A second order upper bound for the ground state energy of a hard-sphere gas in the Gross-Pitaevskii regime

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


Introduction and main result
In [24], Lee-Huang-Yang predicted that the ground state energy per particle of a system of N bosons moving in a box with volume N/ρ and interacting through a potential with scattering length a is given, as N → ∞, by e(ρ) = 4πaρ 1 + 128 15 √ π (ρa 3 ) 1/2 + . . . (1.1) up to corrections that are small, in the low density limit ρa 3 ≪ 1 (see [32,28] for the heuristics behind this formula and its relation with the expected occurrence of Bose-Einstein condensation in dilute Bose gases). At leading order, the validity of (1.1) follows from the upper bound obtained in [16] and from the matching lower bound established in [29]. Recently, also the second order term on the r.h.s. of (1.1) has been rigorously justified. The upper bound has been shown in [35] (through a clever modification of a quasi-free trial state proposed in [17]) and (for a larger class of interactions and using a simpler trial state) in [3]. As for the lower bound, it has been first obtained in [20] for integrable potentials and then in [21], for particles interacting through general potentials, including hard-spheres. The upper bound for the case of hard-sphere potential is still an open question. An alternative approach to the study of the ground state energy of the zero temperature Bose gas, still not justified rigorously but possibly valid beyond the dilute regime, has been proposed in [26] and recently revived in [12,13,14]. Trapped Bose gases can be described as systems of N bosons, confined by external fields in a volume of order one and interacting through a radial, repulsive potential V with scattering length of the order N −1 ; this scaling limit is known as the Gross-Pitaevskii regime (see [28,Chapter 6] for an introduction, and [33,34] for reviews of more recent results). Focussing for simplicity on systems trapped in the unit torus Λ, the Hamilton operator takes the form (1.2) and acts on L 2 s (Λ N ), the subspace of L 2 (Λ N ) consisting of functions that are symmetric w.r.t. permutations of the N particles. Note that x i − x j is here the difference between the position vectors of particles i and j on the torus. Equivalently, we can think of x i −x j as the difference in R 3 ; however, in this case, V has to be replaced by its periodisation. As proven in [29,27,30], the ground state energy E N of (1.2) is given, to leading order, by E N = 4πaN + o(N ) (1. 3) in the limit N → ∞. For V ∈ L 3 (R 3 ), more precise information on the low-energy spectrum of (1.2) has been determined in [8]. Here, the ground state energy was proven to satisfy Additionally, the spectrum of H N − E N below a threshold ζ > 0 was shown to consist of eigenvalues having the form A new and simpler proof of (1.4), (1.6) was recently obtained in [22], for V ∈ L 2 (Λ). Moreover, these results have been also extended to the non-homogeneous case of Bose gases trapped by external fields in [31,11].
While the approach of [31] applies to V ∈ L 1 (R 3 ), the validity of (1.4), (1.6) for bosons interacting through non-integrable potentials is still an open question. The goal of this paper is to prove that (1.4) remains valid, as an upper bound, for particles interacting through a hard-sphere potential.
We consider N bosons in Λ = [− 1 2 , 1 2 ] 3 ⊂ R 3 , with periodic boundary conditions. We assume particles to interact through a hard-sphere potential, with radius a/N , for some a > 0. We are interested in the ground state energy of the system, defined by where the infimum is taken over all normalized wave functions Ψ ∈ L 2 s (Λ N ) satisfying the hard-core condition Ψ(x 1 , . . . , x N ) = 0 (1.8) almost everywhere on the set N i<j (x 1 , . . . , x N ) ∈ R 3N : |x i − x j | ≤ a/N .  (1.9) for all N large enough, with e Λ defined as in (1.5). Remarks.
1) Theorem 1.1 establishes an upper bound for the ground state energy (1.7). With minor modifications, it would also be possible to obtain upper bounds for low-energy excited eigenvalues, agreeing with (1.6). To conclude the proof of the estimates (1.4), (1.6) for particles interacting through hard-sphere potentials, we would need to establish matching lower bounds. A possible approach to achieve this goal (at least for the ground state energy) consists in taking the lower bound established in [21], for particles in the thermodynamic limit, and to translate it to the Gross-Pitaevskii regime.
