Bose-Einstein condensation for two dimensional bosons in the Gross-Pitaevskii regime

We consider systems of N bosons trapped on the two-dimensional unit torus, in the Gross-Pitaevskii regime, where the scattering length of the repulsive interaction is exponentially small in the number of particles. We show that low-energy states exhibit complete Bose-Einstein condensation, with almost optimal bounds on the number of orthogonal excitations.


Introduction
We consider N ∈ N bosons trapped in the two-dimensional box Λ = [−1/2; 1/2] 2 with periodic boundary conditions. In the Gross-Pitaevskii regime, particles interact through a repulsive pair potential, with a scattering length exponentially small in N . The Hamilton operator is given by and acts on a dense subspace of L 2 s (Λ N ), the Hilbert space consisting of functions in L 2 (Λ N ) that are invariant with respect to permutations of the N particles. We assume here V ∈ L 3 (R 2 ) to be compactly supported and pointwise non-negative (i.e. V (x) ≥ 0 for almost all x ∈ R 2 ).
We denote by a the scattering length of the unscaled potential V . We recall that in two dimensions and for a potential V with finite range R 0 , the scattering length is defined by 2π log(R/a) = inf φ BR where R > R 0 , B R is the disk of radius R centered at the origin and the infimum is taken over functions φ ∈ H 1 (B R ) with φ(x) = 1 for all x with |x| = R. The unique minimizer of the variational problem on the r.h.s. of (1.2) is non-negative, radially symmetric and satisfies the scattering equation in the sense of distributions. For R 0 < |x| ≤ R, we have φ (R) (x) = log(|x|/a) log(R/a) .
By scaling, φ N (x) := φ (e N R) (e N x) is such that We have for all x ∈ R 2 with e −N R 0 < |x| ≤ R. Here a N = e −N a. The properties of trapped two dimensional bosons in the Gross-Pitaevskii regime (in the more general case where the bosons are confined by external trapping potentials) have been first studied in [10,11,12]. These results can be translated to the Hamilton operator (1.1), defined on the torus, with no external potential. They imply that the ground state energy E N of (1.1) is such that Moreover, they imply Bose-Einstein condensation in the zero-momentum mode ϕ 0 (x) = 1 for all x ∈ Λ, for any approximate ground state of (1.1). More precisely, it follows from [11] that, for any sequence ψ N ∈ L 2 s (Λ N ) with ψ N = 1 and lim N →∞ 1 N ψ N , H N ψ N = 2π, (1.4) the one-particle reduced density matrix γ N = tr 2,...,N |ψ N ψ N | is such that for a sufficiently smallδ > 0. The estimate (1.5) states that, in many-body states satisfying (1.4) (approximate ground states), almost all particles are described by the one-particle orbital ϕ 0 , with at most N 1−δ ≪ N orthogonal excitations. For V ∈ L 3 (R 2 ), our main theorem improves (1.3) and (1.5) by providing more precise bounds on the ground state energy and on the number of excitations. Theorem 1.1. Let V ∈ L 3 (R 2 ) have compact support, be spherically symmetric and pointwise non-negative. Then there exists a constant C > 0 such that the ground state energy E N of (1.1) satisfies Furthermore, consider a sequence ψ N ∈ L 2 s (Λ N ) with ψ N = 1 and such that ψ N , H N ψ N ≤ 2πN + K for a K > 0. Then the reduced density matrix γ N = tr 2,...,N |ψ N ψ N | associated with ψ N is such that for all N ∈ N large enough.
It is interesting to compare the Gross-Pitaevskii regime with the thermodynamic limit, where a Bose gas of N particles interacting through a fixed potential with scattering length a is confined in a box with volume L 2 , so that N, L → ∞ with the density ρ = N/L 2 kept fixed. Let b = | log(ρa 2 )| −1 . Then, in the dilute limit ρa 2 ≪ 1, the ground state energy per particle in the thermodynamic limit is expected to satisfy e 0 (ρ) = 4πρ 2 b 1 + b log b + 1/2 + 2γ + log π b + o(b) , (1.8) with γ the Euler's constant. The leading order term on the r.h.s. of (1.8) has been first derived in [17] and then rigorously established in [14], with an error rate b −1/5 . The corrections up to order b have been predicted in [1,15,16]. To date, there is no rigorous proof of (1.8). Some partial result, based on the restriction to quasi-free states, has been recently obtained in [7,Theorem 1]. Extrapolating from (1.8), in the Gross-Pitaevskii regime we expect |E N −2πN | ≤ C. While our estimate (1.6) captures the correct lower bound, the upper bound is off by a logarithmic correction. Eq. (1.7), on the other hand, is expected to be optimal (but of course, by (1.6), we need to choose K = C log N to be sure that (1.1) can be satisfied). This bound can be used as starting point to investigate the validity of Bogoliubov theory for two dimensional bosons in the Gross-Pitaevskii regime, following the strategy developed in [4] for the three dimensional case; we plan to proceed in this direction in a separate paper.
