_{i'}^\dagger a_0^{4\dagger}a_0^4 a_i. \end{align} $$

In the following we are going to verify that the most significant term $\sum _{ijk}\overline {\zeta _{ijk}}(a_0^\dagger )^4 a_i a_j \tilde {c}_k a_0 $ in Eq. (119) satisfies the desired bound. By Cauchy-Schwarz, we have for $\epsilon>0$

$$ \begin{align*} &\left(\sum_{ijk}\overline{\zeta_{ijk}}(a_0^\dagger )^4 a_i a_j \tilde{c}_k a_0 +\mathrm{H.c.}\right)\leq \epsilon \sum_{k}|k|^{2} \tilde{c}_k^\dagger \tilde{c}_k+\epsilon^{-1}\sum_k \frac{1}{|k|^{2}}\left|\sum_{ij}\zeta_{ijk} a_0^\dagger a_0^4 a_i^\dagger a_j^\dagger \right|^2\\& \quad =\epsilon \sum_{k}|k|^{2} \tilde{c}_k^\dagger \tilde{c}_k+\epsilon^{-1}G^{(0)} X+\epsilon^{-1}\sum_{i,i'} (G^{(1)})_{i,i'}a_i^\dagger a_{i'} X+\epsilon^{-1}\sum_{ij,i'j'}(G^{(2)})_{ij,i'j'}a_i^\dagger a^\dagger_j X a_{j'} a_{i'} \end{align*} $$

with $G^{(0)}:=N^5\sum _{ijk}\frac {|\zeta _{ijk}|^2}{|k|^{2}}$ , $G^{(1)}_{i,i'}:=N^5\delta _{i,i'}\sum _{jk}\frac {|\zeta _{ijk}|^2+\overline {\zeta _{ijk}}\zeta _{jik}}{|k|^{2}}$ and $G^{(2)}_{ij,i' j'}:=N^5\sum _k \frac {\zeta _{ijk}\zeta _{i' j' k}}{|k|^{2}}$ , and $X:=N^{-5}a_0^{4\dagger }a_0 a_0^\dagger a_0^4$ . Using again the regularity of V and Eq. (107), as well as the bounds on $T_4$ in Eq. (109), yields

$$ \begin{align*} |\zeta_{ijk}|\leq C N^{-\frac{7}{2}}\delta_{i+j+k=0}\left(1+\frac{|i|^2+|j|^2+|k|^2}{N}\right)^{-3}, \end{align*} $$

and therefore $|G^{(0)}|\lesssim 1 $ and $\|G^{(1)}\|\lesssim N^{-\frac {3}{2}}$ . The choice $\epsilon :=N{-\frac {1}{4}}$ then yields

$$ \begin{align*} \epsilon^{-1}G^{(0)}X+\epsilon^{-1}\sum_{i,i'} G^{(1)}_{i,i'}a_i^\dagger a_{i'}X\lesssim N^{\frac{1}{4}}+N^{-\frac{5}{4}}\mathcal{N}. \end{align*} $$

Finally $\|G^{(2)}\|\leq N^{-\frac {3}{2}}$ , and therefore

$$ \begin{align*} \sum_{ij,i'j'}G^{(2)}_{ij,i'j'}a_i^\dagger a^\dagger_j X a_{j'} a_{i'}\lesssim N^{-\frac{3}{2}}\mathcal{N}^2\leq N^{-\frac{1}{2}}\mathcal{N}. \end{align*} $$

This concludes the proof together with Eq. (116).

4.2 Proof of the lower bound in Theorem 1

In this subsection, we are going to verify the lower bound in Theorem 1 making use of the sequence of states $\Phi _N$ constructed in Corollary 2, which simultaneously satisfies

Starting point for our investigations is then the lower bound

$$ \begin{align*} H_N&\geq \sum_{k}|k|^{2}d_k^\dagger d_k+ (a_0^\dagger )^3 a_0^3 \lambda_{0,0}+N^{-4}(a_0^\dagger )^4 a_0^4\, (\gamma_N - \sigma_N) -N^{-2}(a_0^\dagger )^2 a_0^2\, \mu_N +\mathbb{Q}_K\\& \quad - \left(\sum_{ijk, \ell m n}(\Theta)_{ijk} \, \widetilde{\psi}_{ijk}^\dagger \, a_0^\dagger a_0^4 + \mathrm{H.c.}\right) + \left(\mathcal{E}_*+\mathcal{E}_*^\dagger \right)- \widetilde{\mathcal{E}}, \end{align*} $$

see Eq. (105). As is proven in Section 7, the coefficients $\gamma _N,\mu _N$ and $\sigma _N$ converge to the corresponding constants $\gamma (V)$ , $\mu (V)$ and $\sigma (V)$ introduced in Eq. (10), Eq. (5) and Eq. (9)

(120) $$ \begin{align} \gamma_N & =\gamma(V)\sqrt{N}+O_{N\rightarrow \infty} \left(N^{-\frac{1}{4}}\right), \end{align} $$

(121) $$ \begin{align} \mu_N & = \mu(V)\sqrt{N}+O_{N\rightarrow \infty}\left(1\right), \end{align} $$

(122) $$ \begin{align} \sigma_N & =\sigma(V)\sqrt{N}+O_{N\rightarrow \infty} \left(N^{\frac{1}{4}}\right), \end{align} $$

see Lemma 17. Given $\epsilon>0$ , assume that K is large enough such that the function $\epsilon (K)$ from Lemma 8 satisfies $\epsilon (K)\leq \epsilon $ . Making use of the fact that

we immediately obtain for C and $\widetilde {C}$ large enough

$$ \begin{align*} |\langle \Phi_N,\widetilde{\mathcal{E}} \Phi_N\rangle| \leq C N^{-\frac{1}{4}} \left\langle \Phi_N, \sum_k |k|^2 c_k^\dagger c_k \Phi_N \right\rangle + \epsilon \left\langle \Phi_N, \mathcal{N} \Phi_N \right\rangle + CN^{\frac{1}{4}} \leq \widetilde{C}N^{\frac{1}{4}} + \epsilon \left\langle \Phi_N, \mathcal{N} \Phi_N \right\rangle . \end{align*} $$

Similarly we obtain by Lemma 10 and Lemma 4 for suitable $C,\widetilde {C}>0$

$$ \begin{align*} &\left|\langle \Phi_N, \left(\sum_{ijk, \ell m n}(\Theta)_{ijk} \, \widetilde{\psi}_{ijk}^\dagger \, a_0^\dagger a_0^4 + \mathrm{H.c.}\right) \Phi_N\rangle\right|\\& \quad \leq C N^{-\frac{1}{4}}\left(\left\langle \Phi_N, \sum_k |k|^2 c_k^\dagger c_k \Phi_N \right\rangle+\sqrt{N}\right)+C N^{\frac{1}{4}}\leq \widetilde{C}N^{\frac{1}{4}},\\& \qquad \left|\langle \Phi_N, \left(\mathcal{E}_*+\mathcal{E}_*^\dagger \right) \Phi_N\rangle\right|\leq CN^{\frac{1}{4}}. \end{align*} $$