2) We believe that the statement of Theorem 1.1 and its proof can also be extended to bosons in the Gross-Pitaevskii regime interacting through a larger class of potentials, combining a hard-sphere potential at short distances and an integrable potential at larger distances. This would require the extension of Lemma 2.1 to more general interactions. To keep our analysis as simple as possible, we focus here on hard-sphere bosons.
3) Theorem 1.1 and its proof could also be extended to systems of N bosons interacting through a hard-sphere potential with radius of the order N −1+κ for sufficiently small κ > 0 (results for integrable potentials with scattering length of the order N −1+κ have been recently discussed in [1,9,19,2]).
The proof of (1.4), (1.6) obtained in [8] is based on a rigorous version of Bogoliubov theory, developed in [5,6,7]. The starting point of Bogoliubov theory is the observation that, at low energies, the Bose gas exhibits complete condensation; all particles, up to a fraction vanishing in the limit N → ∞, can be described by the same zero-momentum orbital ϕ 0 defined by ϕ 0 (x) = 1, for all x ∈ Λ. This, however, does not mean that the factorized wave function ϕ ⊗N 0 is a good approximation for the ground state of (1.2); in fact, its energy does not even approximate the ground state energy to leading order. To decrease the energy and approach (1.3), correlations are crucial. The strategy developed in [5,6,7,8] is based on the idea that most correlations can be inserted through the action of (generalized) Bogoliubov transformations, having the form where the (modified) creation and annihilation operators b * p , b p act on the Fock space of orthogonal excitations of the Bose-Einstein condensate; the precise definitions are given below, in Section 5 (to be more precise, the action of (1.10) has to be corrected through an additional unitary operator, given by the exponential of a cubic, rather than quadratic, expression in creation and annihilation operators; see [8] for details). An important feature of (generalized) Bogoliubov transformations of the form (1.10), which plays a major role in the derivation of (1.4), (1.6), is the fact that their action on creation and annihilation operators is (almost) explicit. This makes computations relatively easy and it gives the possibility of including correlations also at very large length scales.
Unfortunately, Bogoliubov transformations of the form (1.10) do not seem compatible with the hard-core condition (1.8). As a consequence, they do not seem appropriate to construct trial states approximating the ground state energy of a system of particles interacting through a hard-sphere potential. A different class of trial states, for which (1.8) can be easily verified, consists of products having the form for a function f satisfying f (x) = 0, for all |x| < a/N (as mentioned after (1.2), also here x i − x j is interpreted as difference on the torus). Such an ansatz was first used in the physics literature in [4,15,23]; it is often known as Jastrow factor. In order for (1.11) to provide a good approximation for the ground state energy, f must describe two-particle correlations. Probably the simplest possible choice of f is given by the solution of the zero-energy scattering equation −∆f = 0, with the hard-core requirement f (x) = 0 for |x| < a/N and the boundary condition f (x) → 1, as |x| → ∞. The problem with this choice is the fact that f has long tails; as a consequence, it is extremely difficult to control the product (1.11). To make computations possible, we need to cutoff f at some intermediate length scale a/N ≪ ℓ ≪ 1, requiring that f (x) = 1 for |x| ≥ ℓ (the cutoff can be implemented in different ways; below, we will choose f as the solution of a Neumann problem on the ball |x| ≤ ℓ and we will keep it constant outside the ball). Choosing ℓ small enough (in particular, smaller than the typical distance among particles, which is of the order N −1/3 ), the Jastrow factor becomes more manageable and it is not too difficult to show that its energy matches, to leading order, the ground state energy (1.3). In the thermodynamic limit, this was first verified in [16], using a modification of (1.11), considering only correlations among neighbouring particles.
While Jastrow factors can lead to the correct leading order term in the ground state energy, it seems much more difficult to use (1.11) to obtain an upper bound matching also the second order term on the r.h.s. of (1.9). The point is that the second order corrections are generated by correlations at much larger length scales; to produce the term on the second line of (1.9) we would need to take ℓ of order one, making computations very difficult.
In order to prove Theorem 1.1, we will therefore consider a trial state given by the product of a Jastrow factor (1.11), describing correlations up to a sufficiently small length scale 1/N ≪ ℓ ≪ 1, and of a wave function Φ N , constructed through a Bogoliubov transformation, describing correlations on length scales larger than ℓ. This allows us to combine the nice features of the Jastrow factor (in particular, the fact that it automatically takes care of the hard core condition (1.8)) and of the Bogoliubov transformation (in particular, their (almost) explicit action on creation and annihilation operators, which enables us to insert correlations at large length scales).