The proof of Theorem 1.1 follows the strategy that has been recently introduced in [3] to prove condensation for three-dimensional bosons in the Gross-Pitaevskii limit. There are, however, additional obstacles in the two-dimensional case, requiring new ideas. To appreciate the difference between the Gross-Pitaevskii regime in twoand three-dimensions, we can compute the energy of the trivial wave function ψ N ≡ 1. The expectation of (1.1) in this state is of order N 2 . It is only through correlations that the energy can approach (1.6). Also in three dimensions, uncorrelated manybody wave functions have large energy, but in that case the difference with respect to the ground state energy is only of order N (N V (0)/2 rather than 4πa 0 N ). This observation is a sign that correlations in two-dimensions are stronger and play a more important role than in three dimensions (this creates problems in handling error terms that, in the three dimensional setting, were simply estimated in terms of the integral of the potential).
The paper is organized as follows. In Sec. 2 we introduce our setting, based on a description of orthogonal excitations of the condensate on a truncated Fock space. In Sec. 3 and 4 we show how to renormalize the excitation Hamiltonian, to regularise the singular interaction. Finally, in Sec. 5, we show our main theorem. Sec. 6 and App. A contain the proofs of the results stated in 3 and 4, respectively. Finally, in App. B we establish some properties of the solution of the Neumann problem associated with the two-body potential V .
Acknowledgment. We are thankful to A. Olgiati for discussions on the two dimensional scattering equation. C.C. and S.C. gratefully acknowledge the support from the GNFM Gruppo Nazionale per la Fisica Matematica -INDAM through the project "Derivation of effective theories for large quantum systems". B. S. gratefully acknowledges partial support from the NCCR SwissMAP, from the Swiss National Science Foundation through the Grant "Dynamical and energetic properties of Bose-Einstein condensates" and from the European Research Council through the ERC-AdG CLaQS.

The Excitation Hamiltonian
Low-energy states of (1.1) exhibit condensation in the zero-momentum mode ϕ 0 defined by ϕ 0 (x) = 1 for all x ∈ Λ = [−1/2; 1/2] 2 . Similarly as in [8,2,3], we are going to describe excitations of the condensate on the truncated bosonic Fock space constructed on the orthogonal complement L 2 ⊥ (Λ) of ϕ 0 in L 2 (Λ). To reach this goal, we define a unitary map U N : With the usual creation and annihilation operators, we can write and that U * N U N = 1, i.e. U N is unitary. With U N , we can define the excitation Hamiltonian L N := U N H N U * N , acting on a dense subspace of F ≤N + . To compute the operator L N , we first write the Hamiltonian (1.1) in momentum space, in terms of creation and annihilation operators a * p , a p , for momenta p ∈ Λ * = 2πZ 2 . We find is the Fourier transform of V , defined for all k ∈ R 2 (in fact, (1.1) is the restriction of (2.1) to the N -particle sector of the Fock space). We can now determine L N using the following rules, describing the action of the unitary operator U N on products of a creation and an annihilation operator (products of the form a * p a q can be thought of as operators mapping L 2 s (Λ N ) to itself). For any p, q ∈ Λ * + = 2πZ 2 \{0}, we find (see [8]): (2.2) where N + = p∈Λ * + a * p a p is the number of particles operator on F ≤N + . We conclude that p,q∈Λ * + ,r∈Λ * : r =−p,−q V (r/e N )a * p+r a * q a p a q+r , where we introduced generalized creation and annihilation operators for all p ∈ Λ * + . On states exhibiting complete Bose-Einstein condensation in the zero-momentum mode ϕ 0 , we have a 0 , a * 0 ≃ √ N and we can therefore expect that b * p ≃ a * p and that b p ≃ a p . From the canonical commutation relations for the standard creation and annihilation operators a p , a * p , we find It is also useful to notice that the operators b * p , b p , like the standard creation and annihilation operators a * p , a p , can be bounded by the square root of the number of particles operators; we find

Quadratic renormalization
From (2.4) we see that conjugation with U N extracts, from the original quartic interaction in (2.1), some large constant and quadratic contributions, collected in L N respectively. In particular, the expectation of L N on the vacuum state Ω is of order N 2 , this being an indication of the fact that there are still large contributions to the energy hidden among cubic and quartic terms in L Since U N only removes products of the zero-energy mode ϕ 0 , correlations among particles remain in the excitation vector U N ψ N . Indeed, correlations play a crucial role in the two dimensional Gross-Pitaevskii regime and carry an energy of order N 2 .
