A new second order upper bound for the ground state energy of dilute Bose gases
aa r X i v : . [ m a t h - ph ] M a r A new second order upper bound for theground state energy of dilute Bose gases
Giulia Basti ∗ , Serena Cenatiempo ∗ , Benjamin Schlein † March 4, 2021
Abstract
We establish an upper bound for the ground state energy per unit volume ofa dilute Bose gas in the thermodynamic limit, capturing the correct second orderterm, as predicted by the Lee-Huang-Yang formula. This result has been first es-tablished in [17] by H.-T. Yau and J. Yin. Our proof, which applies to repulsiveand compactly supported V ∈ L ( R ), gives better rates and, in our opinion, issubstantially simpler. We consider N bosons in a finite box Λ L = [ − L , L ] ⊂ R , interacting via a two-bodynon negative, radial, compactly supported potential V with scattering length a . TheHamilton operator has the form H L = − N X i =1 ∆ i + X ≤ i 1, the specific ground state energy (1.2) is so that e ( ρ ) = 4 π a ρ (cid:20) √ π ( ρ a ) / + o (( ρ a ) / ) (cid:21) (1.3) ∗ Gran Sasso Science Institute, Viale Francesco Crispi 7, 67100 L’Aquila, Italy † Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich. 1n particular, up to lower order corrections, it only depends on the interaction potentialthrough the scattering length a .The validity of the leading term on the r.h.s. of (1.3) was established by Dyson, whoobtained an upper bound in [7], and by Lieb-Yngvason, who proved the matching lowerbound in [13]. A rigorous upper bound with the correct second order contribution wasfirst derived in [17] by Yau-Yin for regular potentials, improving a previous estimatefrom [8], which only recovered the correct formula (as an upper bound) in the limit ofweak coupling. The approach of [17] has been reviewed and adapted to a grand canonicalsetting in [1]. As for the lower bound, preliminary results have been obtained in [11]and [5], where (1.3) was shown in particular regimes, where the potential scales with thedensity ρ . Finally, a rigorous lower bound matching (1.3) has been recently obtained,for L potentials, by Fournais-Solovej in [9] (for hard core potentials, the best availablelower bound is given in [6] and it matches (1.3), up to corrections of the same size asthe second order term).Our goal, in this paper, is to show a new upper bound for (1.3). With respect to theupper bound established in [17], our result holds for a larger class of potentials (in [17],the upper bound is proven for smooth potentials), it gives a better rate (although stillfar from optimal) and, most importantly in our opinion, it relies on a simpler proof. Theorem 1.1. Let V ∈ L ( R ) be non-negative, radially symmetric, with supp( V ) ⊂ B R (0) and scattering length a ≤ R . Then, the specific ground state energy e ( ρ ) of theHamilton operator H L defined in (1.1) satisfies e ( ρ ) ≤ πρ a h √ π ( ρ a ) / i + Cρ / / (1.4) for some C > and for ρ small enough.Remark: since Dirichlet boundary conditions lead to the largest energy, the upperbound (1.4) holds in fact for arbitrary boundary conditions. Remark: at the cost of a longer proof, we could improve the bound on the error, upto the order ρ / / (this is the rate determined by Lemma 5.1).The proof of 1.4 is based on the construction of an appropriate trial state. However,we do not construct directly a trial state in L s (Λ NL ) for the Hamiltonian (1.1). Instead,to simplify the analysis, it is very convenient to 1) consider smaller boxes (rather thanletting N, L → ∞ first and considering small ρ at the end, we will consider a diagonallimit, with L = ρ − γ , for some γ > F (Λ L ) = M n ≥ L s (Λ nL ) = M n ≥ L (Λ L ) ⊗ s n L s (Λ nL ) is the subspace of L (Λ nL ) consisting of wave functions that are symmetricw.r.t. permutations. On F (Λ L ), we consider the number of particles operator N definedthrough ( N ψ ) ( n ) = nψ ( n ) . Moreover, we introduce the Hamiltonian operator H , setting( H ψ ) ( n ) = H ( n ) ψ ( n ) (1.5)with H ( n ) = n X j =1 − ∆ x j + X ≤ i Let e ( ρ ) be defined as in (1.2) , with Dirichlet boundary conditions.Let R < ℓ < L , with R the radius of the support of the potential V , as defined in Theorem1.1. Then, for any normalized Ψ L ∈ F (Λ L ) satisfying periodic boundary conditions andsuch that h Ψ L , N Ψ L i ≥ ρ (1 + c ′ ρ )( L + 2 ℓ + R ) , h Ψ L , N Ψ L i ≤ C ′ ρ ( L + 2 ℓ + R ) (1.6) for some c ′ , C ′ > . Then we have e ( ρ ) ≤ h Ψ L , H Ψ L i L + CL ℓ h Ψ L , N Ψ L i (1.7) for a universal constant C > . The bulk of the paper contains the proof of the following proposition, establishingthe existence of a trial state with the correct energy per unit volume and the correctexpected number of particles, on boxes of size L = ˜ ρ − γ . We use here the notation ˜ ρ forthe density to stress the fact the upper bound (1.9) will be inserted in (1.7) to provean upper bound for the specific ground state energy e ( ρ ), for a slightly different density ρ < ˜ ρ (to make up for the corrections on the r.h.s. of (1.6)). Proposition 1.3. As in Theorem 1.1 assume that V ∈ L ( R ) is non-negative, radiallysymmetric with supp V ⊂ B R (0) and scattering length a ≤ R . For γ > and ˜ ρ > let L = ˜ ρ − γ . Then, for every < ε < / , there exists Ψ ˜ ρ ∈ F (Λ L ) satisfying periodicboundary conditions such that h Ψ ˜ ρ , N Ψ ˜ ρ i ≥ ˜ ρL , h Ψ ˜ ρ , N Ψ ˜ ρ i ≤ C ˜ ρ L (1.8) and (cid:10) Ψ ˜ ρ , H Ψ ˜ ρ (cid:11) L ≤ π a ˜ ρ (cid:18) √ π ( a ˜ ρ ) / (cid:19) + E , (1.9) with E ≤ C ˜ ρ / · max { ˜ ρ ε , ˜ ρ − γ − ε , ˜ ρ / − γ/ − ε } . emark: the condition γ > γ = 11 / 10 to optimize the rate, our analysis allowsus to take any 1 < γ < / 3. With a longer proof, our techniques could be extended toall 1 < γ < / 3. This suggests that our trial state captures the correct correlations ofthe ground state, up to length scales of the order ρ − / .With Prop. 1.2 and Prop. 1.3 we can prove Theorem 1.1. Proof of Theorem 1.1. For given ρ > 0, we would like to choose ˜ ρ , or equivalently L =˜ ρ − γ , so that (1.8) implies (1.6). Fixing c ′ > ℓ = L α , for some α ∈ (0; 1), this leadsto the implicit equation L = ˜ ρ − γ = (cid:2) ρ (1 + c ′ ρ )(1 + 2 L α − + RL − ) (cid:3) − γ . (1.10)Setting L = (cid:0) ρ (1 + c ′ ρ ) (cid:1) − γ x , we rewrite (1.10) as x = (cid:0) (cid:0) ρ (1 + c ′ ρ ) (cid:1) γ (1 − α ) /x − α + R (cid:0) ρ (1 + c ′ ρ ) (cid:1) γ /x (cid:1) − γ and we conclude that the existence of a solution L = L ( ρ ) of (1.10) follows from theimplicit function theorem, if ρ > x = 1 for ρ = 0). By construction L = ˜ ρ − γ , with˜ ρ = ρ (1 + c ′ ρ )(1 + 2˜ ρ γ (1 − α ) + R ˜ ρ γ ) and thus ρ ≤ ˜ ρ ≤ ρ (1 + Cρ + Cρ γ (1 − α ) ) . (1.11)From Prop. 1.3, we find Ψ ˜ ρ ∈ F (Λ L ) such that (1.8) and (1.9) hold true. In particular,(1.8) implies (1.6) (with ℓ = L α , C ′ = C ). Thus, from Prop. 1.2 we conclude e ( ρ ) ≤ h Ψ ˜ ρ , H Ψ ˜ ρ i L + CL ℓ h Ψ ˜ ρ , N Ψ ˜ ρ i Inserting (1.9) and (1.8), we obtain (since (1.8) also implies that h Ψ ˜ ρ , N Ψ ˜ ρ i ≤ C e ρL ) e ( ρ ) ≤ π a ˜ ρ (cid:20) √ π ( a ˜ ρ ) / (cid:21) + C ˜ ρ γ (1+ α ) + C ˜ ρ / · max { ˜ ρ ε , ˜ ρ − γ − ε , ˜ ρ / − γ/ − ε } . With (1.11), we conclude that e ( ρ ) ≤ π a ρ (cid:20) √ π ( a ρ ) / (cid:21) + Cρ / · max { ρ γ (1 − α ) − / , ρ γ (1+ α ) − / , ρ ε , ρ − γ − ε , ρ / − γ/ − ε } where we neglected errors of order Cρ , which are subleading being ε ∈ (0; 1 / α = 1 / (2 γ ) . Comparing instead thirdand fourth errors, we set ε = (4 − γ ) / ∈ (0; 1) and ε ∈ (0; 1 / γ > e ( ρ ) ≤ π a ρ (cid:20) √ π ( a ρ ) / (cid:21) + Cρ / · max { ρ γ − , ρ (4 − γ ) / } . Choosing γ = 11 / 10, we find (1.4).The proof of Prop. 1.3 occupies the rest of the paper (excluding Appendix A, wherewe show Prop. 1.2). In Section 2 we define our trial state. To this end, we will startwith a coherent state describing the Bose-Einstein condensate. Similarly as in [10, 8],we will then act on the coherent state with a Bogoliubov transformation to add the ex-pected correlation structure. Finally, we will apply the exponential of a cubic expressionin creation operators. While the Bogoliubov transformation creates pairs of excitationswith opposite momenta p, − p , the cubic operator creates three excitations at a time,two with large momenta r + v, − r and one with low momentum v . This last step isessential, since, as follows from [8, 14], quasi-free states cannot approximate the groundstate energy to the precision of (1.3). We remark that the idea of creating triples ofexcitations originally appeared in the work of Yau-Yin [17] (a brief comparison with thetrial state of [17] can be found after the precise definition of our trial state in (2.29)).Recently, it has been also applied to establish the validity of Bogoliubov theory in theGross-Pitaevskii regime in [2, 3]; while our approach is inspired by these papers, we needhere new tools to deal with the large boxes considered in Prop.1.3 (a simple computa-tion shows that the Gross-Pitaevskii regime corresponds to the exponent γ = 1 / 2; tocontrol localization errors, we need instead to choose γ > Acknowledgment. We are grateful to C. Boccato and S. Fournais for valuable dis-cussions. G.B. and S.C. gratefully acknowledge the support from the GNFM GruppoNazionale per la Fisica Matematica. B. S. gratefully acknowledges partial support fromthe NCCR SwissMAP, from the Swiss National Science Foundation through the Grant“Dynamical and energetic properties of Bose-Einstein