The paper is organised as follows. In Section 2, we define our trial state Ψ N as the product of a Jastrow factor and an N -particle wave function Φ N , to be specified later on, and we compute its energy. One of the contributions to the energy of Ψ N is a three-body term; under certain conditions on Φ N (see (3.1)), we show that this term is negligible in Sect. 3. In Sect. 4 we then prove that the remaining contributions to the energy can be reduced (again under suitable assumptions on Φ N ; see (4.4)) to the expectation of an effective Hamiltonian H eff N , defined in (4.3). Sects. 5 and 6 are devoted to the study of H eff N ; the goal is to find Φ N so that the expectation of H eff N produces the energy on the r.h.s of (1.9), up to negligible errors. Here, we use the approach developed in [5,6,7]. In Sect. 7, we show that the chosen wave function Φ N satisfies the bounds that were used in Sects. 3 and 4. Finally, in Sect. 8, we put all ingredients together to conclude the proof of Theorem 1.1. The proof of important properties concerning the solution of the scattering equations is deferred to Appendix A.

The Jastrow factor and its energy
As explained in the introduction, our trial state involves a Jastrow factor, to describe short-distance correlations. To define the Jastrow factor, we choose 1/N ≪ ℓ ≪ 1 and we consider the ground state solution of the Neumann problem on the ball B ℓ = {x ∈ R 3 : |x| ≤ ℓ}, with the hard-core condition f ℓ (x) = 0 for |x| ≤ a/N and the normalization f ℓ (x) = 1 for |x| = ℓ (we denote here by ∂ r the radial derivative). We extend f ℓ to Λ setting f ℓ (x) = 1 for |x| ∈ Λ\B ℓ . We have where χ ℓ denotes the characteristic function of B ℓ . The following lemma establishes properties of λ ℓ , f ℓ , of the difference ω ℓ (x) = 1 − f ℓ (x) and of its Fourier coefficients ω ℓ (p) = e ip·x ω ℓ (x)dx defined for p ∈ Λ * = 2πZ 3 (since ω ℓ has compact support inside [−1/2; 1/2] 3 , we can think of the integral as being over R 3 ).
With the solution f ℓ of the Neumann problem (2.1), we consider trial states of the form for Φ N ∈ L 2 s (Λ N ) to be specified later on. Again, x i − x j should be interpreted as difference on the torus (or f ℓ should be replaced with its periodic extension). Note that a similar trial state has been used in [27]. However, for us the wave function Φ N serves a completely different purpose (in our analysis, Φ N carries correlations on length scales larger than ℓ; in [27], on the other hand, it was a product state, describing the condensate trapped in an external potential).
We compute where the sum in the last term runs over i, j, m ∈ {1, . . . , N } all different. Noticing that the operator on the first line is the Laplacian with respect to the measure defined by (the square of) the Jastrow factor, and using (2.2) in the second line, we conclude that where we introduced the notation x = (x 1 , . . . , x N ) ∈ Λ N .

Estimating the three-body term
In the next proposition, we control the last term on the r.h.s. of (2.12). To this end, we need to assume some regularity on the N -particle wave function Φ N , appearing in (2.11) (we will later make sure that our choice of Φ N satisfies these estimates).
and define Ψ N as in (2.11). Then, for every δ > 0, there exists C > 0 such that To prove this proposition, we will use the following lemma.
Then W can be extended to a periodic function (i.e. a function on the torus Λ) satisfying, on L 2 (Λ) ⊗ L 2 (Λ), the operator inequalities for a constant C > 0, independent on W . Moreover, for every δ ∈ [0, 1/2) there exists C > 0 such that Additionally, for any r > 1, there exists C > 0 such that Proof. The proof is an adaptation to the torus of arguments that are, by now, standard on R 3 . For example, (3.3) follows by writing, in momentum space To show (3.4), we proceed similarly, writing where 1/r + 1/r ′ = 1 and where we used the bound uniformly in p, for any r > 1.
We are now ready to show Proposition 3.1.