To take into account the short scale correlation structure on top of the condensate, we consider the ground state f ℓ of the Neumann problem on the ball |x| ≤ e N ℓ, normalized so that f ℓ (x) = 1 for |x| = e N ℓ. Here and in the following we omit the N -dependence in the notation for f ℓ and for λ ℓ . By scaling, we observe that f ℓ (e N ·) satisfies on the ball |x| ≤ ℓ. We choose 0 < ℓ < 1/2, so that the ball of radius ℓ is contained in the box Λ = [−1/2; 1/2] 2 . We extend then f ℓ (e N .) to Λ, by setting f N,ℓ (x) = f ℓ (e N x), if |x| ≤ ℓ and f N,ℓ (x) = 1 for x ∈ Λ, with |x| > ℓ. Then where χ ℓ is the characteristic function of the ball of radius ℓ. The Fourier coefficients of the function f N,ℓ are given by for all p ∈ Λ * . We introduce also the function w ℓ (x) = 1 − f ℓ (x) for |x| ≤ e N ℓ and extend it by setting w ℓ (x) = 0 for |x| > e N ℓ. Its re-scaled version is defined by The Fourier coefficients of the re-scaled function w N,ℓ are given by (3.4) where we used the notation χ ℓ for the Fourier coefficients of the characteristic function on the ball of radius ℓ. Note that χ ℓ (p) = ℓ 2 χ(ℓp) with χ(p) the Fourier coefficients of the characteristic function on the ball of radius one.
In the next lemma, we collect some important properties of the solution of (3.1).
be non-negative, compactly supported (with range R 0 ) and spherically symmetric, and denote its scattering length by a. Fix 0 < ℓ < 1/2, N sufficiently large and let f ℓ denote the solution of (3.2). Then ii) We have iii) There exist a constant C > 0 such that iv) There exists a constant C > 0 such that . Then the Fourier coefficients of the function w N,ℓ defined in (3.3) are such that Proof. The proof of points i)-iv) is deferred in Appendix B. To prove point v) we use the scattering equation (3.4): Using the fact that e 2N λ ℓ ≤ Cℓ −2 | ln(e N ℓ/a)| −1 and that 0 ≤ f ℓ ≤ 1, we end up with We now defineη : Λ → R througȟ and in particular, recalling that e −N R 0 < ℓ ≤ 1/2, for all x ∈ Λ. Using (3.10) we find In the following we choose ℓ = N −α , for some α > 0 to be fixed later, so that This choice of ℓ will be crucial for our analysis, as commented below. Notice, on the other hand, that the H 1 -norms of η diverge, as N → ∞. From (3.9) and Lemma 3.1, part iv) we find for N ∈ N large enough. We denote with η : Λ * → R the Fourier transform ofη, or equivalently for all p ∈ Λ * + = 2πZ 2 \{0}, and for some constant C > 0 independent of N , if N is large enough. From (3.12) we also have (3.15) Moreover, (3.4) implies the relation (3.16) or equivalently, expressing also the other terms through the coefficients η p , (3.17) We will mostly use the coefficients η p with p = 0. Sometimes, however, it will be useful to have an estimate on η 0 (because Eq. (3.17) involves η 0 ). From (3.13) and Lemma 3.1, part iv) we find With the coefficients (3.13) we define the antisymmetric operator and we consider the unitary operator We refer to operators of the form (3.20) as generalized Bogoliubov transformations.
In contrast with the standard Bogoliubov transformations defined in terms of the standard creation and annihilation operators, operators of the form (3.20) leave the truncated Fock space F ≤N + invariant. On the other hand, while the action of standard Bogoliubov transformation on creation and annihilation operators is explicitly given by there is no such formula describing the action of generalized Bogoliubov transformations.
Conjugation with (3.20) leaves the number of particles essentially invariant, as confirmed by the following lemma.
Lemma 3.2. Assume B is defined as in (3.19), with η ∈ ℓ 2 (Λ * ) and η p = η −p for all p ∈ Λ * + . Then, for every n ∈ N there exists a constant C > 0 such that, on F ≤N + , as an operator inequality on F ≤N + . The proof of (3.22) can be found in [5, Lemma 3.1] (a similar result has been previously established in [19]).