condensates” and from the Euro-pean Research Council through the ERC-AdG CLaQS. To show Prop. 1.3, we find it convenient to work with rescaled variables. We considerthe transformation x j → x j /L , and, motivated by the choice L = ˜ ρ − γ in Prop. 1.3, weset N = ˜ ρ − γ (we will look for trial states with expected number of particles close to N to make sure that (1.8) holds true). It follows that the Hamiltonian (1.5) is unitarilyequivalent to the operator L − H N = ˜ ρ γ H N , with H N acting on the Fock space F (Λ)5efined over the unit box Λ = Λ = [ − / 2; 1 / (with periodic boundary conditions) sothat ( H N Ψ) ( n ) = H ( n ) N Ψ ( n ) , with H ( n ) N = n X j =1 − ∆ x j + X ≤ i,j ≤ n N − κ V ( N − κ ( x i − x j ))and κ = (2 γ − / (3 γ − γ > κ ∈ (1 / 2; 2 / p ∈ Λ ∗ = 2 π Z , we introduce on the Fock space F (Λ) = L n ≥ L s (Λ n ), the operators a ∗ p , a p , creating and, respectively, annihilating a particlewith momentum p . Creation and annihilation operators satisfy the canonical commuta-tion relations (cid:2) a p , a ∗ q (cid:3) = δ pq , [ a p , a q ] = (cid:2) a ∗ p , a ∗ q (cid:3) = 0 . (2.1)On F (Λ), we define the number of particles operator N = P p ∈ Λ ∗ a ∗ p a p . Expressed interms of creation and annihilation operators, the Hamiltonian H N takes the form H N = X p ∈ Λ ∗ p a ∗ p a p + 12 N − κ X p,q,r ∈ Λ ∗ b V ( r/N − κ ) a ∗ p + r a ∗ q a q + r a p . (2.2)We construct now our trial state. To generate a condensate, we use a Weyl operator W N = exp (cid:2)p N a ∗ − p N a (cid:3) (2.3)with a parameter N to be specified later on. While W N leaves a p , a ∗ p invariant, for all p ∈ Λ ∗ \{ } , it produces shifts of a , a ∗ ; in other words W ∗ N a W N = a + p N , W ∗ N a ∗ W N = a ∗ + p N . (2.4)When acting on the vacuum vector Ω = { , , . . . } , (2.3) generates a coherent state inthe zero-momentum mode ϕ ( x ) ≡ 1, with expected number of particles N .It turns out, however, that the coherent state does not approximate the ground stateenergy, not even to leading order. To get closer to the ground state energy, it is crucialto add correlations among particles. To this end, we fix 0 < ℓ < / f ℓ of the Neumann problem (cid:20) − ∆ + 12 V (cid:21) f ℓ = λ ℓ f ℓ (2.5)on the ball | x | ≤ N − κ ℓ , with the normalization f ℓ ( x ) = 1 if | x | = N − κ ℓ . Furthermore,by rescaling, we define f N ( x ) := f ℓ (cid:0) N − κ x (cid:1) for | x | ≤ ℓ . We extend f N to a function onΛ, by fixing f N ( x ) = 1, for all x ∈ Λ, with | x | > ℓ . Then (cid:20) − ∆ + 12 N − κ V ( N − κ x ) (cid:21) f N ( x ) = N − κ λ ℓ f N ( x ) χ ℓ ( x ) (2.6)6or all x ∈ Λ, where χ ℓ denotes the characteristic function of the ball of radius ℓ . Wedenote by b f N ( p ) the Fourier coefficients of the function f N , for p ∈ Λ ∗ . We also define w ℓ ( x ) = 1 − f ℓ ( x ) (with w ℓ ( x ) = 0 for | x | > N − κ ℓ ) and its rescaled version w N : Λ → R through w N ( x ) = w ℓ ( N − κ x ) = 1 − f N ( x ). The Fourier coefficients of w N are given by b w N ( p ) = Z Λ w ℓ ( N − κ x ) e − ip · x dx = 1 N − κ b w ℓ (cid:0) p/N − κ (cid:1) where b w ℓ ( k ) denotes the Fourier transform of the (compactly supported) function w ℓ .Some important properties of the solution of the eigenvalue problem (2.5) are summa-rized in the following lemma, whose proof can be found in [2, Appendix A] (replacing N ∈ N by N − κ ). Lemma 2.1. Let V ∈ L ( R ) be non-negative, compactly supported and sphericallysymmetric. Fix ℓ > and let f ℓ denote the solution of (2.5) . For N ∈ N large enoughthe following properties hold true.i) We have (cid:12)(cid:12)(cid:12)(cid:12) λ ℓ − a N − κ ℓ (cid:12)(cid:12)(cid:12)(cid:12) ≤ N − κ ℓ C a ℓN − κ . ii) We have ≤ f ℓ , w ℓ ≤ . Moreover there exists a constant C > such that (cid:12)(cid:12)(cid:12)(cid:12)Z V ( x ) f ℓ ( x ) dx − π a (cid:12)(cid:12)(cid:12)(cid:12) ≤ C a ℓN − κ . iii) There exists a constant C > such that, for all x ∈ R , w ℓ ( x ) ≤ C | x | + 1 and |∇ w ℓ ( x ) | ≤ Cx + 1 . iv) There exists a constant C > such that, for all p ∈ R , | b w N ( p ) | ≤ CN − κ p . We consider the coefficients η : Λ ∗ → R defined through η p = − N b w N ( p ) = − N κ N − κ b ω ℓ ( p/N − κ ) . (2.7)Lemma 2.1 implies that | η p | ≤ CN κ p (2.8)for all p ∈ Λ ∗ + = 2 π Z \{ } , and for some constant C > N ∈ N (for N ∈ N large enough). From (2.6), we find the relation p η p + N κ b V ( p/N − κ ) + 12 N X q ∈ Λ ∗ N κ b V (( p − q ) /N − κ ) η q = N − κ λ ℓ ( b χ ℓ ∗ b f N )( p ) . (2.9)7rom Lemma 2.1, part iii), we also obtain | η | ≤ N − κ Z R w ℓ ( x ) dx ≤ CN κ . (2.10)The coefficients η p will be used to model, through a Bogoliubov transformation,short-distance correlations among particles. To reach this goal, it is enough to act onmomenta | p | ≫ N κ/ . On low momenta, the Bogoliubov transformation is needed todiagonalize the (renormalized) quadratic part of the Hamiltonian. For ε > P L = n p ∈ Λ ∗ + : | p | ≤ N κ/ ε o , (2.11)of low momenta (the condition on ε makes sure that the two sets are disjoint). We willdenote its complement by P cL = Λ ∗ + \ P L . For p ∈ Λ ∗ + we set ν p = τ p χ ( p ∈ P L ) + η p χ ( p ∈ P cL )with η p defined in (2.7) and τ p ∈ R defined bytanh(2 τ p ) = − π a N κ p + 8 π a N κ . (2.12)With these coefficients, we define the Bogoliubov transformation T ν = exp (cid:18) X p ∈ Λ ∗ + ν p (cid:0) a ∗ p a ∗− p − h.c. (cid:1) (cid:19) . (2.13)For any p = 0 we have T ∗ ν a p T ν = γ p a p + σ p a ∗− p (2.14)with the notation γ p = cosh( ν p ) and σ p = sinh( ν p ).With the Weyl operator (2.3) and the Bogoliubov transformation (2.13), we obtainthe “squeezed” coherent state e Ψ N = W N T ν Ω. Choosing N so that N = N + k σ L k ,one can show that this trial state has approximately N particles and, to leading order,the correct ground state energy. However, as observed in [8] (for a similar trial state)and later in [14], the energy of the quasi-free state e Ψ N does not match the second ordercorrection in (1.3). To prove Prop. 1.3, we need therefore to modify the trial state. Wedo so by replacing the vacuum Ω in the definition of e Ψ N by the normalized Fock spacevector ξ ν / k ξ ν k , with ξ ν = e A ν Ω and the cubic phase A ν = 1 √ N X r ∈ P H ,v ∈ P S : r + v ∈ P H η r σ v a ∗ r + v a ∗− r a ∗− v Θ r,v . (2.15)Here, we introduced the momentum sets P H = { p ∈ Λ ∗ + : | p | > N − κ − ε } ,P S = n p ∈ Λ ∗ + : N κ/ − ε ≤ | p | ≤ N κ/ ε o . (2.16)8otice that P S ⊂ P L . On the other hand, to make sure that P H ∩ P L = ∅ , we willrequire, from now on, that ε > κ − ε < η L , η L c , η S , η H the restriction of η : Λ ∗ → R to the set P L , P cL , P S and,respectively, P H . Similarly, we define γ L , γ L c , γ H , γ S and σ L , σ L c , σ H , σ S . The followinglemma collects important bounds for these functions. Lemma 2.2. We have k η L c k ≤ CN κ/ − ε , k η L c k H ≤ CN κ , k η L c k ∞ ≤ CN − ε k η H k ≤ CN κ − ε , k η H k H ≤ CN κ , k η H k ∞ ≤ CN κ − ε . In particular, this implies that k γ H k ∞ , k σ H k ∞ ≤ C . Moreover we have k γ L k ∞ , k σ L k ∞ ≤ CN κ/ , k γ L σ L k ≤ CN κ/ ε and k γ L k ≤ CN κ/ ε , k σ L k ≤ CN κ/ , k σ L k H ≤ CN κ/ ε . Finally, we observe that k σ S k ≤ CN κ/ , k σ S k H ≤ CN κ/ ε , k γ S k ∞ , k σ S k ∞ ≤ CN ε . Proof. The bounds for k η L c k , k η H k , k η L c k ∞ and k η H k ∞ follow from (2.8). On theother hand, with the notation ˇ η ( x ) = − N w ℓ ( N − κ x ) for the function on Λ with Fouriercoefficients η p , we find from Lemma 2.1, part iii), k η L c k H ≤ C X p ∈ Λ ∗ p | η p | ≤ C Z |∇ ˇ η ( x ) | dx ≤ CN κ Z R | x | + 1) dx ≤ CN κ . To show bounds for σ L , γ L we observe that, from (2.12), σ p = p + 8 π a N κ − p | p | + 16 π a N κ p p | p | + 16 π a N κ p , σ p γ p = − π a N κ p | p | + 16 π a N κ p . (2.17)Recalling that P L = { p ∈ Λ ∗ + : | p | ≤ N κ/ ε } , we find k σ L k ∞ ≤ sup p ∈ P L : | p |≤ N κ/ C N κ/ | p | + sup p ∈ P L : | p |≥ N κ/ C N κ | p | ≤ CN κ/ , (2.18)Moreover by definition of P S we also get k σ S k ∞ ≤ sup N κ/ − ε ≤| p |≤ N κ/ C N κ/ | p | + sup p ∈ P L : | p |≥ N κ/ C N κ | p | ≤ CN ε . γ p = 1 + σ p , we find k γ L k ∞ ≤ CN κ/ and k γ S k ∞ ≤ CN ε . Similarly, we obtain k σ L k ≤ C X p ∈ P L : | p |≤ N κ/ N κ/ | p | + C X p ∈ P L : | p | >N κ/ N κ | p | ≤ CN κ/ (2.19)and thus, using again γ p = 1 + σ p , also k γ L k ≤ CN κ/ ε . Moreover, we have k σ L k H ≤ C X p ∈ P L : | p |≤ N κ/ | p | N κ/ + C X p ∈ P L : | p |≥ N κ/ N κ p ≤ CN κ/ ε . (2.20)The bound for k σ L γ L k is proved similarly, using the expression for γ p σ p in (2.17).Finally, we note that the estimates (2.19) and (2.20) do not improve when we considerthe restriction of σ p to P S ⊂ P L , hence k σ S k ≤ CN κ/ and k σ S k H ≤ CN κ/ ε .In (2.15), we included, for every r ∈ P H and every v ∈ P S , the cutoffΘ r,v = Y s ∈ P H h − χ ( N s > χ ( N − s + v > i × Y w ∈ P S h − χ ( N w > χ ( N r − w + N − r − v − w > i where N p = a ∗ p a p and χ ( t > 0) is the characteristic function of the set t > Remark: It is easy to check that the computation of the energy and the numberof particles of the trial state we are constructing would not change substantially (andwould still lead to a proof of Prop. 1.3), if in the definition (2.15) of A ν we restricted thesum over r to the finite set P H ∩ { p ∈ Λ ∗ + : | p | < N − κ + ε } = { p ∈ Λ ∗ + : N − κ − ε < | p | 3, the cutoff Θ r m ,v m implements additional restrictions10nvolving three indices of the form − p i + v j = p k with p ℓ ∈ {− r ℓ , r ℓ + v ℓ } , ℓ = i, j, k where i, j, k = 1 , . . . , m, i = j = k , so that exactly one of the three indices is m . Weconclude that, for any m ≥ A mν Ω = 1 N m/ X r ∈ P H ,v ∈ P S : r + v ∈ P H · · · X r m ∈ P H ,v m ∈ P S : r m + v m ∈ P H m Y i =1 η r i σ v i × θ ( { r j , v j } mj =1 ) a ∗ r m + v m a ∗− r m a ∗− v m . . . a ∗ r + v a ∗− r a ∗ v Ω (2.21)where θ (cid:0) { r j , v j } mj =1 (cid:1) = m Y i,j,k =1 j = k Y p i ∈{− r i ,r i + v i } p k ∈{− r k ,r k + v k } δ − p i + v j = p k . (2.22)To understand the reason for the introduction of the cutoff, let us compute the norm k ξ ν k of the vector ξ ν = e A ν Ω. With (2.21), we find k ξ ν k = X m ≥ m !) k ( A ν ) m Ω k = X m ≥ m !) N m X v , ˜ v ∈ P S r , ˜ r ∈ P H : r + v , ˜ r +˜ v ∈ P H · · · X v m , ˜ v m ∈ P S r m , ˜ r m ∈ P H : r m + v m , ˜ r m +˜ v m ∈ P H θ (cid:0) { ˜ r j , ˜ v j } mj =1 (cid:1) θ (cid:0) { r j , v j } mj =1 (cid:1) × m Y i =1 η r i η ˜ r i σ v i σ ˜ v i (cid:10) a ∗ r m + v m a ∗− r m a ∗− v m . . . a ∗− v Ω , a ∗ ˜ r m +˜ v m a ∗− ˜ r m a ∗− ˜ v m . . . a ∗− ˜ v Ω (cid:11) . (2.23)Clearly, for the expectation on the last line not to vanish, all creation and annihilationoperators with momenta in P S must be contracted among themselves. Since, on thesupport of θ ( { r j , v j } mj =1 ), v i = v j for all i = j (and similarly ˜ v i = ˜ v j for all i = j on thesupport of θ ( { e r j , e v j } mj =1 )) we have ( m !) identical contributions arising from this pairing.We end up with k ξ ν k = X m ≥ m ! 1 N m X v ∈ P S ,r , ˜ r ∈ P H : r + v , ˜ r + v ∈ P H · · · X v m ∈ P S ,r m , ˜ r m ∈ P H : r m + v m , ˜ r m + v m ∈ P H θ (cid:0) { r j , v j } mj =1 (cid:1) θ (cid:0) { ˜ r j , v j } mj =1 (cid:1) × m Y i =1 η r i η ˜ r i σ v i (cid:10) Ω , A r ,v . . . A r m ,v m A ∗ ˜ r m ,v m . . . A ∗ ˜ r ,v Ω (cid:11) (2.24)where we have introduced the notation A r i ,v i = a r i + v i a − r i .It is now important to observe that, because of the presence of the cutoffs, theannihilation operators in A r j ,v j must be contracted with the creation operators in A e r j ,v j .In fact, if this was not the case, we would have − r j = − e r ℓ or − r j = e r ℓ + v ℓ and11 j + v j = − e r k or r j + v j = e r k + v k , with at least one of the two indices ℓ, k differentfrom j . This would imply one of the four relations e r ℓ + v j = − e r k , e r ℓ + v j = e r k + v k , e r ℓ + v ℓ = e r k + v j , − e r ℓ − v ℓ + v j = e r k + v k , all of which are forbidden by the cutoff θ (cid:0) { e r j , e v j } mj =1 (cid:1) . We conclude that (cid:10) Ω , A r ,v . . . A r m ,v m A ∗ ˜ r m ,v m . . . A ∗ ˜ r ,v Ω (cid:11) θ (cid:0) { r j , v j } mj =1 (cid:1) θ (cid:0) { ˜ r j , ˜ v j } mj =1 (cid:1) = m Y i =1 (cid:16) δ ˜ r i ,r i + δ − ˜ r i ,r i + v i (cid:17) θ (cid:0) { r j , v j } mj =1 (cid:1) , (2.25)(after identification of the momenta, the second cutoff becomes superfluous). From(2.24), we obtain k ξ ν k = X m ≥ m ! 1 N m X v ∈ P S ,r ∈ P H : r + v ∈ P H · · · X v m ∈ P S ,r m ∈ P H : r m + v m ∈ P H θ (cid:0) { r j , v j } mj =1 (cid:1) m Y i =1 η r i (cid:0) η r i + η r i + v i (cid:1) σ v i = X m ≥ m m ! 1 N m X v ∈ P S ,r ∈ P H : r + v ∈ P H · · · X v m ∈ P S ,r m ∈ P H : r m + v m ∈ P H θ (cid:0) { r j , v j } mj =1 (cid:1) m Y i =1 (cid:0) η r i + η r i + v i (cid:1) σ v i . (2.26)where we used the invariance of θ , w.r.t. − r i → r i + v i . The cutoffs have been usedfirst to exclude coinciding momenta in v , . . . , v m and in e v , . . . , e v m (which implies that,up to permutations, the pairing of the momenta in P S is unique) and then in (2.25) tomake sure that annihilation operators in A r j ,v j can only be contracted with the creationoperators in A ∗ e r j ,v j . This substantially simplifies computations. Similar simplificationswill arise in the computation of the energy of our trial state.A part from the formula (2.26) for the norm k ξ ν k , we will also need bounds on theexpectation, in the state ξ ν / k ξ ν k , of the number of particles operator N , of N , of thekinetic energy operator K and of the product KN . These bounds are collected in thenext proposition, whose proof will be discussed in Sec. 5. Proposition 2.3. Let ξ ν = e A ν Ω with e A ν defined in (2.15) with κ ∈ (1 / 2; 2 / and ε > such that κ − ε < . Then, under the assumptions of Theorem 1.1 we have h ξ ν , N j ξ ν ik ξ ν k ≤ CN (9 κ/ − ε ) j (2.27) and (cid:10) ξ ν , KN j − ξ ν (cid:11) k ξ ν k ≤ CN κ/ N (9 κ/ − ε )( j − (2.28) for j = 1 , . Using the Weyl operator W N from (2.3), the Bogoliubov transformation T ν definedin (2.13) and the cubic phase A ν introduced in (2.15) (or, equivalently, the vector ξ ν =12 A ν Ω), we can now define our trial stateΨ N = W N T ν e A ν Ω k W N T ν e A ν Ω k = W N T ν ξ ν k ξ ν k . (2.29)Here, we choose N > N = N + k σ L k (2.30)where we recall that σ L is the restriction to the set P L of the coefficients σ p = sinh( ν p ),with ν p defining the Bogoliubov transformation T ν ; see (2.13).Let us briefly compare our trial state with the one of [17]. We both perturb thecondensate with operators creating double and triple excitations, the latter having twoparticles with high momenta and one particle with low momentum. Moreover, we bothimpose cutoffs making sure that each low momentum appears only once. In contrastto [17], here we also impose cutoffs on high momenta. Moreover, we have a clearerseparation between creation of pairs (obtained through the Bogoliubov transformation T ν ) and creation of triples. Finally, in our approach, we create triple excitations throughthe action of e A ν on the vacuum; the algebraic structure of the exponential makes theanalysis and the combinatorics much simpler.As shown in the next proposition, the choice (2.30) of N guarantees that Ψ N hasthe right expected number of particles. Proposition 2.4. Let Ψ N be defined in (2.29) with the parameter N appearing in (2.3) defined by (2.30) . Let κ ∈ (1 / 2; 2 / and ε > so that κ − ε < . Then h Ψ N , N Ψ N i ≥ N , h Ψ N , N Ψ N i ≤ CN (2.31) for all N large enough.Proof. From (2.4) and (2.14) we get T ∗ ν W ∗ N N W N T ν = N + X p ∈ Λ ∗ + σ p + p N ( a + a ∗ ) + a ∗ a + X p ∈ Λ ∗ + (cid:2) ( σ p + γ p ) a ∗ p a p + γ p σ p ( a p a − p + h.c.) (cid:3) . (2.32)Using that, by definition of ξ ν , a ξ ν = 0 and h ξ ν , a p a − p ξ ν i = 0 for every p ∈ Λ ∗ , as wellas the definition N = N + k σ L k , we obtain that h Ψ N , N Ψ N i = h ξ ν , T ∗ ν W ∗ N N W N T ν ξ ν ik ξ ν k = N + X p ∈ P cL σ p + X p ∈ P S ∪ P H ( σ p + γ p ) h ξ ν , a ∗ p a p ξ ν ik ξ ν k . (2.33)This immediately implies that h Ψ N , N Ψ N i ≥ N .13ith a ξ ν = 0 and the assumption 3 κ − ε < 0, (2.32) also implies that h ξ ν , T ∗ ν W ∗ N N W N T ν ξ ν i ≤ CN + C X p,q ∈ Λ ∗ + ( σ p + γ p )( σ q + γ q ) h ξ ν , a ∗ p a p a ∗ q a q ξ ν i + C X p,q ∈ Λ ∗ + γ p σ p γ q σ q h ξ ν , ( a p a − p + a ∗ p a ∗− p )( a q a − q + a ∗ q a ∗− q ) ξ ν i . Since ξ ν is a superposition of states with 3 m particles with momenta in P H ∪ P S , weobtain, writing a p a − p a ∗ q a ∗− q = a ∗ q a p a ∗− q a − p + ( δ p,q + δ − p,q )( a ∗ p a p + 1) + δ p,q a ∗− p a − p andsimilarly for a ∗ p a ∗− p a q a − q , h ξ ν , T ∗ ν W ∗ N N W N T ν ξ ν i ≤ CN + CN ε h ξ ν , ( N + 1) ξ ν i + C X p ∈ P H ∪ P S γ p σ p h ξ ν , a ∗ p a p ξ ν i + C X p ∈ Λ ∗ + γ p σ p k ξ ν k + C X p,q ∈ P H ∪ P S γ p σ p γ q σ q h ξ ν , a ∗ p a q a ∗− p a − q ξ ν i . Estimating the last term through (cid:12)(cid:12)(cid:12) X p,q ∈ P H ∪ P S γ p σ p γ q σ q h ξ ν , a ∗ p a q a ∗− p a − q ξ ν i (cid:12)(cid:12)(cid:12) ≤ X p,q ∈ P H ∪ P S | γ p || σ p || γ q || σ q | k a ∗ q a p ξ ν kk a ∗− p a − q ξ ν k≤ X p,q ∈ P H ∪ P S | γ p || σ p || γ q || σ q | (cid:2) k a q a p ξ ν k + k a p ξ ν k (cid:3)(cid:2) k a − p a − q ξ ν k + k a − q ξ ν k (cid:3) ≤ C k γ S ∪ H k ∞ k σ S ∪ H k ∞ k ( N + 1) ξ ν k + C k γ S ∪ H k ∞ k σ S ∪ H k kN / ξ ν k + C k γ S ∪ H k ∞ k σ S ∪ H k ∞ k σ S ∪ H kk ( N + 1) ξ ν kkN / ξ ν k we conclude with the bounds in Lemma 2.2 and in Prop. 2.3 that, for 3 κ − ε < , h ξ ν , T ∗ ν W ∗ N N W N T ν ξ ν i ≤ CN + CN ε h ξ ν , N ξ ν i + CN ε k σ k h ξ ν , N ξ ν i + C k γ k ∞ k σ k ≤ CN + CN κ − ε + CN κ − ε + CN κ ≤ CN . The next theorem, whose proof occupies the rest of the paper, determines the energyof Ψ N and, combined with Prop. 2.4, allows us to conclude the proof of Prop. 1.3. Theorem 2.5. Let H N and Ψ N ∈ F be defined as in (2.2) and (2.29) respectively, andlet E Ψ N = h Ψ N , H N Ψ N i . Let κ ∈ (1 / 2; 2 / , ε > so small that κ − ε < . Then,under the assumption of Theorem 1.1, we have E Ψ N ≤ π a N κ (cid:18) √ π ( a N κ − ) / (cid:19) + CN κ/ max { N − ε , N κ − ε , N κ/ − ε } (2.34) for all N large enough. emark: Eq. (2.34) gives the correct second order term for all κ < / ε > γ < / κ < / 12 (corresponding to γ < / Proof of Prop. 1.3. Prop. 1.3 follows from Prop.2.4 and Theorem 2.5, recalling that(1.5) is unitarily equivalent to L − H N , with H N as defined in (2.2), L = ˜ ρ − γ , N =˜ ρL = ˜ ρ − γ and κ = (2 γ − / (3 γ − ε (3 γ − → ε . In this section we prove Thm. 2.5. With (2.29) and introducing the notations G N = T ∗ ν L N T ν , with L N = W ∗ N H N W N . (3.1)we write E Ψ N = h Ψ N , H N Ψ N i = h ξ ν , G N ξ ν ik ξ ν k with ξ ν = e A ν Ω, as defined before (2.15). With (2.2), and recalling from (2.4) that W ∗ N a p W N = a p + p N δ p, (3.2)we obtain L N = L (0) N + L (1) N + L (2) N + L (3) N + L (4) N , with L (0) N = N N κ − b V (0) L (1) N = N / N κ − b V (0)( a + h.c.) L (2) N = X p ∈ Λ ∗ p a ∗ p a p + N N X p ∈ Λ ∗ N κ (cid:0) b V ( p/N − κ ) + b V (0) (cid:1) a ∗ p a p + N N X p ∈ Λ ∗ N κ b V ( p/N − κ )( a ∗ p a ∗− p + h.c.) L (3) N = √ N N X p,r ∈ Λ ∗ N κ b V ( r/N − κ )( a ∗ p a ∗ r a p + r + h.c.) L (4) N = 12 N X p,q,r ∈ Λ ∗ N κ b V ( r/N − κ ) a ∗ p + r a ∗ q a p a q + r . (3.3)To compute G N , we have to conjugate the operators in (3.3) with the Bogoliubovtransformation T ν . The result is described in the following proposition, whose proof willbe discussed in Sec. 4. 