Proof. Writing again
for all s ∈ (0; 1). Thus, we obtain the upper bound where kin + E kin + E kin . (4.8) Consider the first error term on the r.h.s. of (4.8). Writing p = |ϕ 0 ϕ 0 | for the orthogonal projection onto the condensate wave function ϕ 0 (x) ≡ 1, p j for p acting on the j-th particle and q j = 1 − p j , we find (4.9) With Lemma 3.2, and observing that, on the range of q, (1 − ∆) ≤ −C∆, we obtain The term E (2) kin can be treated like E (1) kin . Proceeding analogously, we also find, with (3.4), E for any p ≥ 1. From the assumption (4.4), we find Choosing δ > 0 sufficiently small and r > 1 sufficiently close to 1, we conclude that there exist C, ε > 0 such that Let us now consider the potential energy. From (4.1), we can estimate With (4.6) (applied now to the product over 3 ≤ i < j), we obtain pot .
Proceeding similarly to (4.9) (introducing the projections p j , q j ), we can bound . From the assumption (4.4) and from (4.10), we obtain Thus, choosing δ > 0 small enough, we can find C, ε > 0 such that Finally, we consider the denominator on the r.h.s. of (4.2). With the lower bound in (4.6) (and the assumption Φ N 2 = 1), we find Observing that, by (3.3), (4.10) and by the assumption (4.4), we conclude, choosing δ > 0 sufficiently small and recalling that for ε > 0 small enough. Combining the last equation with (4.7), (4.11) we arrive at (recall the assumption Φ N , H eff

Properties of the effective Hamiltonian
Keeping in mind that, by (2.4), λ ℓ ≃ 3a/N ℓ 3 and that 1/N ≪ ℓ ≪ 1, (5.1) looks like the Hamilton operator of a Bose gas in an intermediate scaling regime, interpolating between mean-field and Gross-Pitaevskii limits. The validity of Bogoliubov theory in such regimes has been recently established in [6]. The goal of this section is to apply the strategy of [6] to the Hamilton operator (4.3). This will lead to bounds for the operator H eff N and, eventually, to an ansatz for Φ N . While part of our analysis in this section can be taken over from [6], we need additional work to control the effect of the difference u ℓ = 1 − f 2 ℓ , appearing in the kinetic and the potential energy in the effective Hamiltonian (4.3).
To determine the spectrum of (4.3), it is useful to factor out the condensate and to focus instead on its orthogonal excitations. To this end, following [25], we define a unitary map U N : Here ϕ 0 (x) ≡ 1 for all x ∈ Λ denotes the condensate wave function, and L 2 ⊥ (Λ) is the orthogonal complement of ϕ 0 in L 2 (Λ). The action of the unitary operator U N is determined by the rules where N + denotes the number of particles operator on F ≤N + (it measures therefore the number of excitations of the condensate) and where we introduced modified creation and annihilation operators b * p , b p satisfying the commutation relations 3) in momentum space, using the formalism of second quantization, as Then, we apply (5.3). This will produce a constant term, as well as contributions that are quadratic, cubic and quartic in (modified) creation and annihilation operators. Following Bogoliubov's method, we would like to eliminate cubic and quartic terms. This would reduce L eff N to a quadratic expression, whose spectrum could be computed through diagonalization with a (generalized) Bogoliubov transformation. As explained in [6], though, cubic and quartic terms in L eff N are not negligible (they contribute to the energy to order ℓ −1 ). Before proceeding with the diagonalization, we need to extract relevant contributions to the energy from cubic and quartic terms. As in [6], we do so by conjugating L eff N with a (generalized) Bogoliubov transformation removing short-distance correlations characterising low-energy states. To reach this goal, we fix ℓ 0 ≫ ℓ, small, but of order one, independent of N . Similarly as in (2.1), we define f ℓ 0 to be the ground state solution of the Neumann problem for the hard sphere potential in the ball B ℓ 0 . Extending f ℓ 0 to the box Λ, we find with f ℓ 0 (x) = 0 for |x| = a/N (the eigenvalue λ ℓ 0 is approximately given by (2.4), of course with ℓ replaced by ℓ 0 ). For a/N ≤ |x| ≤ ℓ 0 , we can then define g ℓ 0 (x) = f ℓ 0 (x)/f ℓ (x). We can also extend g ℓ 0 to Λ, setting g ℓ 0 (x) = lim |y|↓a/N g ℓ 0 (y) for all |x| ≤ a/N and g ℓ 0 (x) = 1 for all x ∈ Λ\B ℓ 0 . A simple computation shows that g ℓ 0 solves the equation with the Neumann boundary condition ∂ r g ℓ 0 (x) = 0 for |x| = ℓ 0 (this follows easily from the observation that, for ℓ ≤ |x| ≤ ℓ 0 , g ℓ 0 (x) = f ℓ 0 (x)). Conversely, it is interesting to observe that, integrating (5.7) against g ℓ 0 , we find ). Some properties of g ℓ 0 ,η and of their Fourier coefficients are collected in the next lemma, whose proof is deferred to Appendix A. We introduce here the notation and, analogously, and denote by η p the Fourier coefficients ofη. Then (5.7) takes the form or, equivalently, with the definition (5.10), We have and, for p ∈ Λ * + , In particular, this implies for all 1 < r < 5.