By Lemma 2 we furthermore obtain for $\tau ,\epsilon>0$ and K large enough, and a suitable $C>0$ ,

$$ \begin{align*} & \langle \Phi_N, \left((a_0^\dagger )^3 a_0^3 \lambda_{0,0}+\mathbb{Q}_K\right)\Phi_N\rangle\geq \frac{1}{6}b_{\mathcal{M}}(V)N-\epsilon\left\langle\Phi_N, \sum_k |k|^{2\tau}a_k^\dagger a_k\Phi_N \right\rangle -C. \end{align*} $$

Moreover we note that we have by Eq. (120)-(122)

$$ \begin{align*} & \langle \Phi_N, N^{-4}(a_0^\dagger )^4 a_0^4\, (\sigma_N-\gamma_N) \Phi_N\rangle\leq (\sigma_N-\gamma_N) + |\sigma_N-\gamma_N|\langle \Phi_N, \left(1-N^{-4}(a_0^\dagger )^4 a_0^4\right) \Phi_N\rangle\\ & \ \ \ \ \ \ \ \leq \sigma_N-\gamma_N + |\sigma_N-\gamma_N|\left\langle \Phi_N,\frac{\mathcal{N}}{N}\Phi_N\right\rangle \leq (\sigma(V)-\gamma(V))\sqrt{N} + o_{N\rightarrow \infty} \left(N^{\frac{1}{4}}\right), \end{align*} $$

and similarly . Finally by Lemma 9

(123) $$ \begin{align} \left\langle \Phi_N, \mathcal{N} \Phi_N \right\rangle\leq \left\langle\Phi_N, \sum_k |k|^{2\tau}a_k^\dagger a_k\Phi_N \right\rangle\leq C\left\langle\Phi_N, \sum_k |k|^{2}d_k^\dagger d_k\Phi_N \right\rangle+CN^\tau. \end{align} $$

Choosing $\tau <\frac {1}{4}$ and $\epsilon <\frac {1}{2C}$ concludes the proof, since

$$ \begin{align*} E_N +C & \geq\langle \Phi_N, H_N \Phi_N\rangle\geq \frac{1}{6}b_{\mathcal{M}}(V)N+\left(\gamma-\sigma-\mu\right) \sqrt{N}\\ & \ \ \ \ -CN^{\frac{1}{4}}+\left(1-2C\epsilon\right)\left\langle\Phi_N, \sum_k |k|^{2}d_k^\dagger d_k\Phi_N \right\rangle. \end{align*} $$

5 Second-order upper bound

It is the goal of this Section to introduce a trial state $\Phi $ , which simultaneously annihilates the variables $d_k$ for $k\neq 0$ and $\xi _{\ell m n}$ in case $(\ell ,m ,m)\neq 0$ , at least in an approximate sense. We are then going to use this trial state $\Phi $ to verify the upper bound in Theorem 1. For the rest of this Section we specify the parameter K introduced above the definition of $\pi _K$ in Eq. (17) as $K:=0$ . In order to find $\Phi $ , we define $\alpha _{jk}:=(T_2-1)_{jk,00}$ and $\beta _{u i j k}:=(T_4-1)_{uijk,0000}$ , and the generator

(124) $$ \begin{align} \nonumber & \mathcal{G}_2:=\frac{1}{2}\sum_{j k}\alpha_{jk} a_j^\dagger a_k^\dagger a^2_0,\\ & \mathcal{G}_4:=\frac{1}{24}\sum_{u i j k}\beta_{u i j k}a_u^\dagger a_i^\dagger a_j^\dagger a_k^\dagger a^4_0 \end{align} $$

of a unitary group $W_s:=e^{s(\mathcal {G}_2+\mathcal {G}_4)^\dagger - s (\mathcal {G}_2+\mathcal {G}_4)}$ and $W:=W_1$ . We note at this point, that the action of the unitary operator W only creates an $O(1)$ amount of particles, in the sense that

(125) $$ \begin{align} W_{-s} \, \mathcal{N}^m W_s\leq e^{C_m |s|} (\mathcal{N}+1)^m, \end{align} $$

as is proven in Appendix A, see Lemma A1. Applying Duhamel’s formula, we can express $W^{-1}a_{i_1} a_{i_2} a_{i_3} W$ as

(126) $$ \begin{align} & W^{-1}a_{i_1} a_{i_2} a_{i_3} W=a_{i_1} a_{i_2} a_{i_3} - \int_0^1 W_{-s} [a_{i_1} a_{i_2} a_{i_3},\mathcal{G}_4]W_s\mathrm{d}s + \int_0^1 W_{-s} [a_{i_1} a_{i_2} a_{i_3},\mathcal{G}_2^\dagger + \mathcal{G}_4^\dagger - \mathcal{G}_2]W_s\mathrm{d}s. \end{align} $$

Furthermore, note that we can write

(127) $$ \begin{align} [a_{i_1} a_{i_2} a_{i_3},\mathcal{G}_4]=\sum_u \beta_{u i_1 i_2 i_3} a_u^\dagger a_0^4+(\delta \xi)_{i_1 i_2 i_3}, \end{align} $$

where we define the error term

$$ \begin{align*} (\delta \xi)_{i_1 i_2 i_3}:=\frac{1}{4} \sum_{\sigma \in S_3}\sum_{jk} \beta_{i_{\sigma_2}i_{\sigma_3}j k}a_j^\dagger a_k^\dagger a_{i_{\sigma_1}}a_0^4 + \frac{1}{12} \sum_{\sigma \in S_3}\sum_{ijk} \beta_{i_{\sigma_3}i j k}a_i^\dagger a_j^\dagger a_k^\dagger a_{i_{\sigma_1}}a_{i_{\sigma_2}}a_0^4. \end{align*} $$