15 roposition 3.1. Let K = X p ∈ Λ ∗ + p a ∗ p a p , V ( H ) N = 12 N X r ∈ Λ ∗ , p,q ∈ P H : p + r,q + r ∈ P H N κ b V ( r/N − κ ) a ∗ p + r a ∗ q a p a q + r . (3.4) and (recalling from (2.30) that N = N − k σ L k ) C N = √ N N X p,r ∈ P H p + r ∈ P S N κ b V ( r/N − κ ) σ p + r γ p γ r ( a ∗ p + r a ∗− p a ∗− r + h.c. ) . (3.5) Moreover, let C G N = N κ b V (0) + X p ∈ Λ ∗ + p σ p + X p ∈ Λ ∗ + N κ b V ( p/N − κ ) σ p γ p + X p ∈ P L N κ b V ( p/N − κ ) σ p + 12 N X p,r ∈ Λ ∗ + r = p N κ b V ( r/N − κ ) σ p σ p − r γ p γ p − r − N X v ∈ P L σ v X p ∈ P cL N κ b V ( p/N − κ ) η p . (3.6) Then, under the assumptions of Theorem 1.1, for all κ ∈ (1 / 2; 2 / and ε > with κ − ε < and N sufficiently large, we have h ξ ν , G N ξ ν ik ξ ν k ≤ C G N + h ξ ν , ( K + V ( H ) N + C N ) ξ ν ik ξ ν k + CN κ/ · max { N − ε , N κ − ε , N κ/ − ε } . (3.7)The expectation of the operators K , V ( H ) N , C N in the state ξ ν / k ξ ν k , appearing on ther.h.s. of (3.7) is determined by the next proposition, which will be shown in Sec. 5. Proposition 3.2. Under the assumptions of Theorem 1.1, we have h ξ ν , ( K + V ( H ) N + C N ) ξ ν ik ξ ν k ≤ N X v ∈ P L σ v X r ∈ P H N κ b V ( r/N − κ )( η r + η r + v )+ CN κ/ · max { N − ε , N κ − ε } for all κ ∈ (1 / 2; 2 / and all ε > so small that κ − ε < . Let us now use the statements of Prop. 3.1 and Prop. 3.2 to obtain an upper boundfor the energy of the trial state Ψ N and prove Thm. 2.5. From Prop. 3.1 and Prop. 3.216e find E Ψ N ≤ N κ b V (0) + X p ∈ Λ ∗ + p σ p + X p ∈ Λ ∗ + N κ b V ( p/N − κ ) γ p σ p + X v ∈ P L N κ b V ( v/N − κ ) σ v + 12 N X p,q ∈ Λ ∗ + p = q N κ b V (( p − q ) /N − κ ) γ p γ q σ p σ q − N X v ∈ P L σ v X p ∈ P cL N κ b V ( p/N − κ ) η p + 1 N X v ∈ P L σ v X r ∈ P H N κ b V ( r/N − κ )( η r + η r + v ) + E (3.8)with E ≤ CN κ/ · max { N − ε , N κ − ε , N κ/ − ε } for all κ ∈ (1 / 2; 2 / 3) and all ε > κ − ε < κ − ε < κ − ε ). Since | σ p − η p | ≤ C | η p | for all p ∈ P cL , with (2.8) we find X p ∈ Λ ∗ + p σ p ≤ X v ∈ P L v σ v + X p ∈ P cL p η p + CN κ/ − ε . (3.9)Similarly, with | γ p σ p − η p | ≤ Cη p for all p ∈ P cL , the last term on the first line of (3.8)can be written as X p ∈ Λ ∗ + N κ b V ( p/N − κ ) γ p σ p ≤ X v ∈ P L N κ b V ( v/N − κ ) γ v σ v + X p ∈ P cL N κ b V ( p/N − κ ) η p + CN κ/ − ε . (3.10)Next, we focus on the last term on the second line of (3.8). We write12 N X p,q ∈ Λ ∗ + p = q N κ b V (( p − q ) /N − κ ) γ p γ q σ p σ q = 1 N X p ∈ P cL ,q ∈ P L N κ b V (( p − q ) /N − κ ) η p γ q σ q + 12 N X p,q ∈ P cL p = q N κ b V (( p − q ) /N − κ ) η p η q + E . (3.11)Using again | γ p σ p − η p | ≤ C | η p | for all p ∈ P cL , the estimatesup r ∈ Λ ∗ X s ∈ Λ ∗ + N κ | b V (( r − s ) /N − κ ) || η s | ≤ CN κ (3.12)and the bounds from Lemma 2.2, we conclude (using the condition 3 κ − ε < 0) that E ≤ C (cid:2) N κ k η L c k ∞ k η L c k + N κ − k η L c k ∞ k η L c k k σ L γ L k + N κ − k σ L γ L k (cid:3) ≤ CN κ/ − ε . 17o prove (3.12) we use (2.8) and we remark that X s ∈ Λ ∗ : | s |≤ N − κ N κ | b V (( r − s ) /N − κ ) || η s | ≤ CN κ X s ∈ Λ ∗ + : | s |≤ N − κ | s | − ≤ CN κ and that, rescaling variables (setting ˜ r = r/N − κ ) and using an integral approximation, X s ∈ Λ ∗ + : | s | >N − κ N κ | b V (( r − s ) /N − κ ) || η s | ≤ CN κ X s ∈ Λ ∗ /N − κ : | s | > N − − κ ) | b V (˜ r − s ) || s | − ≤ CN κ Z | s | > | b V (˜ r − s ) || s | − ds ≤ C q N κ k b V k q for any q < 3. With the assumption V ∈ L q ′ ( R ), for some q ′ > / 2, (3.12) follows bythe Hausdorff-Young inequality.Finally, we remark that the terms in the last two lines of (3.8) can be combined,using (2.8) and the bound k σ L k ≤ CN κ/ from Lemma 2.2 into − N X v ∈ P L σ v X p ∈ P cL N κ b V ( p/N − κ ) η p + 1 N X v ∈ P L σ v X r ∈ P H N κ b V ( r/N − κ )( η r + η r + v ) ≤ N X v ∈ P L σ v X r ∈ P cL N κ b V ( r/N − κ ) η r + v + CN κ/ − ε . (3.13)Inserting (3.9), (3.10), (3.11) and (3.13) into (3.8), we obtain E Ψ N ≤ N κ b V (0) + X p ∈ P cL (cid:2) p η p + N κ b V ( p/N − κ ) + 12 N X r ∈ P cL N κ b V (( r − p ) /N κ ) η r (cid:3) η p + X v ∈ P L h v σ v + (cid:0) σ v + γ v σ v (cid:1)(cid:16) N κ b V ( v/N − κ ) + 1 N X r ∈ P cL N κ b V (( r − v ) /N − κ ) η r (cid:17)i + CN κ/ max { N − ε , N κ − ε , N κ/ − ε } . (3.14)Let us now consider the first square bracket on the r.h.s. of (3.14). Using thescattering equation (2.9) we obtain X p ∈ P cL h p η p + N κ b V ( p/N − κ ) + 12 N X r ∈ P cL N κ b V (( r − p ) /N − κ ) η r i η p = 12 X p ∈ P cL N κ b V ( p/N − κ ) η p − N X p ∈ P cL v ∈ P L N κ b V (( p − v ) /N − κ ) η v η p + E (3.15)with E = N − κ λ ℓ X p ∈ P cL (cid:0) b χ ℓ ∗ b f N (cid:1) p η p − N X p ∈ P cL N κ b V ( p/N − κ ) η p η . N − κ λ ℓ ≤ C (Lemma 2.1), k b χ ℓ ∗ b f N k = k χ ℓ f N k ≤ C and k η L c k ≤ CN κ/ − ε (Lemma 2.2) in the first term, (2.10) and (3.12) in the second term, we find E ≤ CN κ/ − ε (using that 3 κ − ε < κ > / X v ∈ P L (cid:0) σ v + γ v σ v (cid:1)(cid:16) N κ b V ( v/N − κ ) + 1 N X r ∈ P cL N κ b V (( r − v ) /N − κ ) η r (cid:17) = X v ∈ P L (cid:0) σ v + γ v σ v (cid:1) N κ (cid:0) b V ( · /N − κ ) ∗ b f N (cid:1) v + E (3.16)with E = − N X v, r ∈ P L N κ b V (( r − v ) /N − κ ) (cid:0) σ v + γ v σ v (cid:1) η r − η N X v ∈ P L N κ b V N ( v/N − κ ) (cid:0) σ v + γ v σ v (cid:1) . With (2.8), Lemma 2.2, | η | ≤ CN κ and the assumption 3 κ − ε < 0, we obtain E ≤ N κ/ − ε .Inserting (3.15) and (3.16) in (3.14) and completing sums over p on the r.h.s. of(3.15), we arrive at E Ψ N ≤ N (cid:0) N κ b V N ( · /N − κ ) ∗ b f N (cid:1) + X v ∈ P L h v σ v + (cid:0) σ v + γ v σ v (cid:1)(cid:0) N κ b V ( · /N − κ ) ∗ b f N (cid:1) v − N κ (cid:0) b V ( · /N − κ ) ∗ b f N (cid:1) v η v i + CN κ/ max { N − ε , N κ − ε , N κ/ − ε } . (3.17)Let us now introduce the notation ˆ g p = ( N κ b V ( · /N − κ ) ∗ b f N ) p . Notice that (cid:12)(cid:12) b g − π a N κ (cid:12)(cid:12) ≤ CN κ − , (cid:12)(cid:12) b g p − b g (cid:12)(cid:12) ≤ C | p | N κ − . (3.18)With the expression (2.17), we obtain X v ∈ P L h v σ v + (cid:0) σ v + γ v σ v (cid:1)b g v i = 12 X v ∈ P L (cid:20) − v − b g v + v + v (8 π a N κ + b g v ) √ v + 16 π a N κ v (cid:21) = 12 X v ∈ P L hp v + 16 π a N κ v − v − π a N κ i + E where, with (3.18) and 3 κ − ε < 0, we find E = 12 X v ∈ P L " π a N κ − b g v + b g v − π a N κ p π a N κ /v ≤ C X v ∈ P L | π a N κ − b g v | | p π a N κ /v − | p π a N κ /v ≤ CN κ − h X | v | 0. Hence | (cid:10) ξ ν , E ξ ν (cid:11) | ≤ CN κ kN / ξ ν k . (4.1)We consider now the contribution from L (2 ,V ) N . Using again h ξ ν , a ∗ p a ∗− p ξ ν i = 0 for all p ∈ Λ ∗ + and h ξ ν , a ∗ p a p ξ ν i = 0 for all p ∈ Λ ∗ + \ ( P S ∪ P H ), a straightforward computationshows that h ξ ν , T ∗ ν L (2 ,V ) N T ν ξ ν ik ξ ν k = N N X p ∈ Λ ∗ + N κ b V ( p/N − κ ) γ p σ p + N N X p ∈ Λ ∗ + N κ (cid:0) b V (0) + b V ( p/N − κ ) (cid:1) σ p + N N X p ∈ P S ∪ P H N κ (cid:2) b V ( p/N κ − )( γ p + σ p ) + b V (0)( γ p + σ p ) (cid:3) h ξ ν , a ∗ p a p ξ ν ik ξ ν k . With the bounds k γ S k ∞ , k σ S k ∞ , k σ H k ∞ , k γ H k ∞ ≤ CN ε from Lemma 2.2, with (4.1)and with the estimate h ξ ν , N ξ ν i ≤ CN κ/ − ε k ξ ν k from Prop. 2.3, we conclude that h ξ ν , G (2) N ξ ν ik ξ ν k ≤ h ξ ν , K ξ ν ik ξ ν k + X p ∈ Λ ∗ + p σ p + N N X p ∈ Λ ∗ + N κ b V ( p/N − κ ) γ p σ p + N N X p ∈ Λ ∗ + N κ (cid:0) b V (0) + b V ( p/N − κ ) (cid:1) σ p + CN κ/ − ε (4.2)using again the condition 3 κ − ε < 0. 21ext, we study the contribution of G (3) N = T ∗ ν L (3) N T ν , with L (3) N as in (3.3). Recall theoperator C N , defined in (3.5). Taking into account the fact that ξ ν is a superposition ofvectors with 2 m particles with momenta in P H and m particles with momenta in P S ,for m ∈ N , we obtain that h ξ ν , G (3) N ξ ν i = h ξ ν , C N ξ ν i + X j =1 [ h ξ ν , F j ξ ν i + h.c.]withF = √ N N X p,r ∈ P H : p + r ∈ P S N κ b V ( r/N − κ ) γ p + r σ p σ r a ∗ p + r a ∗− p a ∗− r F = √ N N X p ∈ P H ,r ∈ P S : p + r ∈ P H N κ h b V ( r/N − κ ) + b V ( p/N − κ ) i γ p + r σ p σ r a ∗ p + r a ∗− p a ∗− r F = √ N N X p ∈ P H ,r ∈ P S : p + r ∈ P H N κ h b V ( r/N − κ ) + b V ( p/N − κ ) i σ p + r γ p γ r a − p − r a p a r . Using k a ∗− r ( N + 1) / ξ ν k ≤ k a − r ( N + 1) / ξ ν k + k ( N + 1) / ξ ν k , we can bound h ξ ν , F ξ ν i ≤ CN κ − / k γ S k ∞ X p,r ∈ P H | σ r || σ p | k a p + r a − p ( N + 1) − / ξ ν k× h k a − r ( N + 1) / ξ ν k + k ( N + 1) / ξ ν k i ≤ CN κ − / k γ S k ∞ k σ H k ∞ k σ H k k ( N + 1) / ξ ν kk ( N + 1) ξ ν k + CN κ − / k γ S k ∞ k σ H k k ( N + 1) / ξ ν k . With Lemma 2.2 and Prop. 2.3, we obtain h ξ ν , F ξ ν ik ξ ν k ≤ CN κ/ − ε/ + CN κ/ − / ε/ ≤ CN κ/ · N κ/ − / ε/ from the assumption that 3 κ − ε < 0. Similarly, we find h ξ ν , F ξ ν i ≤ CN κ − / k γ H k ∞ k σ S k ∞ k σ H k k ( N + 1) / ξ ν kk ( N + 1) ξ ν k + CN κ − / k γ H k ∞ k σ S kk σ H k k ( N + 1) / ξ ν k ≤ C (cid:2) N κ/ − ε + N κ/ − ε/ (cid:3) k ξ ν k ≤ CN κ/ · N κ/ − ε/ k ξ ν k and also h ξ ν , F ξ ν i ≤ CN κ − / k γ H k ∞ k γ S k ∞ k σ H k k ( N + 1) / ξ ν kk ( N + 1) ξ ν k + CN κ − / k γ H k ∞ k γ S kk σ H k k ( N + 1) / ξ ν k ≤ CN κ/ · N κ/ − ε k ξ ν k . h ξ ν , G (3) N ξ ν ik ξ ν k ≤ h ξ ν , C N ξ ν ik ξ ν k + CN κ/ · N κ/ − ε . (4.3)Finally, let us consider G (4) N = T ∗ ν L (4) N T ν . We decompose h ξ ν , G (4) N ξ ν i = P j =1 h ξ ν , G j ξ ν i with G = 12 N X r ∈ Λ ∗ , p,q ∈ Λ ∗ + − r = q,p N κ b V ( r/N − κ ) γ p γ q γ p + r γ q + r a ∗ p + r a ∗ q a p a q + r G = 12 N X r ∈ Λ ∗ , p,q ∈ Λ ∗ + r = q, − p N κ b V ( r/N − κ ) (cid:0) γ p + r σ q a ∗ p + r a − q + σ p + r γ q a − p − r a ∗ q (cid:1) × (cid:0) γ p σ q + r a p a ∗− q − r + σ p γ q + r a ∗− p a q + r (cid:1) G = 12 N X r ∈ Λ ∗ , p,q ∈ Λ ∗ + r = q, − p N κ b V ( r/N − κ ) σ p σ q σ p + r σ q + r a p + r a q a ∗ p a ∗ q + r . To estimate contributions from G , we arrange terms in normal order. We findG = 12 N X r ∈ Λ ∗ , p,q ∈ Λ ∗ + − r = q,p N κ b V ( r/N − κ ) σ p σ q σ p + r σ q + r a ∗ p a ∗ q + r a p + r a q + 12 N X r ∈ Λ ∗ , p ∈ Λ ∗ + p = − r N κ b V ( r/N − κ ) σ p σ p + r (cid:0) a ∗ p a p + a ∗ p + r a p + r (cid:1) + 1 N X p,q ∈ Λ ∗ + N κ b V (0) σ p σ q a ∗ p a p + 12 N X r ∈ Λ ∗ , p ∈ Λ ∗ + : p = − r N κ b V ( r/N − κ ) σ p σ p + r + 12 N X p,q ∈ Λ ∗ + N κ b V (0) σ p σ q . Since a p ξ ν = 0 if p ∈ Λ ∗ + \ ( P S ∪ P H ) and k σ H k ∞ ≤ k σ S k ∞ we find, by Cauchy-Schwarz, h ξ ν , G ξ ν i ≤ CN κ − (cid:2) k σ S k ∞ k σ k k ( N + 1) ξ ν k + k σ k k ξ ν k (cid:3) ≤ CN κ/ · max { N − ε , N κ − ε }k ξ ν k (4.4)using Prop. 2.3 and 3 κ − ε < 0. We proceed similarly for G . Through normal23rdering, we getG = 1 N X r ∈ Λ ∗ , p,q ∈ Λ ∗ + − r = q,p N κ b V ( r/N − κ ) γ p γ p + r σ q σ q + r a ∗ p + r a ∗− q − r a p a − q + 1 N X r ∈ Λ ∗ , p,q ∈ Λ ∗ + − r = q,p N κ b V ( r/N − κ ) γ p + r γ q + r σ p σ q a ∗ p + r a ∗− p a − q a q + r + 1 N X p,q ∈ Λ ∗ + N κ b V (0) γ p σ q a ∗ p a p + 1 N X r ∈ Λ ∗ , p ∈ Λ ∗ + N κ b V ( r/N − κ ) γ p σ p + r a ∗ p a p + 2 N X r ∈ Λ ∗ , p ∈ Λ ∗ + N κ b V ( r/N − κ ) γ p σ p γ p + r σ p + r a ∗ p a p + 12 N X p,r ∈ Λ ∗ + N κ b V ( r/N − κ ) γ p γ p + r σ p σ p + r . Keeping the last contribution intact and estimating the term on the fourth line distin-guishing the two cases ( p + r ) ∈ P S and ( p + r ) ∈ P H , we arrive at h ξ ν , G ξ ν i ≤ N X p,r ∈ Λ ∗ + N κ b V ( r/N − κ ) γ p γ p + r σ p σ p + r k ξ ν k + CN κ − k γ S k ∞ k σ k k ( N + 1) ξ ν k + CN κ − k γ S ∪ H k ∞ k σ S ∪ H k ∞ × " k γ S σ S k + k γ H k ∞ sup p X r ∈ Λ ∗ b V ( r/N − κ ) | η p + r | kN / ξ ν k . With the bounds in Lemma 2.2 and in Prop. 2.3 and with (3.12), we conclude that h ξ ν , G ξ ν i ≤ N X p,r ∈ Λ ∗ + N κ b V ( r/N − κ ) γ p γ p + r σ p σ p + r k ξ ν k + CN κ/ · N κ − ε k ξ ν k . (4.5)Finally, we consider G . Recalling that a p ξ ν = 0 if p ∈ Λ ∗ + \ ( P S ∪ P H ) and observingthat h ξ ν , a ∗ p + r a ∗ q a p a q + r ξ ν i 6 = 0 only if the operator a ∗ p + r a ∗ q a p a q + r preserves the numberof particles in P S and in P H , we arrive at h ξ ν , G ξ ν i ≤ N X r ∈ Λ ∗ , p,q ∈ P H : p + r,q + r ∈ P H N κ b V ( r/N − κ ) γ p γ q γ p + r γ q − r h ξ ν , a ∗ p + r a ∗ q a p a q + r ξ ν i + CN κ − k γ S ∪ H k ∞ k γ S k k ( N + 1) ξ k . With | γ p γ q γ p + r γ q + r − | ≤ C k η H k ∞ for all p, q ∈ P H , with ( p + r ) , ( q + r ) ∈ P H , and24sing the estimate (see the proof of (3.12))sup p ∈ Λ ∗ X r ∈ Λ ∗ + : r = p N κ | b V ( r/N − κ ) || p − r | ≤ CN we conclude that h ξ ν , G ξ ν i ≤ h ξ ν , V ( H ) N ξ ν i + C k η H k ∞ kN / K / ξ ν k + CN κ − k γ S ∪ H k ∞ k γ S k kN ξ k with V ( H ) N defined as in (3.4). With Lemma 2.2 and Prop. 2.3, we find (using theassumption 3 κ − ε < h ξ ν , G ξ ν i ≤ h ξ ν , V ( H ) N ξ ν i + CN κ/ · N κ − ε k ξ ν k . With (4.4) and (4.5), we have shown that h ξ ν , G (4) N ξ ν ik ξ ν k ≤ h ξ ν , V ( H ) N ξ ν ik ξ ν k + CN κ/ · max { N − ε , N κ − ε } . Combining the last bound with (4.2) and (4.3), we obtain h ξ ν , G N ξ ν ik ξ ν k ≤ e C N + h ξ ν , ( K + V ( H ) N + C N ) ξ ν ik ξ ν k + CN κ/ · max { N − ε , N κ − ε , N κ/ − ε } where we defined e C N = N N N κ b V (0) + X p ∈ Λ ∗ + p σ p + N N X p ∈ Λ ∗ + N κ (cid:0) b V ( p/N − κ ) + b V (0) (cid:1) σ p + N N X p ∈ Λ ∗ + N κ b V ( p/N − κ ) σ p γ p + 12 N X p,r ∈ Λ ∗ + r = p N κ b V ( r/N − κ ) σ p σ p + r γ p γ p + r . (4.6)Inserting N = N − k σ L k and recalling from Lemma 2.2 that k σ L k ≤ CN κ/ and k σ L c k ≤ CN κ/ − ε , we obtain e C N = C G N + O ( N κ/ − ε ), with C G N as defined in (3.6)(with the assumption 3 κ − ε < | σ p γ p − η p | ≤ Cη p ≤ CN κ / | p | , for p ∈ P cL . This completes theproof of Prop. 3.1. In this section we prove Prop. 2.3 and Prop. 3.2, which is a conequence of the followinglemma. 25 emma 5.1. Let A ν be defined in (2.15) , and K , V ( H ) N and C N be defined in (3.4) and (3.5) respectively. Then, for ξ ν = e A ν Ω , h ξ ν , K ξ ν ik ξ ν k ≤ N X v ∈ P S ,r ∈ P H : r + v ∈ P H r η r ( η r + η r + v ) σ v + E , (5.1) h ξ ν , C N ξ ν ik ξ ν k ≤ N X v ∈ P S ,r ∈ P H : r + v ∈ P H N κ b V ( r/N − κ )( η r + η r + v ) σ v + E , (5.2) h ξ ν , V ( H ) N ξ ν ik ξ ν k ≤ N X v ∈ P S ,r ∈ P H : r + v ∈ P H (cid:0) N κ b V ( · /N − κ ) ∗ η (cid:1) r ( η r + η r + v ) σ v + E , (5.3) with E ≤ CN κ/ · max { N − ε , N κ − ε } for all κ ∈ (1 / 2; 2 / , ε > so small that κ − ε < and N large enough. With Lemma 5.1, we can immediately show Prop. 3.2. Proof of Proposition 3.2. From Lemma 5.1 we have h ξ ν , ( K + V ( H ) N + C N ) ξ ν ik ξ ν k ≤ N X v ∈ P S σ v X r ∈ P H : r + v ∈ P H h r η r + N κ b V ( r/N − κ ) + N κ N (cid:0) b V ( · /N − κ ) ∗ η (cid:1) r i ( η r + η r + v ) + E with E ≤ CN κ/ · max { N − ε , N κ − ε } . With the scattering equation (2.9), we obtain h ξ ν , ( K + V ( H ) N + C N ) ξ ν ik ξ ν k ≤ N X v ∈ P S σ v X r ∈ P H : r + v ∈ P H N κ b V ( r/N − κ )( η r + η r + v ) + E ′ with E ′ ≤ N X v ∈ P S σ v X r ∈ P H : r + v ∈ P H N − κ λ ℓ ( b χ ℓ ∗ b f N ) r η r + E . Using | N − κ λ ℓ | ≤ C and k b χ ℓ ∗ b f N k ≤ C , we conclude h ξ ν , ( K + V ( H ) N + C N ) ξ ν ik ξ ν k ≤ N X v ∈ P S σ v X r ∈ P H : r + v ∈ P H N κ b V ( r/N − κ )( η r + η r + v ) + CN κ/ · max { N − ε , N κ − ε } . Finally, with (3.12) and the expression (2.17) for σ v , we can extend the sum over v ∈ P S to a sum over all v ∈ P L , without changing the size of the error. This completes theproof of Prop. 3.2. 26e still have to show Prop. 2.3 and Lemma 5.1. In this section we prove (5.1) and Prop 2.3. We start by computing the expectation h ξ ν , K ξ ν i . We proceed as we did in (2.23)-(2.26) to compute k ξ ν k . With K a ∗ r + v a ∗− r a ∗− v = a ∗ r + v a ∗− r a ∗− v ( K + ( r + v ) + r + v ) we obtain (cid:10) ξ ν , K ξ ν (cid:11) = X m ≥ m ( m − N m X v ∈ P S ,r ∈ P H : r + v ∈ P H · · · X v m ∈ P S ,r m ∈ P H : r m + v m ∈ P H θ (cid:0) { r j , v j } mj =1 (cid:1) × [ r m + v m + ( r m + v m ) ] m Y i =1 ( η r i + η r i + v i ) σ v i . with the cutoff θ introduced in (2.22). Since all terms are positive, we can find an upperbound for h ξ ν , K ξ ν i by replacing θ ( { r j , v j } mj =1 ) with θ ( { r j , v j } m − j =1 ), removing conditionsinvolving momenta with index m . Recalling (2.26), we find (cid:10) ξ ν , K ξ ν (cid:11) ≤ N X v ∈ P S ,r ∈ P H : r + v ∈ P H [ r + v + ( r + v ) ]( η r + η r + v ) σ v k ξ ν k ≤ N X v ∈ P S ,r ∈ P H r η r ( η r + η r + v ) σ v k ξ ν k + E with (using Lemma 2.2 and the assumption 3 κ − ε < Ek ξ ν k = 2 N X v ∈ P S ,r ∈ P H : r + v ∈ P H ( v + r · v ) η r ( η r + η r + v ) σ v ≤ CN ( k σ S k H k η H k + k σ S kk η H kk σ S k H k η H k H ) ≤ CN κ − ε ≤ CN κ/ − ε . This proves (5.1). In particular, (5.1) implies, together with Lemma 2.2, that h ξ ν , K ξ ν ik ξ ν k ≤ CN − k η H k H k σ S k ≤ CN κ/ (5.4)which shows (2.28) with j = 1 in Prop. 2.3.Analogously, we find h ξ ν , KN ξ ν i ≤ X m ≥ m m ( m − N m X v ∈ P S ,r ∈ P H : r + v ∈ P H · · · X v m ∈ P S ,r m ∈ P H : r m + v m ∈ P H θ (cid:0) { r j , v j } mj =1 (cid:1) × [ r m + v m + ( r m + v m ) ] m Y i =1 ( η r i + η r i + v i ) σ v i . m = 1 + ( m − θ ( { r j , v j } mj =1 ) by θ ( { r j , v j } m − j =1 ), we obtain h ξ ν , KN ξ ν i ≤ h ξ ν , K ξ ν i + 34 N X r,r ′ ∈ P H ,v,v ′ ∈ P S [ r + v + ( r + v ) ] × ( η r + η r + v ) ( η r ′ + η r ′ + v ′ ) σ v σ v ′ k ξ ν k . With (5.4) and with the bounds for k η H k H , k η H k , k σ S k from Lemma 2.2), we find h ξ ν , KN ξ ν ik ξ ν k ≤ CN κ/ · N κ/ − ε (5.5)which shows (2.28) with j = 2.To show (2.27) we observe that, by (2.15), the operator A ν only creates particleswith momenta in P S ∪ P H and for each particle with momentum in P S , it creates twoparticles with momenta in P H . Since | p | > N − κ − ε for all p ∈ P H , we find, by (5.4), h ξ ν , N ξ ν i = X p ∈ P S ∪ P H h ξ ν , a ∗ p a p ξ ν i = 32 X p ∈ P H h ξ ν , a ∗ p a p ξ ν i≤ CN − κ +2 ε h ξ ν , K ξ ν i ≤ N κ/ − ε k ξ ν k proving (2.27) for j = 1. Analogously, we find h ξ ν , N ξ ν i = X p ∈ P S ∪ P H hN / ξ ν , a ∗ p a p N / ξ ν i≤ X p ∈ P H hN / ξ ν , a ∗ p a p N / ξ ν i ≤ CN − κ +2 ε h ξ ν , KN ξ ν i . By (5.5), we obtain (2.27) with j = 2. This completes the proof of Prop. 2.3. The goal of this section is to show (5.2). From (3.5), we have (using the reality of η p , γ p , σ p ) h ξ ν , C N ξ ν i = 2 √ N N X m ≥ m !