Using the coefficients η p , for p ∈ Λ * + , we define now and we introduce the renormalized excitation Hamiltonian As explained in [6], conjugation with the generalized Bogoliubov transformation e B(η) models correlations up to scales of order one (determined by the radius ℓ 0 of the ball used to define g ℓ 0 ). It extracts important contributions to the energy from terms in L eff N that are quartic in creation and annihilation operators. This will allow us to approximate G eff N by the sum of a constant and of a quadratic expression in creation and annihilation operators, whose ground state energy will be computed by simple diagonalization (through a second Bogoliubov transformation). Unfortunately, conjugation with e B(η) also produces several error terms, which need to be bounded. For 1 < r < 5, we consider the positive operator acting on F ≤N + . The growth of P (r) (and of products of P (r) with moments of the number fo particles operator) under the action of B(η) is controlled by the next lemma.
Lemma 5.2. Let B(η) be defined as in (5.20). Then, for every n ∈ N and r ∈ (1; 5) there is C > 0 such that, for all t ∈ [0; 1], Proof. The proof of the first bound in (5.23) is standard and can be found for example in [10, Lemma 6.1]. As for the second inequality, let us consider the case n = 0. For any ξ ∈ F ≤N By Cauchy-Schwarz's inequality and (5.19) we get Inserting this into (5.24) and using Gronwall's Lemma, we obtain the desired bound. The proof for n ≥ 1 is similar, we omit further details.
With Lemma 5.2 we are ready to establish the form of G eff N,ℓ , up to errors which are negligible on our trial state. We use the notation (recall the definition (5.10) of V ℓ ) .21), with B(η) as in (5.20), with ℓ ≥ N −1+ν for some ν > 0 and ℓ 0 > 0 small enough (but fixed, independent of N ). Let P (r) be defined as in (5.22). Then, for any 0 < κ < ν/2 we have On the other hand, using the notation γ p = cosh(η p ) and σ p = sinh(η p ), let with D p defined in (5.14). Denote also where, and K and V ℓ are defined in (5.25).
Proof. According to (5.6) we can decompose and We can compute G eff N,ℓ with tools developed in [6]. From Propositions 7.4 -7.7 of [6], we obtain, on the one hand, the lower bound and, on the other hand, the approximation Some care is required here when we apply results from [6]. First of all, the interaction potential considered in [6] has the form N 3β W (N β x), for some 0 < β < 1. The potential (5.31) has this form only if we approximate f ℓ ≃ 1 and λ ℓ ≃ 3a/(N ℓ 3 ). A closer inspection to [6] shows, however, that (5.34) does not rely on the precise form of the interaction potential but instead only on the bounds which are the analog of [6, Eq. (7.5) and (7.75)] and follow from V ℓ ∞ ≤ C and V ℓ 2 ≤ Cℓ −3/2 . Moreover, the estimate (5.34) was proven in [6] under the assumption that W = λV , for a sufficiently small λ > 0. This assumption was used in [6] to make sure that the ℓ 2 -norm of η is sufficiently small. As later shown in [8], smallness of η can also be achieved by choosing the parameter ℓ 0 small enough, with no restriction on the size of the interaction potential 1 . Finally, in [6], the choice of η was slightly different from the definition given after (5.7) (the presence of the second term on the r.h.s. of (5.6) affects the choice of η, as we will see shortly). However, the derivation of (5.34) does not depend on the exact form of η, but rather on bounds, proven in Lemma 5.1, that holds for both choices of η. This explains why (5.34) holds true, for sufficiently small values of ℓ 0 .