Therefore we can write the transformed operators $W^{-1}\xi _{i_1 i_2 i_3}W$ as

(128) $$ \begin{align} \nonumber W^{-1}\xi_{i_1 i_2 i_3}W&=( \psi + \delta_1 \psi)_{i_1 i_2 i_3} - \int_0^1 W_s^{-1 }(\delta \xi)_{i_1 i_2 i_3} W_s\, \mathrm{d}s + \int_0^1 W_{-s} [a_i a_j a_k,\mathcal{G}_2^\dagger + \mathcal{G}_4^\dagger - \mathcal{G}_2]W_s\mathrm{d}s\\\nonumber & \quad +\int_0^1 \int_0^s W_t^{-1}\left[\sum_u \beta_{u i_1 i_2 i_3} a_u^\dagger a_0^4,\, \mathcal{G}_2^\dagger +\mathcal{G}_4^\dagger - \mathcal{G}_2 - \mathcal{G}_4 \right]W_t\, \mathrm{d}t \mathrm{d}s\\& \quad + \int_0^1 W_{-s} [\xi_{i_1 i_2 i_3} - a_{i_1}a_{i_2}a_{i_3},\, \mathcal{G}_2^\dagger +\mathcal{G}_4^\dagger - \mathcal{G}_2 - \mathcal{G}_4] W_s\, \mathrm{d}s. \end{align} $$

Recall the definition of $\mathcal {E}_{\mathcal {P}}$ defined in Eq. (75). The following Lemma 11 provides sufficient bounds on the various error terms appearing in Eq. (128).

Lemma 11. There exists a constant $C>0$ , such that

(129) $$ \begin{align} \mathcal{E}_{\mathcal{P}}(\delta \xi) \leq C (\mathcal{N}+1)^6. \end{align} $$

Furthermore, we have $\mathcal {E}_{\mathcal {P}} \left ([a_{i_1} a_{i_2} a_{i_3},\mathcal {G}_2^\dagger + \mathcal {G}_4^\dagger - \mathcal {G}_2]\right )\leq C (\mathcal {N}+1)^6$ and

(130) $$ \begin{align} \kern-2pt & \mathcal{E}_{\mathcal{P}} \left([\xi_{i_1 i_2 i_3} - a_{i_1}a_{i_2}a_{i_3},\, \mathcal{G}_2^\dagger +\mathcal{G}_4^\dagger - \mathcal{G}_2 - \mathcal{G}_4]\right)\leq C (\mathcal{N}+1)^6, \end{align} $$

(131) $$ \begin{align} & \mathcal{E}_{\mathcal{P}} \left(\left[\sum_u \beta_{u i_1 i_2 i_3} a_u^\dagger a_0^4,\, \mathcal{G}_2^\dagger +\mathcal{G}_4^\dagger - \mathcal{G}_2 - \mathcal{G}_4 \right]\right)\leq C (\mathcal{N}+1)^6. \end{align} $$

Proof. Let us define

$$ \begin{align*} (\delta_1 \xi)_{i_1 i_2 i_3}&:=\frac{1}{4} \sum_{jk} \beta_{i_{2}i_{3}j k}a_j^\dagger a_k^\dagger a_{i_{1}}a_0^4,\\ C_N&:=\sup_{i_1}\sum_{jk,i_2 i_3,i^{\prime}_1 i^{\prime}_2 i^{\prime}_3}\left|(V_N)_{i_1 i_2 i_3,i^{\prime}_1 i^{\prime}_2 i^{\prime}_3} \beta_{i_{2}i_{3}j k}\beta_{i^{\prime}_{2}i^{\prime}_{3}j k}\right|\lesssim N^{-5}, \end{align*} $$

where we have used Eq. (109) to estimate $C_N$ . Applying Cauchy-Schwarz yields

(132) $$ \begin{align} \sum_{(i_1 i_2 i_3),(i_1' i_2' i_3')\in A}(V_N)_{i_1 i_2 i_3,i_1' i_2' i_3'}(\delta_1 \xi)_{i_1 i_2 i_3}^\dagger (\delta_1 \xi)_{i_1' i_2' i_3'}\leq C_N (a_0^\dagger )^4a_0^4 \left(\sum_{i_1} a_{i_1}^\dagger a_{i_1}\right)(\mathcal{N}+1)^2. \end{align} $$

Using the fact that $C_N(a_0^\dagger )^4a_0^4 \left (\sum _{i_1} a_{i_1}^\dagger a_{i_1}\right )\leq C_N N^5\lesssim 1$ , we observe that the quantity in Eq. (132) is bounded by the right-hand side of Eq. (129). Let us furthermore define

$$ \begin{align*} (\delta_2 \xi)_{i_1 i_2 i_3}:=\frac{1}{12} \sum_{ijk} \beta_{i_{3}i j k}a_i^\dagger a_j^\dagger a_k^\dagger a_{i_{1}}a_{i_{2}}a_0^4. \end{align*} $$

In the following we want to distinguish between the cases $A':=\{(i_1 i_2 i_3)\in A: i_1,i_2\neq 0\}$ and $A":=A\setminus A'$ , leading to the definition

where we have again used Eq. (109). Applying Cauchy-Schwarz leads to the estimate

$$ \begin{align*} \sum_{(i_1 i_2 i_3),(i_1' i_2' i_3')\in A}(V_N)_{i_1 i_2 i_3,i_1' i_2' i_3'}(\delta_2 \xi)_{i_1 i_2 i_3}^\dagger (\delta_2 \xi)_{i_1' i_2' i_3'}\leq C_N' N^4 (\mathcal{N}+1)^5 + C_N" N^6 (\mathcal{N}+1)^3, \end{align*} $$

which is bounded by the right-hand side of Eq. (129). Finally we use that $V_N$ is permutation-symmetric and non-negative, and therefore the left-hand side of Eq. (129) is bounded by

$$ \begin{align*} & 6 \sum_{(i_1 i_2 i_3),(i_1' i_2' i_3')\in A}(V_N)_{i_1 i_2 i_3,i_1' i_2' i_3'}(\delta_1 \xi+\delta_2 \xi)_{i_1 i_2 i_3}^\dagger (\delta_1 \xi+\delta_2 \xi)_{i_1' i_2' i_3'}\\& \quad\leq 12 \sum_{(i_1 i_2 i_3),(i_1' i_2' i_3')\in A} (V_N)_{i_1 i_2 i_3,i_1' i_2' i_3'}(\delta_1 \xi)_{i_1 i_2 i_3}^\dagger (\delta_1 \xi)_{i_1' i_2' i_3'}+ 12 \sum_{(i_1 i_2 i_3),(i_1' i_2' i_3')\in A} (V_N)_{i_1 i_2 i_3,i_1' i_2' i_3'}(\delta_2 \xi)_{i_1 i_2 i_3}^\dagger (\delta_2 \xi)_{i_1' i_2' i_3'}. \end{align*} $$

Regarding the term $\mathcal {E}_{\mathcal {P}} \left ([a_{i_1} a_{i_2} a_{i_3},\mathcal {G}_2^\dagger + \mathcal {G}_4^\dagger - \mathcal {G}_2]\right )$ , let us analyze the term involving the commutator with $\mathcal {G}_2$ , the terms involving $\mathcal {G}_2^\dagger + \mathcal {G}_4^\dagger $ can be analyzed in a similar fashion as has been done in Lemma 6. We compute