With the generalized Bogoliubov transformation e B : F ≤N In the next proposition, we collect important properties G N,α . We will use the notation for the kinetic and potential energy operators, restricted on F ≤N + , and H N = K+V N . We also introduce a renormalized interaction potential ω N ∈ L ∞ (Λ), which is defined as the function with Fourier coefficients ω N for any p ∈ Λ * + , and with χ(p) the Fourier coefficients of the characteristic function of the ball of radius one. From (3.5) and ℓ = N −α one has |g N | ≤ C. Note in particular that the potential ω N (p) decays on momenta of order N α , which are much smaller than e N . From Lemma 3.1 parts i) and iii) we find be compactly supported, pointwise non-negative and spherically symmetric. Let G N,α be defined as in (3.23) and define G eff N,α := Then there exists a constant C > 0 such that for all α > 1, ξ ∈ F ≤N + and N ∈ N large enough.
The proof of Prop. 3.3 is very similar to the proof of [4,Prop. 4.2]. For completeness, we discuss the changes in Appendix A.

Cubic Renormalization
Conjugation through the generalized Bogoliubov transformation (3.21) renormalizes constant and off-diagonal quadratic terms on the r.h.s. of (3.28). In order to estimate the number of excitations N + through the energy and show Bose-Einstein condensation, we still need to renormalize the diagonal quadratic term (the part proportional to N V (0)N + , on the first line of (3.28)) and the cubic term on the last line of (3.28). To this end, we conjugate G eff N,α with an additional unitary operator, given by the exponential of the anti-symmetric operator with η p defined in (3.13). An important observation is that while conjugation with e A allows to renormalize the large terms in G N,α , it does not substantially change the number of excitations. The following proposition can be proved similarly to [3, Proposition 5.1].
Proposition 4.1. Suppose that A is defined as in (4.1). Then, for any k ∈ N there exists a constant C > 0 such that the operator inequality e −A (N + + 1) k e A ≤ C(N + + 1) k holds true on F ≤N + , for any α > 0 (recall the choice ℓ = N −α in the definition (3.13) of the coefficients η r ), and N large enough.
We will also need to control the growth of the expectation of the energy H N with respect to the cubic conjugation. This is the content of the following proposition, which is proved in subsection 6.1. for all α ≥ 1, s ∈ [0; 1] and N ∈ N large enough.
We use now the cubic phase e A to introduce a new excitation Hamiltonian, obtained by conjugating the main part G eff N,α of G N,α . We define on a dense subset of F ≤N + . Conjugation with e A renormalizes both the contribution proportional to N + (in the first line in the last line on the r.h.s. of (3.28)) and the cubic term on the r.h.s. of (3.28), effectively replacing the singular potential V (p/e N ) by the renormalized potential ω N (p) defined in (3.25). This follows from the following proposition. Proposition 4.3. Let V ∈ L 3 (R 2 ) be compactly supported, pointwise non-negative and spherically symmetric. Let R N,α be defined in (4.3) and define Then for ℓ = N −α and α > 2 there exists a constant C > 0 such that 5) for N ∈ N sufficiently large.
The proof of Proposition 4.3 will be given in Section 6. We will also need more detailed information on R eff N,α , as contained in the following proposition.
for all α > 2 and N ∈ N large enough.
Moreover, let f, g : R → [0; 1] be smooth, with f 2 (x) + g 2 (x) = 1 for all x ∈ R. For M ∈ N, let f M := f (N + /M ) and g M := g(N + /M ). Then there exists C > 0 such that for all α > 2, M ∈ N and N ∈ N large enough.
Proof. From (4.4), using that | ω N (0)| ≤ C we have For the cubic term on the r.h.s. of (4.8), with (4.10) As for the off-diagonal quadratic term on the r.h.s of (4.8), we combine it with part of the kinetic energy to estimate. For any 0 < µ < 1, we have With the choice µ = C/ log N and with (4.9), we obtain To bound the first terms on the r.h.s. of the last equation, we use the term ω N (0)N + , in (4.8). To this end, we observe that, with (3.27), for every p ∈ Λ * + (notice that |p| ≥ 2π, for every p ∈ Λ * + ) and for N large enough (recall the choice µ = c/ log N ). From (4.8), we find (4.11) Let us now consider the second term on the r.h.s more carefully. Using that, from (3.25), ω N (p) = g N χ(p/N α ), we can bound, for any fixed K > 0, For q ∈ R 2 , let us define h(q) = 1/p 2 , if q is contained in the square of side length 2π centered at p ∈ Λ * + (with an arbitrary choice on the boundary of the squares). We can then estimate, for K large enough, For q in the square centered at p ∈ Λ * + , we bound Inserting in (4.12), we conclude that Combining the last bound with (3.27) (and noticing that the contribution proportional to log N cancels exactly), from (4.11) we obtain which proves (4.6).