Thus, for any κ > 0, we arrive at Combining the last estimate with (5.33), we obtain now with the restriction 0 < κ < ν/2 (from ℓ ≥ N −1+ν , it then follows that N ℓ ≥ N ν ≥ N 2κ ; thus, the first term on the r.h.s. of (5.37) can be controlled by the second).

Diagonalization of the effective Hamiltonian
According to Prop. 5.3, we need to find a good ansatz for the ground state of the quadratic Hamiltonian Q N,ℓ , defined in (5.28). To this end, we are going to conjugate G eff N,ℓ with a second generalized Bogoliubov transformation, diagonalizing Q N,ℓ . In order to define the appropriate Bogoliubov transformation, we first need to establish some properties of the coefficients F p , G p , defined in (5.29). Lemma 6.1. Suppose ℓ ≥ N −1+ν , for some ν > 0. Then there exists a constant C > 0 such that Proof. Recall the notations γ p = cosh(η p ) and σ p = sinh(η p ). With (σ 2 p + γ 2 p ) ≤ C (from the boundedness of η p ) and (5.13) in Lemma 5.1, we immediately obtain F p ≤ C(1 + p 2 ). To prove the lower bound for F p , let us first consider With we conclude that Next, we show |G p | ≤ C/p 2 . With the scattering equation (5.16), we obtain Since and using (5.13) we obtain |G p | ≤ C/p 2 , as claimed. It remains to show |G p | ≤ F p . To this end, we write By Lemma 5.1 we have |D p | ≤ C/(N ℓ). Hence, we find, for N large enough, F p − G p ≥ p 2 − C/(N ℓ) ≥ 0 and, similarly as in the proof of F p ≥ p 2 /2 (distinguishing small and large |p|), F p + G p ≥ Cp 2 − C/(N ℓ) > 0. This shows that F p > |G p | and concludes the proof of the lemma.
With Lemma 6.1, using in particular the bound |G p | < F p , we can define, for every p ∈ Λ * + , τ p ∈ R through the identity Equivalently, for all p ∈ Λ * + . With the coefficients τ p , we define the antisymmetric operator and we consider the generalized Bogoliubov transformation e B(τ ) .
The reason why we are interested in the Bogoliubov transformation e B(τ ) is that it diagonalizes the quadratic operator Q N,ℓ defined as in Prop. 5.3. Lemma 6.3. Let Q N,ℓ be defined as in (5.28), and τ p as in (6.2). Then, we have Proof Proposition 6.4. Let M eff N,ℓ be as defined in (6.6), with B(τ ) as in (6.4) and G eff N,ℓ as in (5.21), with ℓ ≥ N −1+ν for some ν > 0 and ℓ 0 > 0 small enough. Then, we have with e Λ defined as in (1.5).
Proof. With (5.30) and Lemma 6.2, we have With Lemma 6.3 and the assumption ℓ ≥ N −1+ν , we obtain with C N,ℓ , F p and G p defined as in (5.27) and (5.29). We rewrite With the scattering equation (5.16) we find Recalling that V ℓ = 2N λ ℓ χ ℓ f 2 ℓ we obtain, switching to position space, With (5.7) and since g ℓ 0 satisfies Neumann boundary conditions, we notice that Thus, using f ℓ 0 = f ℓ g ℓ 0 , we conclude that 3 To bound the terms on the second line of (6.9), we use Lemma 2.1 to show that Similarly, we find . As for the terms on the fourth line, the last contribution can be bounded, using that To handle the other terms on the fourth line of (6.9), we combine them with the first term in the sum on the r.h.s. of (6.7). Recalling (5.29), we find (using again ℓ ≥ N −1+ν ) where we bounded, using |σ 2 p + γ p σ p − η p | ≤ C|η p | 2 ≤ C/|p| 4 (see (6.1)), As for the remaining term on the r.h.s. of (6.7), we can write From (5.18), we have |D p | ≤ C/(N ℓ). Thus, with (γ p +σ p ) 2 ≤ C and |γ p σ p | ≤ C|p| −2 , we obtain |A p | ≤ C/(N ℓ). Using this bound and the observation that |p| 4 + 2p 2 ( V ℓ * g ℓ 0 )(p) and |p| 4 + 2p 2 ( V ℓ * g ℓ 0 )(p) + A p are positive and bounded away from zero we write .