(133) $$ \begin{align} [a_{i_1} a_{i_2} a_{i_3},\mathcal{G}_2]=\alpha_{i_2 i_3}a_{i_1} a_0^2+\sum_u \alpha_{u i_3}a_u^\dagger a_{i_2} a_{i_3} +\{\mathrm{Permutations}\}. \end{align} $$

In order to analyze the first term on the right-hand side of Eq. (133), let us define $D_N:=\sup _{i_1}\sum _{i_2 i_3, i_1' i_2' i_3'}\left |(V_N)_{i_1 i_2 i_3, i_1' i_2' i_3'} \alpha _{i_2 i_3}\alpha _{i_2' i_3'}\right |$ and note that $D_N\lesssim N^{-3}$ by Eq. (108). Hence

$$ \begin{align*} \sum_{(i_1 i_2 i_3),(i_1' i_2' i_3')\in A}(V_N)_{i_1 i_2 i_3,i_1' i_2' i_3'}\left(\alpha_{i_2 i_3}a_{i_1} a_0^2\right)^\dagger \alpha_{i^{\prime}_2 i^{\prime}_3}a_{i^{\prime}_1} a_0^2\leq D_N N^3 \lesssim 1. \end{align*} $$

Regarding the second term on the right-hand side of Eq. (133), we use again the split $A=A'\cup A"$ and define

where we have used Eq. (108). Consequently

$$ \begin{align*} \sum_{(i_1 i_2 i_3),(i_1' i_2' i_3')\in A}(V_N)_{i_1 i_2 i_3,i_1' i_2' i_3'}\left(\sum_u \alpha_{u i_3}a_u^\dagger a_{i_2} a_{i_3}\right)^\dagger \left(\sum_u \alpha_{u i_3}a_u^\dagger a_{i_2} a_{i_3}\right)\lesssim N^{-\frac{1}{2}}(\mathcal{N}+1)^3+1, \end{align*} $$

and therefore $\mathcal {E}_{\mathcal {P}} \left ([a_{i_1} a_{i_2} a_{i_3},\mathcal {G}_2]\right )\leq 12 \mathcal {E}_{\mathcal {P}} \left (\alpha _{i_2 i_3}a_{i_1} a_0^2\right )+ 12 \mathcal {E}_{\mathcal {P}} \left (\sum _u \alpha _{u i_3}a_u^\dagger a_{i_2} a_{i_3}\right )\lesssim (\mathcal {N}+1)^3$ . The inequalities in Eq. (130) and Eq. (131) can be verified similarly.

With Lemma 11 at hand, we show in the subsequent Corollary 6 that after conjugation with the unitary W, the potential energy of the operators $\xi _{ijk}$ is comparable to the potential energy of $(\psi +\delta _1)_{ijk}$ .

Corollary 6. There exists a constant $C>0$ , such that

$$ \begin{align*} W^{-1}\mathcal{E}_{\mathcal{P}}(\xi)W \leq C\, \mathcal{E}_{\mathcal{P}} \left(\psi+\delta_1 \psi\right) + C\, (\mathcal{N}+1)^6. \end{align*} $$

Proof. Using the sign $V_N\geq 0$ , we obtain by the Cauchy-Schwarz inequality and the representation of $W^{-1}\xi _{i_1 i_2 i_3} W$ in Eq. (128) the estimate

$$ \begin{align*} & W^{-1} \mathcal{E}_{\mathcal{P}}(\xi)W=\mathcal{E}_{\mathcal{P}} \left(W^{-1}\xi W\right)\leq 5\mathcal{E}_{\mathcal{P}} \left(\psi + \delta_1 \right)+5 \int_0^1 W_s^{-1 }\mathcal{E}_{\mathcal{P}} \left( \delta \xi\right) W_s \mathrm{d}s\\& \qquad + 5\int_0^1 W_{-s} \mathcal{E}_{\mathcal{P}} \left([a_i a_j a_k,\mathcal{G}_2^\dagger + \mathcal{G}_4^\dagger - \mathcal{G}_2]\right)\mathrm{d}s W_s \\& \qquad +\frac{5}{2}\int_0^1 \int_0^s W_t^{-1}\mathcal{E}_{\mathcal{P}}\left(\left[\sum_u \beta_{u i_1 i_2 i_3} a_u^\dagger a_0^4,\, \mathcal{G}_2^\dagger +\mathcal{G}_4^\dagger - \mathcal{G}_2 - \mathcal{G}_4 \right]\right)W_t \mathrm{d}t \mathrm{d}s\\& \qquad + 5\int_0^1 W_{-s} \mathcal{E}_{\mathcal{P}} \left([\xi_{i_1 i_2 i_3} - a_{i_1}a_{i_2}a_{i_3},\, \mathcal{G}_2^\dagger +\mathcal{G}_4^\dagger - \mathcal{G}_2 - \mathcal{G}_4] \right) W_s\mathrm{d}s\\& \quad\lesssim \mathcal{E}_{\mathcal{P}} \left(\psi + \delta_1 \right) + \int_0^1 W_{-s}(\mathcal{N} + 1)^6 W_s \, \mathrm{d}s + \int_0^1 \int_0^s W_{-t}(\mathcal{N} + 1)^6 W_t\, \mathrm{d}t\mathrm{d}s \lesssim \mathcal{E}_{\mathcal{P}} \left(\psi + \delta_1 \right) + (\mathcal{N} + 1)^6, \end{align*} $$

where we have first used Lemma 11 and subsequently Eq. (125) in the last line.

Regarding the variable $d_k$ , Duhamel’s formula yields for $k\neq 0$

(134) $$ \begin{align} W^{-1}d_kW = c_k + \int_0^1 \int_0^s W_{-t}\left[\left[a_k,G_2 + G_4\right],\mathcal{G}_2^\dagger + \mathcal{G}^\dagger_4\right]W_t \mathrm{d}t\mathrm{d}s + \int_0^1 W_{-s}\left[d_k - a_k,G_2^\dagger + G_4^\dagger \right]W_s \mathrm{d}s. \end{align} $$

Recall the kinetic energy of an operator-valued one-particle vector $\Theta _k$ defined in Eq. (83). Then the following Lemma 12 provides sufficient bounds on the various error terms appearing in Eq. (134).