( m − X p,r ∈ P H p + r ∈ P S N κ b V ( r/N − κ ) σ p + r γ p γ r h A mν ξ ν , a ∗ p + r a ∗− p a ∗− r A m − ν ξ ν i . Proceeding as in the previous section, we get (cid:10) ξ ν , C N ξ ν (cid:11) = 2 r N N X m ≥ m − ( m − N m X v ∈ P S ,r ∈ P H : r + v ∈ P H · · · X v m ∈ P S ,r m ∈ P H : r m + v m ∈ P H θ (cid:0) { r j , v j } mj =1 (cid:1) × N κ b V ( r m /N − κ ) (cid:0) η r m + η r m + v m (cid:1) γ r m γ r m + v m σ v m m − Y i =1 ( η r i + η r i + v i ) σ v i . 28o reconstruct the norm k ξ ν k on the r.h.s. we need to free the momenta with index m .To this end, we recall the defintion (2.22) to write θ (cid:0) { r j , v j } mj =1 (cid:1) = θ (cid:0) { r j , v j } m − j =1 (cid:1) θ m (cid:0) { r j , v j } mj =1 (cid:1) (5.6)with θ m (cid:0) { r j , v j } mj =1 (cid:1) = m − Y i,j =1 Y p i ,p j ,p m : p ℓ ∈{− r ℓ ,r ℓ + v ℓ } δ p i = − p j + v m δ − p m + v i = p j collecting all conditions involving { r m , v m } . Writing θ m = 1 + [ θ m − h ξ ν , C N ξ ν i = I C + J C with (recall the expression (2.26) for k ξ ν k ) I C = 2 r N N X v ∈ P S , r ∈ P H : r + v ∈ P H N κ − b V ( r/N − κ ) (cid:0) η r + η r + v (cid:1) γ r γ r + v σ v k ξ ν k and J C = 2 r N N X m ≥ m − ( m − N m X v ∈ P S ,r ∈ P H : r + v ∈ P H · · · X v m ∈ P S ,r m ∈ P H : r m + v m ∈ P H θ (cid:0) { r j , v j } m − j =1 (cid:1) × h θ m (cid:0) { r j , v j } mj =1 (cid:1) − i N κ b V ( r m /N − κ ) (cid:0) η r m + η r m + v m (cid:1) γ r m γ r m + v m σ v m × m − Y i =1 ( η r i + η r i + v i ) σ v i . With | p N /N − | ≤ C k σ L k /N and | γ r γ r + v − | ≤ CN κ / | r | for all r ∈ P H , v ∈ P S ,we obtain (using (3.12) and the assumption 3 κ − ε < 0) that I C k ξ ν k ≤ N X v ∈ P S , r ∈ P H : r + v ∈ P H N κ b V ( r/N − κ ) (cid:0) η r + η r + v (cid:1) σ v + CN κ/ − ε . (5.7)To complete the proof of (5.2), we focus now on the error term J C . We observe that | θ m (cid:0) { r j , v j } mj =1 (cid:1) − | ≤ m − X j =1 h δ v j ,v m + X p m ∈{− r m ,r m + v m } p j ∈{− r j ,r j + v j } δ p m ,p j i + m − X j,k =1 j = k h X p j ∈{− r j ,r j + v j } p k ∈{− r k ,r k + v k } δ v m ,p j + p k + X p m ∈{− r m ,r m + v m } p j ∈{− r j ,r j + v j } δ p m , − p j + v k i . (5.8)We bound | J C | ≤ X + X , with X denoting the contribution arising from the first termon the r.h.s. of (5.8) (this term involves two indices, m and j ), and X indicating the29ontribution from the second term on the r.h.s. of (5.8) (this term involves three indices, m, j, k ). We can estimateX ≤ C X m ≥ m − ( m − N m X v ∈ P S ,r ∈ P H : r + v ∈ P H · · · X v m ∈ P S ,r m ∈ P H : r m + v m ∈ P H θ (cid:0) { r j , v j } m − j =1 (cid:1) × N κ | b V ( r m /N − κ ) | (cid:12)(cid:12) η r m + η r m + v m (cid:12)(cid:12) | γ r m || γ r m + v m | σ v m m − Y i =1 ( η r i + η r i + v i ) σ v i × h δ v m ,v m − + X p m − ,p m : p ℓ ∈{− r ℓ ,r ℓ + v ℓ } δ p m ,p m − i . With θ (cid:0) { r j , v j } m − j =1 (cid:1) ≤ θ (cid:0) { r j , v j } m − j =1 (cid:1) , we reconstruct k ξ ν k . Since k γ H k ∞ ≤ C , we endup withX k ξ ν k ≤ CN X r,r ′ ∈ P H ,v,v ′ ∈ P S N κ | b V ( r/N − κ ) | η r + η r + v || η r ′ + η r ′ + v ′ | σ v σ v ′ × h δ v,v ′ + X p ∈{− r,r + v } p ′ ∈{− r ′ ,r ′ + v ′ } δ p,p ′ i ≤ CN κ − k σ S k ∞ k σ S k k η H k X r ∈ P H | b V ( r/N − κ ) | r + CN κ − k σ S k X r ∈ P H | r | − ≤ CN κ/ − ε + CN κ − ε ≤ CN κ/ − ε where we used Lemma 2.2, (3.12), the assumption 3 κ − ε < | η r + v | ≤ CN κ | r | − , for all r ∈ P H and v ∈ P S . We can proceed similarly to estimate X . In the second term on the r.h.s. of (5.8), we have to sum over ( m − m − / j, k . With θ (cid:0) { r j , v j } m − j =1 (cid:1) ≤ θ (cid:0) { r j , v j } m − j =1 (cid:1) and again with Lemma 2.2and(3.12), we arrive atX k ξ ν k ≤ CN X r,r ′ ,r ′′ ∈ P H ,v,v ′ ,v ′′ ∈ P S N κ | b V ( r/N − κ ) | η r + η r + v || η r ′ + η r ′ + v ′ | | η r ′′ + η r ′′ + v ′′ | σ v σ v ′ σ v ′′ × h X p ∈{− r,r + v } ,p ′ ∈{− r ′ ,r ′ + v ′ } δ p, − p ′ + v ′′ + X p ′ ∈{− r ′ ,r ′ + v ′ } ,p ′′ ∈{− r ′′ ,r ′′ + v ′′ } δ v,p ′ + p ′′ i ≤ CN κ − k σ S k k η H k X r ∈ P H | r | − + CN κ − k σ S k X r ∈ P H | b V ( r/N − κ ) | r X r ′ ∈ P H | r ′ | − ≤ CN κ/ − ε ≤ CN κ/ · N κ − ε . Thus, | J C | / k ξ ν k ≤ N κ/ · max { N − ε , N κ − ε } . With (5.7), this implies (5.2).30 .3 Expectation of the quartic term In this section we show the bound (5.3) for the expectation of V ( H ) N . Pairing momentain P S , similarly as we did in (2.24) and in the previous subsections, we obtain (cid:10) ξ ν , V ( H ) N ξ ν (cid:11) = 12 N X m ≥ m ! 1 N m X v ∈ P S , r , ˜ r ∈ P H : r + v , ˜ r + v ∈ P H · · · X v m ∈ P S , r m , ˜ r m ∈ P H : r m + v m , ˜ r m + v m ∈ P H × θ (cid:0) { r j , v j } mj =1 (cid:1) θ (cid:0) { ˜ r j , v j } mj =1 (cid:1) m Y i =1 η r i η ˜ r i σ v i X r ∈ Λ ∗ ,p,q ∈ P H : p + r, q + r ∈ P H N κ b V ( r/N − κ ) × (cid:10) Ω , A r ,v . . . A r m ,v m a ∗ p + r a ∗ q a p a q + r A ∗ ˜ r ,v . . . A ∗ ˜ r m ,v m Ω (cid:11) (5.9)where we use the notation A r i ,v i = a r i + v i a − r i that was already introduced in (2.24). Nextwe observe that, because of the cutoffs θ ( { r j , v j } mj =1 ) and θ ( { ˜ r j , v j } mj =1 ), at most two in-dices i, j ∈ { , . . . , m } can be involved in contractions with the observable a ∗ p + r a ∗ q a p a q + r .We distinguish two possible cases.1) there exists an index i ∈ { , . . . , m } such that a p , a q + r are contracted with A ∗ ˜ r i ,v i and a ∗ q , a ∗ p + r are contracted with A r i ,v i 2) there are two indices i = j ∈ { , . . . , m } such that the operators a p and a q + r arecontracted with a ∗ ˜ p i and a ∗ ˜ p j for some ˜ p ℓ ∈ {− ˜ r ℓ , ˜ r ℓ + v ℓ } , ℓ = i, j and the operators a ∗ q , a ∗ p + r are contracted with a p i , a p j , with p ℓ ∈ {− r ℓ , r ℓ + v ℓ } , ℓ = i, j . Note that inthis case the operators a ∗− ˜ p i + v i , a ∗− ˜ p j + v j have to be contracted with a − p i + v i , a − p j + v j . We denote with V and V the contributions to (cid:10) ξ ν , V ( H ) N ξ ν (cid:11) arising from the two casesdescribed above. Let us first consider V . There are m choices (all leading to the samecontribution) for the index i ∈ { , . . . , m } labelling momenta to be contracted withthe observable. Let us fix i = m . Then we have p = ˜ p m , q + r = − ˜ p m + v m with˜ p m ∈ {− ˜ r m , ˜ r m + v m } , and p + r = p m , q = − p m + v m with p m ∈ {− r m , r m + v m } .Note that the choice of p and p + r also determines q and q + r , since we always have q = v m − ( p + r ). The presence of the cutoffs immediately implies that A r j ,v j is fullycontracted with A ∗ ˜ r j ,v j , for all j = m . We find h ξ ν , V ξ ν i = 12 N X m ≥ m − N m X v ∈ P S ,r , ˜ r ∈ P H : r + v , ˜ r + v ∈ P H · · · X v m ∈ P S ,r m , ˜ r m ∈ P H : r m + v m , ˜ r m + v m ∈ P H θ (cid:0) { r j , v j } mj =1 (cid:1) θ (cid:0) { ˜ r j , v j } mj =1 (cid:1) × m − Y j =1 η r j η ˜ r j ( δ r j , ˜ r j + δ − r j , ˜ r j + v j ) σ v j η r m η ˜ r m σ v m X r ∈ Λ ∗ , p ∈ P H : p − v m ,p + r ∈ P H N κ b V ( r/N − κ ) X p m ∈{− r m ,r m + v m } ˜ p m ∈{− ˜ r m , ˜ r m + v m } δ p,p m δ p + r, ˜ p m . (5.10)31ince here (in contrast to the previous subsections) the contraction does not fix ˜ r m to be either r m or − ( r m + v m ), we cannot erase the cutoff θ ( { ˜ r j , v j } mj =1 ). With thedecomposition (5.6), we can replace, on the r.h.s. of (5.10), θ (cid:0) { r j , v j } mj =1 (cid:1) θ (cid:0) { ˜ r j , v j } mj =1 (cid:1) = θ (cid:0) { r j , v j } m − j =1 (cid:1) θ m (cid:0) { r j , v j } mj =1 (cid:1) θ m (cid:0) { ˜ r j , v j } mj =1 (cid:1) . Writing θ m (cid:0) { ˜ r j , v j } mj =1 (cid:1) θ m (cid:0) { ˜ r j , v j } mj =1 (cid:1) = 1 + h θ m (cid:0) { r j , v j } mj =1 (cid:1) θ m (cid:0) { ˜ r j , v j } mj =1 (cid:1) − i we split (similarly as we did in the last subsection) h ξ ν , V ξ ν i = I V + J V , with I V = 1 N X r ∈ Λ ∗ X v ∈ P S ,p ∈ P H : p + r,p − v, p + r − v ∈ P H N κ b V ( r/N − κ ) η p (cid:0) η p + r + η p + r − v (cid:1) σ v k ξ ν k (5.11)and J V = 12 N X m ≥ m − ( m − N m X v ∈ P S ,r ∈ P H : r + v ∈ P H · · · X v m − ∈ P S ,r m − ∈ P H : r m − + v m − ∈ P H X v m ∈ P S ,r m , ˜ r m ∈ P H : r m + v m , ˜ r m + v m ∈ P H × θ (cid:0) { r j , v j } m − j =1 (cid:1)(cid:2) θ m ( { r j , v j } mj =1 ) θ m ( { r ♯j , v j } mj =1 ) − (cid:3) m − Y i =1 ( η r i + η r i + v i ) σ v i × η r m η ˜ r m σ v m X r ∈ Λ ∗ N κ b V ( r/N − κ ) X p ∈ P H : p + r,p − v m , p + r − v m ∈ P H X p m ∈{− r m ,r m + v m } ˜ p m ∈{− ˜ r m , ˜ r m + v m } δ p,p m δ p + r, ˜ p m (5.12)where r ♯j = r j for j = 1 , . . . , m − r ♯m = ˜ r m in the argument of θ m . Observing that,with Lemma 2.2 and (3.12),1 N X r ∈ Λ ∗ X v ∈ P S ,p ∈ Λ ∗ : p ∈ P cH or p − v ∈ P cH N κ | b V ( r/N − κ ) || η p | (cid:0) | η p + r | + | η p + r − v | (cid:1) σ v ≤ CN − κ k σ S k h X | p |≤ N − κ − ε | p | − i sup p ∈ Λ ∗ X r ∈ Λ ∗ | b V ( r/N − κ ) || η p + r | ≤ CN κ/ − ε we conclude from (5.11) (switching p + r → p and v → − v ) that I V k ξ ν k ≤ N X v ∈ P S ,p ∈ P H : p + v ∈ P H (cid:0) N κ b V ( · /N − κ ) ∗ η (cid:1) p ( η p + η p + v ) σ v + CN κ/ − ε . (5.13)32et us now focus on the term J V . With (cid:12)(cid:12)(cid:12) θ m (cid:0) { r j , v j } mj =1 (cid:1) θ m (cid:0) { ˜ r j , v j } mj =1 (cid:1) − (cid:12)(cid:12)(cid:12) ≤ m − X j =1 δ v m ,v j + m − X j =1 h X p j ∈{− r j ,r j + v j } p m ∈{− r m ,r m + v m } δ p m ,p j + X p j ∈{− r j ,r j + v j } ˜ p m ∈{− ˜ r m , ˜ r m + v m } δ ˜ p m ,p j i + m − X j,k =1 j = k h X p j ∈{− r j ,r j + v j } p m ∈{− r m ,r m + v m } δ p m , − p j + v k + X p j ∈{− r