Bounds on the trial state
We introduce some operators to control the regularity of our trial state. First of all, we recall the definition of the operator P (r) , defined in (5.23) for 1 < r < 5. Furthermore, we need some observables acting of several particles. For n ∈ N, we define T n = p 1 ,...,pn∈Λ * + p 2 1 . . . p 2 n a * p 1 . . . a * pn a pn . . . a p 1 .
Since η has limited decay in momentum space (see (5.18)), we will only be able to control the expectation of T n for n = 2, 3, 4. To control some error terms, it is also important to use less derivatives on each particle. We define, for δ > 0 small enough (we will later impose the condition δ ∈ (0; 1/6)), We will be able to control the expectation of A n , for all n ∈ N. Additionally, we will also need the observable All these operators act on the excitation Fock space F ≤N + . In order to bound their expectation on our trial state, we need to control their growth under the action of B(η), similarly as we did in Lemma 5.2 for P (r) . Lemma 7.1. For n ∈ N\{0} and 0 < δ < 1/6, we consider A (δ) n as in (7.2). We define recursively the sequence α n (depending on the parameter δ) by setting α 1 = 1/2 + δ, α 2 = 2 + 2δ and α n = α n−1 + α n−2 /2 + 7/4 + 3δ/2 .
Remark. The sequence α n defined in (7.4) is given explicitly by Proof. We begin with (7.5). We consider k = 0; the case k > 0 can be handled similarly. For n ≥ 1 and 0 < δ < 1/6, we set For n ≥ 2, we compute With the identity Thus . . a p 1 .

Proof of Theorem 1.1
With the unitary operator U N as in (5.2), with η as introduced after (5.7) and τ as in We recall that we assumed N −1+ν ≤ ℓ ≤ N −3/4−ν (see Prop. 4.1) and ℓ 0 > 0 small enough (independent of N ). From Prop. 6.4, we find that for a sufficiently small ε > 0. Additionally, with Lemma 7.1 we obtain important regularity estimates for Φ N . From (7.7) (and from (5.23) in Lemma 5.2), we find C > 0 such that From (7.5) we find, for n ∈ N and 0 < δ < 1/6, a constant C > 0 such that (8.4) From (7.6) in Lemma 7.1, we find, for n ∈ N and for every ε ∈ (−1; 3), δ ∈ (0; 1/6) such that ε + 2δ < 3/2 n−1 , a constant C > 0 such that (8.5) Let us prove (8.5), the other bounds can be shown similarly. First of all, we symmetrize the expectation on the l.h.s. of (8.5), writing Next, we express the observable in second quantized form and we apply the rules (5.3). We find With (7.6), we conclude that To control the growth of S (ε,δ) j , we can proceed exactly as in the proof of Lemma 7.1; the difference is that, by (6.3), |τ p | ≤ C/|p| 4 , uniformly in N, ℓ (this should be compared with the bound (5.18), for the coefficients η p ). As a consequence, for 0 < r < 5, we find p∈Λ * + |p| r |τ p | 2 ≤ C and thus the analog of the bounds in Lemma 7.1, with B(η) replaced by B(τ ), holds uniformly in ℓ. This observation leads to (8.5).
With Φ N as in (8.1), we define the trial function Ψ N ∈ L 2 s (Λ N ) by The presence of the Jastrow factor guarantees that Ψ N satisfies the hard-sphere condition (1.8 To conclude the proof of Theorem 1.1, we still have to show that the contribution on the last line is negligible, in the limit N → ∞. From (5.26) in Prop. 5.3, we find for 0 < κ < ν/2. Notice here that both sides of the equation are operators on the Hilbert space L 2 s (Λ N −2 ) describing states with (N − 2) particles. For µ > 0 to be chosen small enough, we can estimate for any m ∈ N. Thus, the contribution arising from the first term in the parenthesis on the r.h.s. of (8.8) can be bounded by Using u ℓ 1 ≤ Cℓ 2 /N and (3.3) in the first and u ℓ ∞ ≤ C in the second term (by Lemma 2.1), we obtain Here we used Lemma 5.2 to control the growth of (N + + 1) m+1 under the action of B(η). Moreover, with q = 1−|ϕ 0 ϕ 0 | denoting the projection onto the orthogonal complement to the condensate wave function ϕ 0 in L 2 (Λ) and with q j = 1 ⊗ · · · ⊗ q ⊗ · · · ⊗ 1 acting as q on the j-th particle, we estimated, on the N -particle space L 2 s (Λ N ), (with a slight abuse of notation, N + denotes the number of particles operators on F ≤(N −2) + on the l.h.s. and the number of particles operator on F ≤N + on the r.h.s.). Using again Lemma 5.2 (and Lemma 6.2, for the action of B(τ )), together with the bounds in (8.3), we conclude that choosing first µ > 0 small enough and then m ∈ N large enough.