Lemma 12. There exists a constant $C>0$ , such that for $m\in \mathbb {N}$

$$ \begin{align*} \mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m \left[d_k- a_k,G_2^\dagger + G_4^\dagger \right]\right)&\leq \frac{C}{N}(\mathcal{N}+1)^{5+2m},\\\mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m\left[\left[a_k,G_2+G_4\right],\mathcal{G}_2^\dagger + \mathcal{G}^\dagger_4\right]\right)&\leq \frac{C}{N}(\mathcal{N}+1)^{5+2m}. \end{align*} $$

Proof. Let us compute as an example for $k\neq 0$

$$ \begin{align*} &\left[\left[a_k,G_4\right],\mathcal{G}^\dagger_4\right]= \frac{1}{24}\sum_{i_1 i_2 i_3}\beta_{k i_1 i_2 i_3}\left[ a_{i_1}^\dagger a_{i_2}^\dagger a_{i_3}^\dagger a^4_0,\mathcal{G}^\dagger_4\right]\\& \quad =\frac{1}{24}\sum_{i_1 i_2 i_3}\beta_{k i_1 i_2 i_3} a_{i_1}^\dagger a_{i_2}^\dagger \left[a_{i_3}^\dagger,\mathcal{G}^\dagger_4\right] a^4_0+\frac{1}{24}\sum_{i_1 i_2 i_3}\beta_{k i_1 i_2 i_3} a_{i_1}^\dagger \left[a_{i_2}^\dagger,\mathcal{G}^\dagger_4\right] a_{i_3}^\dagger a^4_0\\& \qquad +\frac{1}{24}\sum_{i_1 i_2 i_3}\beta_{k i_1 i_2 i_3}\left[ a_{i_1}^\dagger,\mathcal{G}^\dagger_4\right] a_{i_2}^\dagger a_{i_3}^\dagger a^4_0+\frac{1}{24}\sum_{i_1 i_2 i_3}\beta_{k i_1 i_2 i_3} a_{i_1}^\dagger a_{i_2}^\dagger a_{i_3}^\dagger \left[a^4_0,\mathcal{G}^\dagger_4\right], \end{align*} $$

and let us focus on the term

$$ \begin{align*} \sum_{i_1 i_2 i_3}\beta_{k i_1 i_2 i_3} a_{i_1}^\dagger a_{i_2}^\dagger \left[a_{i_3}^\dagger,\mathcal{G}^\dagger_4\right] a^4_0=\frac{1}{6} \sum_{i_1 i_2 i_3, j_1 j_2 j_3}\beta_{k i_1 i_2 i_3}\overline{\beta_{i_3 j_1 j_2 j_3}} (a_0^\dagger)^4 a_0^4 a_{i_1}^\dagger a_{i_2}^\dagger a_{j_1}a_{j_2}a_{j_3}. \end{align*} $$

Defining

$$ \begin{align*} C_N:=\sum_{k,i_1 i_2,j_1 j_2 j_3}|k|^2 \left|\sum_{i_3}\beta_{k i_1 i_2 i_3}\overline{\beta_{i_3 j_1 j_2 j_3}}\right|^2\lesssim N^{-\frac{19}{2}}, \end{align*} $$

where we have used Eq. (109), we obtain

$$ \begin{align*} \mathcal{E}_{\mathcal{K}} \left(\sum_{i_1 i_2 i_3}\beta_{k i_1 i_2 i_3} \mathcal{N}^m a_{i_1}^\dagger a_{i_2}^\dagger \left[a_{i_3}^\dagger,\mathcal{G}^\dagger_4\right] a^4_0\right)\lesssim N^{-\frac{19}{2}}\left((a_0^\dagger)^4 a_0^4\right)^2 (\mathcal{N}+1)^{5+2m}\leq N^{-\frac{3}{2}}(\mathcal{N}+1)^{5+2m}. \end{align*} $$

The other estimates in Lemma 12 can be verified similarly.

Similar to Corollary 6, we show in the following Corollary 7 that, after conjugation with the unitary W, the kinetic energy of the operators $ d_k$ is comparable to the one of $c_k$ .

Corollary 7. There exists a constant $C>0$ , such that for $m\in \mathbb {N}$

$$ \begin{align*} W^{-1} \mathcal{E}_K \left(\mathcal{N}^m d\right) W\leq C\mathcal{E}_K \left( c\right) + C \mathcal{E}_K \left(\mathcal{N}^m c\right) + \frac{C}{N} (\mathcal{N}+1)^{5+2m}. \end{align*} $$

Proof. By Eq. (125) we have $(W^{-1}\mathcal {N}^{m}W)^* W^{-1}\mathcal {N}^{m}W=W^{-1}\mathcal {N}^{2m}W\lesssim \mathcal {N}^{2m}+1$ , hence

$$ \begin{align*} W^{-1}\mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m d\right)W=\mathcal{E}_{\mathcal{K}} \left(W^{-1}\mathcal{N}^m W\, W^{-1} d\, W\right)\lesssim \mathcal{E}_{\mathcal{K}} \left( W^{-1} d\, W\right)+\mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m W^{-1} d\, W\right). \end{align*} $$

Following the ideas in Corollary 6, we estimate using Eq. (134)

$$ \begin{align*} & \mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m W^{-1} d\, W\right)\leq 3\, \mathcal{E}_{\mathcal{K}}\left(\mathcal{N}^m c\right)\\& \qquad +\frac{3}{2}\int_0^1 \int_0^s W_{-t}\mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m\left[\left[a_k,G_2 + G_4\right],\mathcal{G}_2^\dagger + \mathcal{G}^\dagger_4\right]\right)W_t\mathrm{d}s \\& \qquad + 3\int_0^1 W_{-s}\mathcal{E}_{\mathcal{K}} \left(\mathcal{N}^m\left[d_k - a_k,G_2^\dagger + G_4^\dagger \right]\right)W_s\mathrm{d}s\\& \quad\lesssim \mathcal{E}_{\mathcal{K}}\left(\mathcal{N}^m c\right) + \frac{1}{N}\int_0^1 \int_0^s W_{-t}(\mathcal{N} + 1)^{5+2m} W_t\mathrm{d}s + \frac{1}{N}\int_0^1 W_{-s}(\mathcal{N} + 1)^5W_s\mathrm{d}s \\& \quad\lesssim \mathcal{E}_{\mathcal{K}}\left(\mathcal{N}^m c\right) + \frac{1}{N}(\mathcal{N} + 1)^{5+2m}, \end{align*} $$

where we have first used Lemma 12 and subsequently Eq. (125) in the last line.

Before we come to the proof of the upper bound in Theorem 1, we are showing in the following Lemma 13 that even without a unitary conjugation, the kinetic energy of $c_k$ is comparable with the one of $d_k$ . The price for dropping the unitary W is that we obtain an order $O_{N\rightarrow \infty } \left (\sqrt {N}\right )$ prefactor in front of the excess term $(\mathcal {N}+3)^{3+2m}$ , instead of a prefactor of the order $O_{N\rightarrow \infty } \left (1\right )$ .