j ,r j + v j } ˜ p m ∈{− ˜ r m , ˜ r m + v m } δ ˜ p m , − p j + v k i + m − X j,k =1 j = k X p j ∈{− r j ,r j + v j } p k ∈{− r k ,r k + v k } δ v m ,p j + p k (5.14)we can bound | J V | ≤ W + W + W + W , with W ℓ indicating the contribution to (5.12)arising from the ℓ -th term, on the r.h.s. of (5.14).The term W contains the sum of ( m − 1) identical contributions, correspondingto j ∈ { , . . . , m − } in the first term on the r.h.s. of (5.14). Let us fix j = m − θ (cid:0) { r j , v j } m − j =1 (cid:1) ≤ θ (cid:0) { r j , v j } m − j =1 (cid:1) and reconstructing the expression (2.26) for k ξ ν k , we can bound (the momenta r ′ , r ′′ , ˜ r ′′ correspond to r m − , r m , ˜ r m )W k ξ ν k ≤ CN − X r ∈ Λ ∗ N κ | b V ( r/N − κ ) | X r ′ ,r ′′ , ˜ r ′′ ∈ P H v ′ ∈ P S ( η r ′ + η r ′ + v ′ ) | η r ′′ || η ˜ r ′′ | σ v ′ X p ′′ ∈{− r ′′ ,r ′′ + v ′ } ˜ p ′′ ∈{− ˜ r ′′ , ˜ r ′′ + v ′ } δ p ′′ + r, ˜ p ′′ . With Lemma 2.2 and with the estimatesup v ∈ P S ∪{ } N X r ∈ Λ ∗ ,q ∈ P H : q − r ∈ P H N κ | b V ( r/N − κ ) || η q − v || η q − r | ≤ CN κ (5.15)which can be shown similarly to (3.12) (using V ∈ L q ( R ), for some q > / 2) we findW k ξ ν k ≤ CN κ − k η H k k σ S k ∞ k σ S k ≤ CN κ/ − ε ≤ CN κ/ − ε (5.16)since 3 κ − ε < 0. Analogously, we bound, with (3.12) and Lemma 2.2W k ξ ν k ≤ CN − X r ∈ Λ ∗ N κ | b V ( r/N − κ ) | X r ′ ,r ′′ , ˜ r ′′ ∈ P H v ′ ,v ′′ ∈ P S ( η r ′ + η r ′ + v ′ ) | η r ′′ || η ˜ r ′′ | σ v ′ σ v ′′ × X p ′′ ∈{− r ′′ ,r ′′ + v ′′ } ˜ p ′′ ∈{− ˜ r ′′ , ˜ r ′′ + v ′′ } δ p ′′ + r, ˜ p ′′ h X p ′ ∈{− r ′ ,r ′ + v ′ } p ′′ ∈{− r ′′ ,r ′′ + v ′′ } δ p ′ ,p ′′ + X p ′ ∈{− r ′ ,r ′ + v ′ } ˜ p ′′ ∈{− ˜ r ′′ , ˜ r ′′ + v ′′ } δ p ′ , ˜ p ′′ i ≤ CN − κ k σ S k h X r ′ ∈ P H | r ′ | − ih sup r ′ ∈ Λ ∗ X r ∈ Λ ∗ ,r = − r ′ | b V ( r/N − κ ) || r + r ′ | i ≤ CN κ/ − ε . (5.17)33s for W , there are ( m − m − 2) possible choices of the indices j, k in (5.14), allleading to the same contribution. We fix j = m − k = m − 2. Estimating now θ (cid:0) { r j , v j } m − j =1 (cid:1) ≤ θ (cid:0) { r j , v j } m − j =1 (cid:1) , we obtain, with (3.12),W k ξ ν k ≤ CN − X r ∈ Λ ∗ N κ | b V ( r/N − κ ) |× X r ′ ,r ′′ ,r ′′′ , ˜ r ′′′ ∈ P H v ′ ,v ′′ ,v ′′′ ∈ P S ( η r ′ + η r ′ + v ′ ) ( η r ′′ + η r ′′ + v ′′ ) | η r ′′′ || η ˜ r ′′′ | σ v ′ σ v ′′ σ v ′′′ × X p ′′′ ∈{− r ′′′ ,r ′′′ + v ′′′ } ˜ p ′′′ ∈{− ˜ r ′′′ , ˜ r ′′′ + v ′′′ } δ p ′′′ + r, ˜ p ′′′ h X p ′ ∈{− r ′ ,r ′ + v ′ } p ′′′ ∈{− r ′′′ ,r ′′′ + v ′′′ } δ p ′′′ , − p ′ + v ′′ + X p ′ ∈{− r ′ ,r ′ + v ′ } ˜ p ′′′ ∈{− ˜ r ′′′ , ˜ r ′′′ + v ′′′ } δ ˜ p ′′′ , − p ′ + v ′′ i ≤ CN − κ k σ S k k η H k X r ′ ∈ P H | r ′ | − ≤ CN κ/ · N κ − ε . (5.18)Analogously, with Lemma 2.2 and (5.15), we findW k ξ ν k ≤ CN − X r ∈ Λ ∗ N κ | b V ( r/N − κ ) |× X r ′ ,r ′′ ,r ′′′ , ˜ r ′′′ ∈ P H v ′ ,v ′′ ,v ′′′ ∈ P S ( η r ′ + η r ′ + v ′ ) ( η r ′′ + η r ′′ + v ′′ ) | η r ′′′ || η ˜ r ′′′ | σ v ′ σ v ′′ σ v ′′′ × X p ′′′ ∈{− r ′′′ ,r ′′′ + v ′′′ } ˜ p ′′′ ∈{− ˜ r ′′′ , ˜ r ′′′ + v ′′′ } δ p ′′′ + r, ˜ p ′′′ X p ′ ∈{− r ′ ,r ′ + v ′ } p ′′ ∈{− r ′′ ,r ′′ + v ′′ } δ v ′′′ ,p ′ + p ′′ ≤ CN − κ k σ S k X r ′ ∈ P H | r ′ | − ≤ CN κ/ · N κ − ε . Together with (5.16), (5.17), (5.18), we conclude that | J V | ≤ CN κ/ · max { N − ε , N κ − ε } . (5.19)Finally, we consider the term V , associated with the second case listed after (5.9).We fix i = m and j = m − a p with a ∗ ˜ p m , of a q + r with a ∗ ˜ p m − and of a ∗ q , a ∗ p + r with a p m , a p m − , where ˜ p ℓ ∈ {− ˜ r ℓ , ˜ r ℓ + v ℓ } and34 ℓ ∈ {− r ℓ , r ℓ + v ℓ } , for ℓ = m, m − 1. We obtain (cid:10) ξ ν , V ξ ν (cid:11) = 12 N X m ≥ m − N m X v ∈ P S ,r , ˜ r ∈ P H : r + v , ˜ r + v ∈ P H · · · X v m ∈ P S ,r m , ˜ r m ∈ P H : r m + v m , ˜ r m + v m ∈ P H θ (cid:0) { r j , v j } mj =1 (cid:1) θ (cid:0) { ˜ r j , v j } mj =1 (cid:1) × m − Y i =1 η r i η ˜ r i (cid:0) δ ˜ r i ,r i + δ − ˜ r i ,r i + v i (cid:1) σ v i Y j = m,m − η r j η ˜ r j σ v j × X r ∈ Λ ∗ , p,q ∈ P H : p − r,q − r ∈ P H N κ b V ( r/N − κ ) X ˜ p ℓ ∈{− ˜ r ℓ , ˜ r ℓ + v ℓ } ℓ = m − ,m δ p, ˜ p m δ q + r, ˜ p m − × X p ℓ ∈{− r ℓ ,r ℓ + v ℓ } ℓ = m − ,m (cid:0) δ q,p m δ p + r,p m − + δ q,p m − δ p + r,p m (cid:1)(cid:0) δ ˜ p m ,p m + δ − ˜ p m + v m , − p m − + v m − (cid:1) . Estimating θ (cid:0) { r j , v j } mj =1 (cid:1) θ (cid:0) { ˜ r j , v j } mj =1 (cid:1) ≤ θ (cid:0) { r j , v j } m − j =1 (cid:1) and using Lemma 2.2 and thecondition 3 κ − ε < 0, we find | (cid:10) ξ ν , V ξ ν (cid:11) |k ξ ν k ≤ CN − X r ∈ Λ ∗ N κ b V ( r/N − κ ) X r ′ , ˜ r ′ ,r ′′ , ˜ r ′′ ∈ P H v ′ ,v ′′ ∈ P S | η r ′ || η ˜ r ′ || η r ′′ || η ˜ r ′′ | σ v ′ σ v ′′ × X p ′ ∈{− r ′ ,r ′ + v ′ } p ′′ ∈{− r ′′ ,r ′′ + v ′′ } X ˜ p ′ ∈{− ˜ r ′ , ˜ r ′ + v ′ } ˜ p ′′ ∈{− ˜ r ′′ , ˜ r ′′ + v ′′ } ( δ ˜ p ′ ,p ′′ + r δ ˜ p ′′ + r,p ′ + δ ˜ p ′ ,p ′ + r δ ˜ p ′′ + r,p ′′ ) × ( δ ˜ p ′′ ,p ′′ + δ − ˜ p ′′ + v ′′ , − p ′ + v ′ ) ≤ CN − κ k η H k k σ S k ≤ CN κ − ε ≤ CN κ/ − ε . With (5.13) and (5.19), we obtain (5.3). A Proof of Proposition 1.2 The proof of Prop. 1.2 is based on standard results, which are collected in this sectionfor the reader convenience. In particular we follow [15] (see Lemma 2.1.3) and [17, Sec.12] for Lemma A.1 and A.2 and the proof of Lemma 3.3.2 in [1] for Lemma A.4 (controlon the second moment of N allows us to avoid the condition imposed in [1] that V isstrictly positive around the origin).The proof of Prop. 1.2 is divided in three parts. First, we show how to switch fromperiodic boundary conditions to Dirichlet boundary conditions, increasing a bit the sizeof the box. In the second step, we replicate the Dirichlet trial state obtained in thefirst step, to obtain an upper bound on the energy in a sequence of boxes, whose size35ncreases to infinity (but with fixed density). In the last step, we show how to pass fromthe grand canonical to the canonical setting.Let Ψ L = { Ψ ( n ) L } n ≥ ∈ F (Λ L ) be a normalized trial state for the Fock-space Hamilto-nian H defined on the box Λ L with periodic boundary conditions (in fact, we denote byΨ ( n ) L ( x , . . . , x n ) the L -periodic extension of Ψ ( n ) L to the whole space R n ). For u ∈ Λ L ,we define Ψ DL +2 ℓ,u ∈ F (Λ uL +2 ℓ ), where Λ uL +2 ℓ = u + Λ L +2 ℓ is a box centered at u , withside length L + 2 ℓ , setting, for any n ∈ N ,(Ψ D L +2 ℓ,u ) ( n ) ( x , . . . , x n ) = Ψ ( n ) L ( x , . . . , x n ) n Y i =1 Q L,ℓ ( x i − u ) (A.1)where Q L,ℓ ( x i ) = Q j =1 q L,ℓ ( x ( j ) i ) with q L,ℓ : R → [0; 1] defined by q L,ℓ ( t ) = cos (cid:0) π ( t + L/ − ℓ )4 ℓ (cid:1) if (cid:12)(cid:12) t + L (cid:12)(cid:12) ≤ ℓ | t | < L − ℓ cos (cid:0) π ( t − L/ ℓ )4 ℓ (cid:1) if (cid:12)(cid:12) t − L (cid:12)(cid:12) ≤ ℓ Dir L +2 ℓ,u ) ( n ) satisfies Dirichlet boundary condition on the box Λ uL +2 ℓ . Thefollowing lemma allows us to compare energy and moments of the number of particlesof Ψ D L +2 ℓ,u with those of Ψ L . Lemma A.1. Under the assumptions of Prop. 1.2, let Ψ D L +2 ℓ,u be defined as in (A.1) with u ∈ Λ L . Then we have k Ψ D L +2 ℓ,u k = 1 . Moreover for all j ∈ N (cid:10) Ψ D L +2 ℓ,u , N j Ψ D L +2 ℓ,u (cid:11) = (cid:10) Ψ L , N j Ψ L (cid:11) , and there exists ¯ u ∈ Λ L such that (cid:10) Ψ D L +2 ℓ, ¯ u , H Ψ D L +2 ℓ, ¯ u (cid:11) ≤ (cid:10) Ψ L , H Ψ L (cid:11) + CLℓ (cid:10) Ψ L , N Ψ L (cid:11) (A.2) for a universal constant C > .Proof. For an arbitrary L -periodic function ψ ∈ L ( R ), we find Z L + ℓ − L − ℓ dt | ψ ( t ) | q ( t ) = Z L − L dt | ψ ( t ) | . (A.3)To prove (A.3), we combine (using the periodicity of ψ ) the integral over [ − L/ − ℓ ; − L/ L/ − ℓ ; L/ 2] and the integral over [ − L/ − L/ ℓ ] with theintegral over [ L/ L/ ℓ ] (using that cos x + cos ( x − π/ 2) = 1).Applying (A.3) (separately on each variable), we obtain that k (Ψ D L +2 ℓ,u ) ( n ) k = k Ψ ( n ) L k ,for all n ∈ N . This implies that k Ψ D L +2 ℓ,u k = k Ψ L k = 1 and that h Ψ D L +2 ℓ,u , N j Ψ D L +2 ℓ,u i = h Ψ L , N j Ψ L i for all j ∈ N . 36o compute the expectation of the kinetic energy in the state Ψ D L +2 ℓ,u , we observethat, for any L -periodic ψ ∈ L ( R ) with ψ ′ ∈ L ( R ), we have (since ψ ′ is also L -periodic) Z L + ℓ − L − ℓ dt | ( qψ ) ′ ( t ) | = Z L L dt | ψ ′ ( t ) | + Z L + ℓ − L − ℓ dt (cid:2) | ψ ( t ) | q ′ ( t ) + q ( t ) q ′ ( t ) ddt | ψ ( t ) | (cid:3) where we used periodicity of ψ ′ and (A.3). Integrating by parts and using q ( ± ( L/ ℓ )) = q ′ ( ± ( L/ − ℓ )) = 0, we get Z L + ℓ − L − ℓ dt | ( qψ ) ′ ( t ) | = Z L − L dt | ψ ′ ( t ) | − Z L + ℓ − L − ℓ dt | ψ ( t ) | q ( t ) q ′′ ( t ) ≤ Z L − L dt | ψ ′ ( t ) | + Cℓ Z R dt | ψ ( t ) | χ L,ℓ ( t ) (A.4)where χ L,ℓ ( t ) = χ ℓ ( t + L/ 2) + χ ℓ ( t − L/ 2) with χ r ( t ) the characteristic function of [ − r, r ]and we used | q ′′ ( t ) | ≤ Cℓ − χ L,ℓ ( t ). Applying (A.4) (separately in every direction), weobtain k∇ x j (Ψ D L +2 ℓ,u ) ( n ) k ≤ k∇ x j (Ψ D L +2 ℓ,u ) ( n ) k + Cℓ Z R dx j e χ L,ℓ ( x j − u ) Z Λ n − L dx . . . dx j − dx j +1 . . . dx n | Ψ ( n ) L ( x , . . . , x n ) | (A.5)where we defined e χ L,ℓ ( x ) = P k =1 χ L,ℓ ( x ( k ) ) Q j = k χ L ( x ( j ) ).To compute the potential energy of ψ L , we have to consider the L -periodic extension V L ( x ) = P m ∈ Z V ( x + mL ) of V . Since we assumed V to be positive and supported in B R (0) and that L > R , we get V ( x ) ≤ V L ( x ) which implies that, for any i = j , | (Ψ D L +2 ℓ,u ) ( n ) ( x , . . . , x n ) | V ( x i − x j ) ≤ (cid:12)(cid:12)(cid:12) Ψ ( n ) L ( x , . . . , x n ) q V L ( x i − x j ) (cid:12)(cid:12)(cid:12) n Y k =1 Q L,ℓ ( x k − u ) . Applying (A.3), we obtain Z (Λ uL +2 ℓ ) n dx . . . dx n | (Ψ D L +2 ℓ,u ) ( n ) ( x , . . . , x n ) | V ( x i − x j ) ≤ Z Λ nL dx . . . dx n | Ψ ( n ) L ( x , . . . , x n ) | V L ( x i − x j ) . (A.6)From (A.5) and (A.6), we conclude (using the bosonic symmetry) (cid:10) Ψ D L +2 ℓ,u , H Ψ D L +2 ℓ,u (cid:11) ≤ (cid:10) Ψ L , H Ψ L (cid:11) + Cℓ X n ≥ n Z R dx e χ L,ℓ ( x − u ) Z Λ n − L dx . . . dx n | Ψ ( n ) L ( x , . . . , x n ) | . u ∈ Λ L we conclude (since k e χ L,ℓ k ≤ CL ℓ ) Z Λ L du (cid:10) Ψ D L +2 ℓ,u , H Ψ D L +2 ℓ,u (cid:11) ≤ L (cid:10) Ψ L , H Ψ L (cid:11) + CL ℓ h Ψ L , N Ψ L i . Hence, there exists ¯ u ∈ Λ L so that (A.2) holds.From now on, let us define Ψ D L +2 ℓ ∈ F (Λ L +2 ℓ ), setting (Ψ D L +2 ℓ ) ( n ) ( x , . . . , x n ) =(Ψ D L +2 ℓ, ¯ u ) ( n ) ( x − ¯ u, . . . , x n − ¯ u ), with Ψ D L +2 ℓ, ¯ u from Lemma A.1. Since Ψ D L +2 ℓ satisfiesDirichlet boundary conditions, we can replicate it into several adjacent copies of Λ L +2 ℓ ,separated by corridors of size R (to avoid interactions between different boxes). Thisallows us to construct a sequence of trial states on boxes, with increasing volume (butkeeping the density fixed).Let t ∈ N and ˜ L = t ( L + 2 ℓ + R ). We think of the large box Λ ˜ L as the (almost)disjoint union of t shifted copies of the small box Λ L +2 ℓ + R , centered at( − ˜ L/ , − ˜ L/ , − ˜ L/ 2) + ( L + 2 ℓ + R ) · ( i − / , i − / , i − / 2) (A.7)with i , i , i ∈ { , . . . , t } . Let { c i } t i =1 denote an enumeration of the centers (A.7). Wedefine Ψ D˜ L ∈ F (Λ ˜ L ) by setting(Ψ D˜ L ) ( m ) ( x , . . . , x m ) = 1 k (Ψ D L +2 ℓ ) ( n ) k t − t Y i =1 (Ψ D L +2 ℓ ) ( n ) ( x ( i − n +1 − c i , . . . , x in − c i )(A.8)if m = nt for an n ∈ N , and (Ψ D˜ L ) ( m ) = 0 otherwise (here we set (Ψ D L +2 ℓ ) ( n ) = 0 ifone of its arguments lies outside Λ L +2 ℓ ). More precisely, (Ψ D˜ L ) ( m ) should be definedas the symmetrization of (A.8) (but we can use (A.8) to compute the expectation ofpermutation symmetric observables). Lemma A.2. Under the assumptions of Prop. 1.2, let Ψ D ˜ L be defined as above. Then k Ψ D ˜ L k = 1 , (cid:10) Ψ D˜ L , N j Ψ D˜ L (cid:11) = t j (cid:10) Ψ D L +2 ℓ , N j Ψ D L +2 ℓ (cid:11) for all j ∈ N , and (cid:10) Ψ D˜ L , H Ψ D˜ L (cid:11) = t (cid:10) Ψ D L +2 ℓ , H Ψ D L +2 ℓ (cid:11) . (A.9) Proof. From the definition (A.8), we have k (Ψ D˜ L ) ( nt ) k = k (Ψ D L +2 ℓ ) ( n ) k for all n ∈ N .Since (Ψ D˜ L ) ( m ) = 0, if m = nt , we conclude that k Ψ D˜ L k = k Ψ D L +2 ℓ k = 1 and also that,for j ∈ N , (cid:10) Ψ D˜ L , N j Ψ D˜ L (cid:11) = X n ≥ ( t n ) j k (Ψ D˜ L ) ( t n ) k = t j X n ≥ n j k (Ψ D L +2 ℓ ) ( n ) k = t j (cid:10) Ψ D L +2 ℓ , N j Ψ D L +2 ℓ (cid:11) . To prove (A.9), we observe, first of all, that for any i = 1 , . . . , nt ,when the operator ∇ x i acts on (Ψ D˜ L ) ( nt ) , it only hits one of the factor (Ψ D L +2 ℓ ) ( n ) on the r.h.s. of (A.8).38imilarly, for any i, j ∈ { , . . . , m } , the operator V ( x i − x j ) has non-zero expectationin the state (Ψ D˜ L ) ( nt ) only if x i , x j are arguments of the same factor (Ψ D L +2 ℓ ) ( n ) onthe r.h.s. of (A.8) (this observation is exactly the reason for introducing corridors ofsize R between the small boxes, where the wave function vanishes). We conclude that (cid:10) Ψ D˜ L , H Ψ D˜ L (cid:11) = t (cid:10) Ψ D L +2 ℓ , H Ψ D L +2 ℓ (cid:11) , as claimed.Finally, in Lemma A.4 we show how to obtain an upper bound for the ground stateenergy per particle in the canonical ensemble, starting from a trial state in the grand-canonical setting. Recall the notation E ( N, L ) for the ground state energy of the Hamil-tonian (1.1), describing N particles in the box Λ L , with Dirichlet boundary conditions.For ρ > ρL ∈ N , we introduce the notation e L ( ρ ) = E ( ρL , L ) L . Comparing with the definition (1.2), we find e ( ρ ) = lim L →∞ e L ( ρ ) (where the limit hasto be taken along sequences of L , with ρL ∈ N ). In the proof of Lemma A.4 we use theexistence of the thermodynamic limit of the specific energy and its convexity (see [16,Thm. 3.5.8 and 3.5.11]), together with the following result on the Legendre transformof convex functions. Lemma A.3. Let D ⊂ R be a closed interval, f : D → R be convex and continuous(also at the boundary of D ). We define the Legendre transform f ∗ : R → R of f by f ∗ ( y ) = sup x ∈ D [ xy − f ( x )] (A.10) Then f ∗ is well-defined (because, by continuity, x → xy − f ( x ) is bounded on D , for all y ∈ R ) and, for all x ∈ D , f ( x ) = sup y ∈ R [ xy − f ∗ ( y )] . (A.11) Proof. By definition of f ∗ , we have f ∗ ( y ) ≥ xy − f ( x ) for all x ∈ D, y ∈ R . This impliesthat f ( x ) ≥ xy − f ∗ ( y ) for all x ∈ D, y ∈ R and therefore that f ( x ) ≥ sup y ∈ R [ xy − f ∗ ( y )] (A.12)for all x ∈ D . On the other hand, fix x ∈ D and t ≤ f ( x ). Then, by convexity of f (and by its continuity at the boundaries of D ), we find a line through ( x , t ) lying belowthe graph of f . In other words, there exists y ∈ R such that f ( x ) ≥ t + y ( x − x ) for all x ∈ D . Thus yx − t ≥ yx − f ( x ) for all x ∈ D , which implies that yx − t ≥ f ∗ ( y )and therefore that t ≤ yx − f ∗ ( y ). In particular, t ≤ sup y ∈ R [ yx − f ∗ ( y )]. Since t ≤ f ( x ) was arbitrary, we conclude that f ( x ) ≤ sup y ∈ R [ yx − f ∗ ( y )]. With (A.12) ,we obtain that f ( x ) = sup y ∈ R [ xy − f ∗ ( y )] for all x ∈ D .39 emma A.4. Under the assumptions of Prop. 1.2, fix ρ > and suppose that thereexists a sequence Ψ D L ∈ F (Λ L ) (parametrized by L with ρL ∈ N ), satisfying Dirichletboundary conditions, such that h Ψ D L , N Ψ D L i ≥ ρ (1 + c ′ ρ ) L , h Ψ D L , N Ψ D L i ≤ C ′ ( ρL ) . (A.13) for some constants c ′ , C ′ > . Then we have e ( ρ ) ≤ lim L →∞ h Ψ D L , H Ψ D L i L . Proof. Using positivity of H , we have, for any µ ≥ M > (cid:10) Ψ D L , H Ψ D L (cid:11) L ≥ µL (cid:10) Ψ D L , N Ψ D L (cid:11) + (cid:10) Ψ D L , ( H − µ N ) χ ( N ≤ M L )Ψ D L (cid:11) L − µL (cid:10) Ψ D L , N χ ( N > M L )Ψ D L (cid:11) ≥ µL (cid:10) Ψ D L , N Ψ D L (cid:11) + ML X m =0 (cid:18) e L (cid:18) mL (cid:19) − µ mL (cid:19)(cid:13)(cid:13)(cid:13) (Ψ D L ) ( m ) (cid:13)(cid:13)(cid:13) − µM L (cid:10) Ψ D L , N Ψ D L (cid:11) , (A.14)where we used the inequality χ ( N > M L ) ≤ N / ( M L ). Hence, with (A.13) and fixing M large enough (depending on c ′ , C ′ ) we find (cid:10) Ψ D L , H Ψ D L (cid:11) L ≥ µρ + ML X m =0 (cid:18) e L (cid:18) mL (cid:19) − µ mL (cid:19)(cid:13)(cid:13)(cid:13) (Ψ D L ) ( m ) (cid:13)(cid:13)(cid:13) . (A.15)Next, we claim that e L ( ρ ) ≥ (cid:16) RL (cid:17) e (cid:18) ρ (cid:16) RL (cid:17) − (cid:19) . (A.16)Indeed, starting from an arbitrary normalized trial state ψ describing N = ρL particlesin a box of side length L , with Dirichlet boundary conditions, we can construct, forany r ∈ N , a trial state describing N ′ = N r = ρL r particles in a box of side length L ′ = r ( L + R ), again with Dirichlet boundary conditions, by placing r copies of thestate ψ in adjacent boxes and using that (thanks to the corridors of size R betweenthe boxes) particles in different boxes do not interact. This construction is very similarto the one presented around Lemma A.2 (the difference is that here we work in thecanonical setting, which makes things slightly simpler). Since N ′ = [ ρ/ (1 + R/L ) ] L ′ ,optimizing the choice of ψ , we obtain that E ([ ρ/ (1 + R/L ) ] L ′ , L ′ ) ≤ r E ( ρL , L ) andtherefore that e L ′ ( ρ/ (1 + R/L ) ) ≤ e L ( ρ ) / (1 + R/L ) . Taking the limit L ′ → ∞ (along the sequence L ′ = r ( L + R ), r ∈ N ), we obtain (A.16).Then (A.15) and (A.16) yield (cid:10) Ψ D L , H Ψ D L (cid:11) L ≥ µρ − (cid:16) RL (cid:17) e ∗ ( µ ) . e ∗ denotes the Legendre transform of e : D → R , defined on the domain D =[0 , M ], as in (A.10) (here we use the convexity of the specific energy e ). It follows thatlim L → + ∞ (cid:10) Ψ D L , H Ψ D L (cid:11) L ≥ µρ − e ∗ ( µ )for all µ ≥ 0. Thuslim L → + ∞ (cid:10) Ψ D L , H Ψ D L (cid:11) L ≥ sup µ ≥ h µρ − e ∗ ( µ ) i = sup µ ∈ R h µρ − e ∗ ( µ ) i = e ( ρ )where we used the fact that e ∗ (0) = 0 (because e ( ρ ) ≥ ρ ≥ e (0) = 0) and e ∗ ( µ ) ≥ − e (0) = 0 for all µ ∈ R in the second step and Lemma A.3 in the third step.With Lemmas A.1, A.2 and A.4 we are ready to show Prop. 1.2. Proof of Prop. 1.2. Given a normalized Ψ L ∈ F (Λ L ) satisfying periodic boundary con-ditions with h Ψ L , N Ψ L i ≥ ρ (1 + c ′ ρ )( L + 2 ℓ + R ) , h Ψ L , N Ψ L i ≤ C ′ ρ ( L + 2 ℓ + R ) we find with Lemma A.1 a normalized Ψ D L +2 ℓ ∈ F (Λ L +2 ℓ ) satisfying Dirichlet conditionssuch that h Ψ L +2 ℓ , N Ψ L +2 ℓ i ≥ ρ (1 + c ′ ρ )( L + 2 ℓ + R ) , h Ψ L +2 ℓ , N Ψ L +2 ℓ i ≤ C ′ ρ ( L + 2 ℓ + R ) and h Ψ D L +2 ℓ , H Ψ D L +2 ℓ i ≤ h Ψ L , H Ψ L i + CLℓ h Ψ L , N Ψ L i . With Lemma A.2, we obtain a sequence Ψ D˜ L ∈ F (Λ ˜ L ), with ˜ L = t ( L + 2 ℓ + R ) for t ∈ N ,such that h Ψ D˜ L , N Ψ D˜ L i ≥ ρ (1 + c ′ ρ ) ˜ L , h Ψ D˜ L , N Ψ D˜ L i ≤ C ′ ρ ˜ L and h Ψ D˜ L , H Ψ D˜ L i ≤ t h Ψ L , H Ψ L i + Ct Lℓ h Ψ L , N Ψ L i . With Lemma A.4, we conclude that e ( ρ ) ≤ lim ˜ L →∞ h Ψ D˜ L , H Ψ D˜ L i ˜ L ≤ ℓ/L + R/L ) (cid:20) h Ψ L , H Ψ L i L + CL ℓ h Ψ L , N Ψ L i (cid:21) ≤ h Ψ L , H Ψ L i L + CL ℓ h Ψ L , N Ψ L i . eferences [1] A. 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