Next we prove (4.7). From (4.8), with the bounds (4.10) and since, by (4.9), where for arbitrary δ > 0, there exists a constant C > 0 such that (4.14) We now note that for f : R → R smooth and bounded and θ N,α defined above, there exists a constant C > 0 such that for all α > 2 and N ∈ N large enough. The proof of (4.15) follows analogously to the one for (4.14), since the bounds leading to (4.14) remain true if we replace the operators b # p , # = {·, * }, and a * p a q with ] respectively, provided we multiply the r.h.s. by an additional factor M −2 f ′ 2 ∞ , since, for example Writing R eff N,α as in (4.13) and using (4.15) we get Its proof makes use of localization in the number of particle and is an adaptation of the proof of [3, Proposition 6.1]. The main difference w.r.t. [3] is that here we need to localize on sectors of F ≤N where the number of particles is o(N ), in the limit N → ∞.
Proposition 5.1. Let V ∈ L 3 (R 2 ) be compactly supported, pointwise non-negative and spherically symmetric. Let G N,α be the renormalized excitation Hamiltonian defined as in (3.23). Then, for every α ≥ 5/2, there exist constants C, c > 0 such that Let us consider the first term on the r.h.s. of (5.2). From Prop. 4.4, for all α > 2 there exist c, C > 0 such that On the other hand, with (4.13) and (4.14) we also find for all α > 2 and N large enough. Moreover, due to the choice M = N 1−ε , we have With the last bound, Eq. (5. 3) implies that for N large enough. Let us next consider the second term on the r.h.s. of (5.2). We claim that there exists a constant c > 0 such that for all N sufficiently large. To prove (5.6) we observe that, since g(x) = 0 for all x ≤ 1/2, for all N large enough. On the other hand, using the definitions of G N,α in (3.28), R N,α and R eff N,α in (4.4), we obtain that the ground state energy E N of the system is given by From the result (1.3) of [10,11,12] inf If we assume by contradiction that (5.7) does not hold true, then we can find a subsequence N j → ∞ with ). This implies that there exists a sequenceξ Nj ∈ F ≤Nj ≥Mj /2 with ξ Nj = 1 for all j ∈ N such that On the other hand, using the relation R eff Hence for ξ Nj = e Aξ Nj we have Eq. (5.9) shows that the sequence ψ N is an approximate ground state of H N . From (1.5), we conclude that ψ N exhibits complete Bose-Einstein condensation in the zero-momentum mode ϕ 0 , and in particular that there existδ > 0 such that Using Lemma 3.2, Prop. 4.1 and the rules (2.2), we observe that Choosing ε <δ and N large enough we get a contradiction with (5.10). This proves (5.7), (5.6) and therefore also Inserting (5.5) and (5.11) on the r.h.s. of (5.2), we obtain that To conclude, we use the relation e −A G N,α e A = R eff N,α + E L and the bound (5.8). We have that for α ≥ 5/2 there exist c, C > 0 such that where we used (5.12) and Prop. 4.1.
We are now ready to show our main theorem.
Proof of Theorem 1.1. Let E N be the ground state energy of H N . Evaluating (3.28) and (3.29) on the vacuum Ω ∈ F ≤N + and using (3.26), we obtain the upper bound With Eq. (5.1) we also find the lower bound E N ≥ 2πN − C. This proves (1.6).
We define the excitation vector ξ N = e −B U N ψ N . Then ξ N = 1 and, recalling that From Eqs. (5.13) and (5.14) we conclude that If γ N denotes the one-particle reduced density matrix associated with ψ N , using Lemma 3.2 we obtain which concludes the proof of (1.7).

Analysis of the excitation Hamiltonian R N
In this section, we show Prop. 4.3, where we establish a lower bound for the operator R N,α = e −A G eff N,α e A , with G eff N,α as defined in (3.28) and with with K and V N as in (3.24), and with We will analyze the conjugation of all terms on the r.h.s. of (6.2) in Subsections 6.2-6.6. The estimates emerging from these subsections will then be combined in Subsection 6.6 to conclude the proof of Prop. 4.3. Throughout the section, we will need Prop. 4.2 to control the growth of the expectation of the energy H N = K + V N under the action of (6.1); the proof of Prop. 4.2 is contained in Subsection 6.1.