Lemma 13. There exists a constant $C>0$ , such that for $m\in \mathbb {N}$

$$ \begin{align*} \mathcal{E}_K \left(\mathcal{N}^m c\right)\leq C \mathcal{E}_K \left(\mathcal{N}^m d\right)+C\sqrt{N}(\mathcal{N}+1)^{3+2m}. \end{align*} $$

Proof. Note that we can write $c_k$ as

$$ \begin{align*} c_k=d_k-2\sum_j \alpha_{jk}\alpha_{jk} a_j^\dagger a_0^2-4\sum_{uij}\beta_{uijk}a^\dagger_u a^\dagger_i a^\dagger_j a_0^4, \end{align*} $$

and therefore

$$ \begin{align*} \mathcal{E}_K \left(\mathcal{N}^m c\right)\leq 3\, \mathcal{E}_K \left(\mathcal{N}^m d\right)+12\, \mathcal{E}_K\left(\sum_j \alpha_{jk}\, \mathcal{N}^m a_j^\dagger a_0^2\right)+48\, \mathcal{E}_K\left(\sum_{uij}\beta_{uijk}\, \mathcal{N}^m a^\dagger_u a^\dagger_i a^\dagger_j a_0^4\right). \end{align*} $$

Defining the constant $C_N:=\sum _{jk}|k|^2 |\alpha _{jk}|^2\lesssim N^{-\frac {3}{2}}$ , which follows from Eq. (108), we obtain

$$ \begin{align*} \mathcal{E}_K\left(\sum_j \alpha_{jk}\, \mathcal{N}^m a_j^\dagger a_0^2\right)\leq C_N (a_0^\dagger)^2 a_0^2 (\mathcal{N}+1)^{m+1}\lesssim \sqrt{N}(\mathcal{N}+1)^{m+1}. \end{align*} $$

Similarly we obtain

$$ \begin{align*} \mathcal{E}_K\left(\sum_{uij}\beta_{uijk}\, \mathcal{N}^m a^\dagger_u a^\dagger_i a^\dagger_j a_0^4\right)\lesssim \sqrt{N}(\mathcal{N}+3)^3\lesssim \sqrt{N}(\mathcal{N}+1)^3.\\[-49pt] \end{align*} $$

Proof of the upper bound in Theorem 1.

Let us define the trial state $\Phi :=W\Gamma $ , where $\Gamma $ is the state defined below Eq. (71), and recall the representation of $H_N$ in Eq. (106)

$$ \begin{align*} \langle \Phi,H_N \Phi\rangle&=\lambda_{0,0}N^3 \langle \Phi,N^{-3}(a_0^\dagger)^3 a_0^3 \Phi\rangle + (\gamma_N - \sigma_N) \langle \Phi, N^{-4}(a_0^\dagger)^4 a_0^4 \Phi\rangle - \mu_N \langle \Phi, N^{-2}(a_0^\dagger)^2 a_0^2 \Phi\rangle\\& \quad + \langle \Phi, \mathcal{E}_{\mathcal{K}}(d) \Phi\rangle+\langle \Phi, \mathcal{E}_{\mathcal{P}}(\xi) \Phi\rangle+ 2\mathfrak{Re}\left\langle \Phi , \mathcal{E}_* \Phi \right\rangle - \left\langle \Phi , \widetilde{\mathcal{E}}\Phi \right\rangle \\& \quad - \left\langle \Phi , \left(\sum_{ijk, \ell m n}(\Theta)_{ijk} \, \widetilde{\psi}_{ijk}^\dagger \, a_0^\dagger a_0^4 + \mathrm{H.c.}\right) \Phi\right\rangle. \end{align*} $$

By Eq. (125) and the fact that $\pm \left (N^{-m}(a_0^\dagger )^m a_0^m-1\right )\lesssim N^{-1}\mathcal {N}$ , we obtain that

$$ \begin{align*} \left| \langle \Phi,N^{-3}(a_0^\dagger)^3 a_0^3 \Phi\rangle-1\right|\lesssim N^{-1}\langle \Phi, \mathcal{N}\Phi\rangle=N^{-1}\langle \Gamma, W^{-1}\mathcal{N}W\Gamma\rangle\lesssim N^{-1}\langle \Gamma, (\mathcal{N}+1)\Gamma\rangle\lesssim N^{-1}, \end{align*} $$

see Eq. (84) for the last estimate. Making use of Lemma 8, Lemma 4 and Lemma 10 yields

$$ \begin{align*} & \left|\left\langle \Phi , \mathcal{E}_* \Phi \right\rangle\right|\lesssim N^{-\frac{1}{4}}\langle \Phi,\mathcal{E}_{\mathcal{K}}(c)\Phi\rangle+N^{\frac{1}{4}}\langle \Phi,\mathcal{N}\Phi\rangle,\\& \quad\Big| \Big\langle \Phi , \left(\sum_{ijk, \ell m n}(\Theta)_{ijk} \, \widetilde{\psi}_{ijk}^\dagger \, a_0^\dagger a_0^4 + \mathrm{H.c.}\right) \Phi \Big\rangle\Big|\lesssim N^{-\frac{1}{4}}\langle \Phi,\mathcal{E}_{\mathcal{K}}(c)\Phi\rangle+N^{-\frac{1}{4}}\langle \Phi,\mathcal{N}\Phi\rangle+N^{\frac{1}{4}},\\& \quad \left| \left\langle \Phi , \widetilde{\mathcal{E}} \Phi \right\rangle\right| \lesssim N^{-\frac{1}{4}}\left\langle \Phi,\mathcal{E}_{\mathcal{K}} \left(\left(\frac{\mathcal{N}}{\sqrt{N}} + 1\right)c\right)\Phi \right\rangle + N^{\frac{1}{4}}\left\langle \Phi, \left(\frac{\mathcal{N}}{\sqrt{N}} + 1\right)^2 \left(\mathcal{N} + \sqrt{N}\right) \Phi\right\rangle + \langle \Phi,\mathcal{N}\Phi\rangle. \end{align*} $$