Applying again Lemma 5.2 (and then Lemma 6.2 for the action of B(τ )), we conclude that As for the term R 1 , we first use (3.3) in Lemma 3.2 to estimate, for δ > 0 small enough, To control R 12 , we apply Lemma 5.2 to bound 2 e B(η) e B(τ ) Ω . With (7.5) and with (7.9) from Lemma 7.1, we conclude that for some ε > 0, if δ is chosen small enough, and 0 < κ < ν/2. It turns out that the term R 11 is more subtle; here we cannot afford the error arising from conjugation of P (2+κ) with e −B(η) . Instead, we have to use the fact that we conjugate back with e B(η) when we take expectation in the state Φ N = e B(η) e B(τ ) Ω. The two generalized Bogoliubov transformations do not cancel identically (because one acts on (N − 2) particles, the other on N ), but of course their combined action produce a much smaller error. We will make use of the following lemma. Moreover, ±X 2 ≤ C(N + + 1) + CN −1 P (r) + ℓ 1−r (N + + 1).
We defer the proof of Lemma 8.1 to the end of the section, showing first how it can be used to estimate the error R 11 and to conclude the proof of Theorem 1.1. Notice first that 2 + κ ≤ 4 since κ < ν/2 and ν is small enough. We can therefore apply Lemma 8.1 to find and that, similarly, Moreover, we find with θ defined by the Fourier coefficientsθ p = |p| 2+κ η p , and with p j denoting the orthogonal projection p = |ϕ 0 ϕ 0 | on the condensate wave function acting on the j-particle. Rewriting the first term in second quantized form (but now, on the N -particle space), we find Therefore, we find Finally, we can estimate the term on the fourth line in (8.18) by Since p 1 θ(x 1 − x 2 )p 1 = p 1θ0 = 0 and, similarly, p 2 θ(x 1 − x 2 )p 2 = 0, we have With θ 2 = θ 2 ≤ Cℓ −3/2−κ , for 0 < κ < 1/2, and with for ℓ ≤ N −2/3 . Since u ℓ 1 ≤ Cℓ 2 /N by Lemma 2.1, the error term R 11 introduced in (8.13) is bounded by R 11 ≤ CN −κ+µ ℓ 1/2−κ ≤ CN µ ℓ 1/2 .
With (8.14), we find for ε > 0 small enough. Combining this bound with (8.12) we conclude, choosing first µ > 0 small enough and then m ∈ N sufficiently large, that for a sufficiently small ε > 0. Together with (8.8) and (8.11), this estimate implies that From (8.7), we obtain We conclude the proof of Theorem 1.1 by giving the proof of Lemma 8.1.
To control the last term, we write a * q a * −q a p a −p = a * q a p a * −q a −p −δ −q,p a * q a −p and we bound, for an arbitrary ξ ∈ F ≤N + , 1 N p,q∈Λ * + |p| r η p η q ξ, a * q a p a * −q a −p ξ ≤ 1 N p,q∈Λ * + |p| r |η p ||η q | a * p a q ξ a * −q a −p ξ ≤ 1 N p,q∈Λ * + |p| r |η p ||η q | a p a q ξ + a q ξ a −q a −p ξ + a −p ξ .
A Properties of one-particle scattering equations In this section we provide the proof of Lemma 2.1 and Lemma 5.1. We start with Lemma 2.1, where we describe properties of the solution of the eigenvalue equation (2.2).
Next, we show Lemma 5.1, devoted to the properties of the solution of (5.7).
The second bound in (5.13) can be proven analogously (on the r.h.s. ℓ is then replaced by ℓ 0 , which is chosen of order one).