In this section, we will always assume that V ∈ L 3 (R 2 ) is compactly supported, pointwise non-negative and spherically symmetric.

A priori bounds on the energy
In this section, we show Prop. 4.2. To this end, we will need the following proposition.
Proposition 6.1. Let V N and A be defined in (3.24) and (4.1) respectively. Then, there exists a constant C > 0 such that for any α > 0, for all ξ ∈ F ≤N + , and N ∈ N large enough.
To bound Θ 1 we switch to position space and apply Cauchy-Schwarz. We find for any ξ ∈ F ≤N + The term Θ 3 can be controlled similarly. We find It remains to bound the term Θ 2 on the r.h.s. of (6.5). Passing to position space we obtain, by Cauchy-Schwarz, To bound the term in the square bracket, we write it in first quantized form and, for any 2 < q < ∞, we apply Hölder inequality and the Sobolev inequality With the bounds (3.11), (3.12), Using Prop. 6 We compute With Prop. 6.1, we have with δ VN satisfying (6.4). Switching to position space and using Prop. 4.1 we find , using (3.11) to bound η ∞ ≤ CN , Together with (6.4) we conclude that for any α > 1/2 if N is large enough. Next, we analyze the first term on the r.h.s. of (6.7). We compute (6.10) With (3.17), we write =: T 11 + T 12 . (6.11) The contribution of T 11 can be estimated similarly as in (6.8); switching to position space and using (3.6), we obtain for any ξ ∈ F ≤N + . The second term in (6.11) can be controlled using that for any Hence, choosing q = log N , With (6.12) and (6.14) we conclude that for all ξ ∈ F ≤N + . As for the second term on the r.h.s. of (6.10) we have for any ξ ∈ F ≤N + . With (6.15) and Prop. 4.1, we conclude that Combining with Eq. (6.9) we obtain With Prop. 4.1 we obtain the differential inequality By Gronwall's Lemma, we find (4.2).

Analysis of e −A O N e A
In this section we study the contribution to R N,α arising from the operator O N , defined in (6.3). To this end, it is convenient to use the following lemma.
Proof. The lemma is analogous to [3,Lemma 8.6]. We estimate where we used Prop. 4.1.
We consider now the action of e A on the operator O N , as defined in (6.3).
Proposition 6.3. Let A be defined in (4.1). Then there exists a constant C > 0 such that for all α > 0, and N ∈ N large enough.
Proof. The proof is very similar to [3,Prop. 8.7]. First of all, with Lemma 6.2 we can bound Moreover, for the contribution quadratic in N + , we can decompose with ξ 1 = e −A N + e A ξ and ξ 2 = N + ξ, and estimate, again with Lemma 6.2,

Contributions from e −A Ke A
In Section 6.6 we will analyse the contributions to R N,α arising from conjugation of the kinetic energy operator K = p∈Λ * + p 2 a * p a p . To this aim we will exploit properties of the commutator [K, A], collected in the following proposition. Proposition 6.4. Let A be defined as in (4.1) and ω N (r) be defined in (3.25). Then there exists a constant C > 0 such that for all α > 1, ξ ∈ F ≤N + , and N ∈ N large enough. Moreover, the operator for all α > 1, ξ ∈ F ≤N + , and N ∈ N large enough.
Let us now focus on (6.18). We have With the commutators from the proof of Prop. 8.8 in [3], we arrive at where and (6.20) To conclude the proof of Prop. 6.4, we show that all operators in (6.19) and (6.20) satisfy (6.18). To study all these terms it is convenient to switch to position space. We recall that ω N (p) = g N χ(ℓp) with |g N | ≤ C and ℓ = N −α . Using (6.13) we find: The expectation of Υ 2 is bounded following the same strategy used to show (6.13). For any 2 ≤ q < ∞ we have where in the last line we chose q = log N . The term Υ 3 is of lower order; using that r ω N (r)η r ≤ χ(./N α ) 2 η 2 ≤ C and Cauchy-Schwarz, we easily obtain The term Υ 4 can be estimated as Υ 1 using (6.13): The term Υ 5 is bounded similarly to Υ 2 ; with q = log N we have The terms Υ 6 and Υ 7 are of smaller order and can be bounded with Cauchy-Schwarz; we have The terms Υ 8 , Υ 11 , Υ 12 are again bounded, as Υ 1 , using (6.13). We find It remains to bound Υ 9 and Υ 10 . The term Υ 9 is bounded analogously to Υ 2 : As for Υ 10 , we find Proceeding as in (6.6), we obtain ξ, Υ 10 ξ ≤ CqN 2α χ ℓ * |η| q ′ K 1/2 ξ 2 ≤ Cq η q ′ K 1/2 ξ 2 for any q > 2, and q ′ < 2 with 1/q + 1/q ′ = 1. Since, for an arbitrary q ′ < 2, We conclude that for any α > 1 ξ, In this subsection, we consider contributions to R N,α arising from conjugation of Z N , as defined in (6.3).
Proposition 6.5. Let A be defined in (4.1). Then, there exists a constant C > 0 such that for all α > 0, and N ∈ N large enough.
Proof. We have (6.21) We compute To bound the first term, we observe, with (4.9), The term Π 3 can be bounded similarly to Π 1 , with (4.9). We find With | ω N (r)| ≤ C, we similarly obtain Finally, we estimate, using again (4.9), With (6.21), we conclude that With Prop. 4.1, Lemma 4.2, we conclude that 6.5 Contributions from e −A C N e A In Section 6.6 we will analyse the contributions to R N,α arising from conjugation of the cubic operator C N defined in (6.3). To this aim we will need some properties of the commutator [C N , A], as established in the following proposition.
Proposition 6.6. Let A be defined in (4.1). Then, there exists a constant C > 0 such that for all α > 0, ξ ∈ F ≤N + , and N ∈ N large enough.
Proof. We consider the commutator As in the proof of Prop. 6.4, we use the commutators from the proof of Prop. 8.8 in [3] to conclude that where as well as To prove the proposition, we have to show that all terms Ξ j , j = 1, . . . , 12, satisfy the bound (6.23). We bound Ξ 1 in position space, with Cauchy-Schwarz, by We can proceed similarly to control Ξ 9 . We obtain The expectations of the terms Ξ 3 and Ξ 12 can be bounded analogously: As for Ξ 4 , we find The terms Ξ 5 and Ξ 6 can be bounded in momentum space, using (A.51). Hence, |r + v| |η r ||r + v| a r+q a v ξ a q a r+v ξ ≤ CN 1/2−α (N + + 1) 1/2 ξ K 1/2 ξ .
Next, we rewrite Ξ 7 , Ξ 8 and Ξ 11 as Thus, we obtain Collecting all the bounds above, we arrive at (6.23).

Proof of Proposition 4.3
With the results of Sections 6.1-6.5, we can now show Proposition 4.3. We assume α > 2. From Eq. (6.2), Prop. 6.3 and Prop. 6.5 we obtain that . From Prop. 6.1, Prop. 6.4 and Prop. 6.6, we can write, for N large enough, We now rewrite 2 r,v∈Λ * and therefore, using Lemma 6.2 and (6.26) On the other hand it is easy to check that e −sA Q 2 e sA is an error term; to this aim we notice that Hence with Props

(6.29)
As for the first term on the second line of (6.24), we use again Prop. 6.6. Using (6.25), (6.27) and (6.28) we have Inserting the bounds (6.27), (6.28), (6.29) and (6.30) into (6.24) we arrive at for N ∈ N sufficiently large.  The analysis in this section follows closely that of [3, Section 7] with some slight modifications due to the different scaling of the interaction potential and the fact that the kernel η p of e B is different from zero for all p ∈ Λ * + (in [3] η p is different from zero only for momenta larger than a sufficiently large cutoff of order one). Moreover, while in three dimensions it was sufficient to choose the function η p appearing in the generalized Bogoliubov transformation with η sufficiently small but of order one, we need here η to be of order N −α for some α > 0 large enough. As discussed in the introduction this is achieved by considering the Neumann problem for the scattering equation in (3.2) on a ball of radius ℓ = N −α ; as a consequence some terms depending on ℓ will be large, compared to the analogous terms in [3].

A.1 Generalized Bogoliubov transformations
In this subsection we collect important properties about the action of unitary operators of the form e B , as defined in (3.20). As shown in [2, Lemma 2.5 and 2.6], we have, if η is sufficiently small, where the series converge absolutely. To confirm the expectation that generalized Bogoliubov transformation act similarly to standard Bogoliubov transformations, on states with few excitations, we define (for η small enough) the remainder operators where q ∈ Λ * + , (♯ m , α m ) = (·, +1) if m is even and (♯ m , α m ) = ( * , −1) if m is odd. It follows then from (A.1) that where we introduced the notation γ q = cosh(η q ) and σ q = sinh(η q ). It will also be useful to introduce remainder operators in position space. For x ∈ Λ, we define the operator valued distributionsď x ,ď * x through whereγ x (y) = q∈Λ * cosh(η q )e iq·(x−y) andσ x (y) = q∈Λ * sinh(η q )e iq·(x−y) . The next lemma is taken from [3, Lemma 3.4].