Observe that and furthermore we have by Lemma 13 and Corollary 7

$$ \begin{align*} & \left\langle \Phi, \mathcal{E}_K \left(\mathcal{N}^m c\right)\Phi \right\rangle\lesssim \left\langle \Phi, \mathcal{E}_K \left(\mathcal{N}^m d\right)\Phi \right\rangle + \sqrt{N}=\left\langle \Gamma , W^{-1}\mathcal{E}_K \left(\mathcal{N}^m d\right)W\Gamma \right\rangle + \sqrt{N}\\ & \ \ \ \ \ \ \ \ \ \lesssim \left\langle \Gamma , \mathcal{E}_K \left( c\right)\Gamma \right\rangle + \left\langle \Gamma , \mathcal{E}_K \left(\mathcal{N}^m c\right)\Gamma \right\rangle + \sqrt{N}\lesssim \sqrt{N}, \end{align*} $$

where we used Corollary 4 in the last estimate. Putting together what we have so far, and utilizing Eq. (120)-(122) again, yields

$$ \begin{align*} \langle \Phi,H_N \Phi\rangle = \frac{1}{6}b_{\mathcal{M}}(V)N + \big(\gamma(V) - \sigma(V) - \mu(V) \big) \sqrt{N} + \langle \Phi, \mathcal{E}_{\mathcal{K}}(d) \Phi\rangle + \langle \Phi, \mathcal{E}_{\mathcal{P}}(\xi) \Phi\rangle + O_{N\rightarrow \infty} \left( N^{\frac{1}{4}} \right) . \end{align*} $$

Using Corollary 6, Corollary 7, Corollary 3 and Corollary 4, we further have

$$ \begin{align*} & 0\leq \langle \Phi, \mathcal{E}_{\mathcal{P}}(\xi) \Phi\rangle=\langle \Gamma , W^{-1} \mathcal{E}_{\mathcal{P}}(\xi)W \Gamma\rangle\lesssim \langle \Gamma , \mathcal{E}_{\mathcal{P}}(\psi+\delta_1 \psi) \Gamma\rangle+1\lesssim 1,\\& 0\leq \langle \Phi, \mathcal{E}_{\mathcal{K}}(d) \Phi\rangle=\langle \Gamma, W^{-1} \mathcal{E}_{\mathcal{K}}(d)W \Gamma\rangle\lesssim \langle \Gamma, \mathcal{E}_{\mathcal{K}}(c) \Gamma\rangle+\frac{1}{N}\lesssim \frac{1}{N}.\\[-41pt] \end{align*} $$

6 Proof of Theorem 2

In the following, we want to verify Theorem 2, claiming that any sequence of states $\Psi _N$ with

$$ \begin{align*} \langle \Psi_N,H_N \Psi\rangle\leq E_N+O_{N\rightarrow \infty} \left(N^{\frac{1}{4}}\right), \end{align*} $$

satisfies complete Bose-Einstein condensation with a rate of the order $N^{-\frac {3}{4}}$ . Together with Theorem 1, the statement follows immediately once we can show that

(135) $$ \begin{align} H_N(\alpha):=H_N-\alpha\mathcal{N}\geq \frac{1}{6}b_{\mathcal{M}}(V)N+\Big( \gamma(V) - \mu(V) - \sigma(V) \Big) \sqrt{N}-CN^{\frac{1}{4}}, \end{align} $$

for some constant C. In order to prove Eq. (135), we observe that by the results in [Reference Nam, Ricaud and Triay23, Section 7], see also the comment below [Reference Nam, Ricaud and Triay25, Theorem 4], the modified operators $H_N(2\alpha )$ satisfy the asymptotic identity

$$ \begin{align*} \lim_{N\rightarrow \infty}\frac{1}{N}\inf \sigma\big(H_N(2\alpha)\big)= \inf_{u\in L^2(\Lambda):\|u\|=1} \mathcal{E}_{\mathrm{GP}}^\alpha(u), \end{align*} $$

where the modified Gross-Pitaevskii functional is defined as

$$ \begin{align*} \mathcal{E}_{\mathrm{GP}}^\alpha(u):=\langle u,(-\Delta)u\rangle + \frac{b_{\mathcal{M}}(V)}{6}\int_\Lambda |u(x)|^6\mathrm{d}x + 2\alpha\|Pu\|^2 - 2\alpha \end{align*} $$

using the projection $P=1-Q$ , with Q being introduced in Eq. (18). Note that in the notation of [Reference Nam, Ricaud and Triay25] the operator $H_N(2\alpha )$ reads

$$ \begin{align*} H_N(2\alpha)=H_N+2\alpha\sum_{j=1}^N P_{x_j} -2\alpha N. \end{align*} $$

Furthermore, for $\alpha <2\pi ^2$ , we have that

$$ \begin{align*} \langle u,(-\Delta)u\rangle\geq 4\pi^2 \|Qu\|^2\geq 2\alpha \|u\|^2-2\alpha\|Pu\|^2 , \end{align*} $$

and by Hölder’s inequality we have

$$ \begin{align*} 1=\left(\int |u|^2\mathrm{d}x\right)^3\leq \left(\int_\Lambda 1\mathrm{d}x\right)^2\int |u|^6\mathrm{d}x=\int |u|^6\mathrm{d}x \end{align*} $$

for any $u\in L^2(\Lambda )$ with $\|u\|=1$ , leading to the lower bound

$$ \begin{align*} \lim_{N\rightarrow \infty}\frac{1}{N}\inf \sigma\big(H_N(2\alpha)\big)\geq \frac{b_{\mathcal{M}}(V)}{6}. \end{align*} $$

Therefore, the ground state $\Psi _{N,\alpha }$ of $H_N(\alpha )$ satisfies

$$ \begin{align*} \langle \Psi_{N,\alpha},H_N(\alpha)\Psi_{N,\alpha}\rangle & = \langle \Psi_{N,\alpha},H_N(2\alpha)\Psi_{N,\alpha}\rangle+\alpha \langle \Psi_{N,\alpha},\mathcal{N }\Psi_{N,\alpha}\rangle\\ & \geq \frac{b_{\mathcal{M}}(V)}{6}N+\alpha \langle \Psi_{N,\alpha},\mathcal{N }\Psi_{N,\alpha}\rangle-o_{N\rightarrow \infty}(N), \end{align*} $$

and by Theorem 1 we obtain the matching upper bound

$$ \begin{align*} \langle \Psi_{N,\alpha},H_N(\alpha)\Psi_{N,\alpha}\rangle=\inf \sigma\big(H_N(\alpha)\big)\leq \inf \sigma\big(H_N\big)\leq \frac{b_{\mathcal{M}}(V)}{6}N+o_{N\rightarrow \infty}(N). \end{align*} $$

As a consequence, the states $\Psi _{N,\alpha }$ satisfy complete Bose-Einstein condensation

$$ \begin{align*} \langle \Psi_{N,\alpha},\mathcal{N }\Psi_{N,\alpha}\rangle=o_{N\rightarrow \infty}(N), \end{align*} $$

and we can proceed exactly as in Corollary 2, as long as the additional condition $\alpha <\delta $ holds with $\delta $ being the constant in Eq. (64). In particular, there exist states $\Phi _{N,\alpha }$ such that

and we have the estimate on the kinetic energy $\left \langle \Phi _{N,\alpha }, \sum _{k}|k|^2 c_k^\dagger c_k \Phi _{N,\alpha } \right \rangle \leq C\sqrt {N}$ . Note that the localization results in Lemma 5 hold without any modification for the operator $H_N(\alpha )$ , since $\mathcal {N}$ commutes with the localization functions . Following Subsection 4.2, we therefore arrive at the lower bound

$$ \begin{align*} \inf \sigma\big(H_N(\alpha)\big) +C & \geq\langle \Phi_{N,\alpha}, H_N(\alpha) \Phi_{N,\alpha}\rangle\geq \frac{1}{6}b_{\mathcal{M}}(V)N+\left(\gamma-\sigma-\mu\right) \sqrt{N}\\ & \quad -\alpha\langle \Phi_{N,\alpha},\mathcal{N}\Phi_{N,\alpha}\rangle-CN^{\frac{1}{4}}+\left(1-2C\epsilon\right)\left\langle\Phi_N, \sum_k |k|^{2}d_k^\dagger d_k\Phi_N \right\rangle. \end{align*} $$

Using again Eq. (123), for $\tau <\frac {1}{4}$ , we obtain for a large enough constant C

$$ \begin{align*} \inf \sigma\big(H_N(\alpha)\big) & \geq \frac{1}{6}b_{\mathcal{M}}(V)N+\left(\gamma-\sigma-\mu\right) \sqrt{N}-CN^{\frac{1}{4}}\\ & \quad +\left(1-2C\epsilon-C\alpha\right)\left\langle\Phi_N, \sum_k |k|^{2}d_k^\dagger d_k\Phi_N \right\rangle. \end{align*} $$

Choosing $\alpha $ and $\epsilon $ small enough such that $2\epsilon +\alpha <\frac {1}{C}$ concludes the proof of Eq. (135).

7 Analysis of the scattering coefficients

This Section is devoted to the study of the variational problems in the definition of $b_{\mathcal {M}}(V)$ in Eq. (4) and the definition of $\sigma (V)$ in Eq. (9) as well as the study of their corresponding minimizers $\omega $ and $\eta $ . Especially we want to compare $\gamma (V),\mu (V)$ , and $\sigma (V)$ defined in Eq. (10), Eq. (5) and Eq. (9) with $\gamma _N,\mu _N$ , and $\sigma _N$ defined in Eq. (97), Eq. (102) and Eq. (98), see Lemma 17. The proof will be based on the observation that the N-dependent quantities can be seen as a counterpart on the three-dimensional torus $\Lambda $ to the N-independent quantities defined in terms of variational problems on the full space $\mathbb {R}^3$ . Similarly we will compare in Lemma 16 the modified scattering length $b_{\mathcal {M}}(V)$ , which can be expressed in terms of the minimizer $\omega $ as

$$ \begin{align*} b_{\mathcal{M}}(V)=\int_{\mathbb{R}^6}(1-\omega)V\, \mathrm{d}x , \end{align*} $$

see [Reference Nam, Ricaud and Triay23], with its counterpart on the torus $\Lambda $ defined in Eq. (29) as

$$ \begin{align*} 6\lambda_{0,0}=\langle u_0 u_0 u_{0},(V_N -V_N R V_N)u_0 u_0 u_0\rangle=\left(V_N -V_N R V_N\right)_{000,000}. \end{align*} $$

The proof of Lemma 16 is based on the observation that $(1-\omega )V=V-\omega V$ is the full space counterpart to the renormalized potential $V_N -V_N R V_N$ .

In the following Lemma 14 we want to derive properties of $\mathcal {Q}$ defined in Eq. (8) as

$$ \begin{align*} \mathcal{Q}(\varphi)= \int_{\mathbb{R}^9}\left\{2\big|\mathcal{M}_*\nabla \varphi(x)\big|^2+\mathbb{V}(x) \left|\frac{f(x)}{\mathbb{V}(x)}-\varphi(x)\right|^2 \right\}\mathrm{d}x \end{align*} $$

and its minimizers. For this purpose it will be useful to introduce for a given cut-off parameter $\ell $ and a smooth function $\chi :\mathbb {R}^6\longrightarrow \mathbb {R}$ function with $\chi (x)=1$ for $|x|_\infty \leq \frac {1}{3}$ and $\chi (x)=0$ for $|x|_\infty>\frac {1}{2}$ , the modified function

$$ \begin{align*} f_\ell(x_1,x_2,x_3):=V(x_1,x_2)\chi(\ell^{-1} x_2,\ell^{-1}x_3)\omega(x_2,x_3). \end{align*} $$

Furthermore, we define the corresponding functional, acting on $\dot {H}^1(\mathbb {R}^9)$ , as

$$ \begin{align*} \mathcal{Q}_\ell(\varphi):=\int_{\mathbb{R}^9}\left\{2\big|\mathcal{M}_*\nabla \varphi(x)\big|^2+\mathbb{V}(x) \left|\frac{f_\ell(x)}{\mathbb{V}(x)}-\varphi(x)\right|^2 \right\}\mathrm{d}x, \end{align*} $$

and $\sigma _\ell (V):=\mathcal {Q}_\ell (0)-\inf _{\varphi \in \dot {H}^1(\mathbb {R}^9)}\mathcal {Q}_\ell (\varphi )$ .

Lemma 14. There exists a unique minimizer $\eta $ of the functional $\mathcal {Q}$ in $\dot {H}^1(\mathbb {R}^9)$ , and $\eta $ satisfies the point-wise bounds $0\leq \eta \leq \frac {1}{-2\Delta _{\mathcal {M}_*}}f$ and $\sigma (V)=\int _{\mathbb {R}^9}f(x)\eta (x)\mathrm {d}x$ , as well as

$$ \begin{align*} ( -2\Delta_{\mathcal{M}_*}+\mathbb{V})\eta=f \end{align*} $$

in the sense of distributions. Furthermore, $\mathcal {Q}_\ell $ has a unique minimizer $\eta _\ell $ , and $\eta _\ell $ satisfyies $0\leq \eta _\ell \leq \frac {1}{-2\Delta _{\mathcal {M}_*}}f_\ell $ and $\sigma _\ell (V)=\int _{\mathbb {R}^9}f_\ell (x)\eta _\ell (x)\mathrm {d}x$ , as well as $( -2\Delta _{\mathcal {M}_*}+\mathbb {V})\eta _\ell =f_\ell $ and

$$ \begin{align*} \sigma(V)=\lim_{\ell\rightarrow \infty}\sigma_\ell(V). \end{align*} $$