Lemma A.1. Let η ∈ ℓ 2 (Λ * + ), n ∈ Z. For p ∈ Λ * + , let d p be defined as in (A.3). If η is small enough, there exists C > 0 such that for all p ∈ Λ * + , ξ ∈ F ≤N + . In position space, withď x defined as in (A.4), we find and, finally, for all ξ ∈ F ≤n + . A first simple application of Lemma A.1 is the following bound on the growth of the expectation of N + .
Proof. We proceed as in the proof of [3,Prop. 7.2]. We write p is defined as in (A.2), with η p replaced by sη p . We find for any ξ ∈ F ≤N + . To bound the term G 3 in (A.11), we switch to position space: Finally, we consider G 2 in (A.11). We split it as G 2 = G 21 + G 22 + G 23 + G 24 , with (A.14) We consider G 21 first. We write (A. 15) and where we introduced the notation d To control the third term in (A.15), we use (3.16) and we switch to position space. We find With (A.7) and |η(x − y)| ≤ CN , we obtain As for E K 232 , with (A.7) and Lemma 3.1 (recalling ℓ = N −α ), we find To bound the last term on the r.h.s. of (A.20) we use Hölder's and Sobolev inequality u q ≤ Cq 1/2 u H 1 , valid for any 2 ≤ q < ∞. We find Choosing q = log N , we get Therefore, for any ξ ∈ F ≤N + ,
Proof. We write (A.28) where γ p = cosh η p , σ p = sinh η p and the operators d p are defined in (A.2). Using |1 − γ p | ≤ η 2 p , |σ p | ≤ C|η p | and using Lemma A.1 for the terms on the second line, we find with ±E V 1 ≤ CN 1−α (N + + 1). Let us now consider the second contribution on the r.h.s. of (A.28). We find With Lemma 3.2, we easily find ±E V 2 ≤ CN −α (N + + 1). Finally, we consider the last term on the r.h.s. of (A.28). With (A.3), we obtain =: F 31 + F 32 + F 33 . (A.31) Using |1 − γ p | ≤ Cη 2 p , |σ p | ≤ C|η p |, we obtain . As for F 32 in (A.31), we divide it into four parts (A.33) We start with F 321 , which we write as and with the notation As for E V 42 , we switch to position space and we use (A.7). We obtain We conclude that To bound the term F 322 in (A.33), we use (A.5) and |σ p | ≤ C|η p |; we obtain Let us now consider the term F 323 on the r.h.s. of (A.33). We write With |γ p − 1| ≤ Cη 2 p and (A.5) we obtain We find, switching to position space and using (A.6), Hence, To estimate the term F 324 in (A.33) we use (A.5) and the bound p∈Λ * Combining the last bounds, we arrive at To control the last contribution F 33 in (A.31), we switch to position space. With (A.8) and (3.11) we obtain The last equation, combined with (A.31), (A.32) and (A.34), implies that Together with (A.29) and with (A.30), and recalling that b * p b p − N −1 a * p a p = a * p a p (1 − N + /N ), we obtain (A.26) with (A.27).

A.4 Analysis of G
(3) We consider here the conjugation of the cubic term L for any α > 1 and N ∈ N large enough.
Proof. This proof is similar to the proof of [3,Prop. 7.5]. Expanding e −B a * −p a q e B , we arrive at where, as usual, γ p = cosh η(p), σ p = sinh η(p) and d p is as in (A.2). We consider E 1 . To this end, we write 11 + E 12 + E 13 .
To show Prop. A.7, we use the following lemma, whose proof can be obtained as in [3,Lemma 7.7].
With |γ Let us now consider E Next, we control the term W 2 , from (A.47). In position space, we find and using Lemma A.8, we obtain Also for the term W 3 in (A.47), we switch to position space. We find With Lemma 3.2, we find Using Lemma A.8, we conclude that The term W 4 in (A.47) can be bounded similarly. In position space, we find withη 2 the function with Fourier coefficients η 2 q , for q ∈ Λ * , and whereη 2 x (y) := η 2 (x − y). Clearly η2 x ≤ C η 2 ≤ CN −2α . With Cauchy-Schwarz and Lemma 3.2, we obtain Applying Lemma A.8 and then Lemma 3.2, we obtain From Lemma 3.1 and estimating χ ℓ = χ ℓ ≤ CN −α , η ≤ CN −α and χ ℓ * η = χ ℓη ≤ η ≤ CN −α , we have Moreover, using (A.51) and the bound (3.18) we find We obtain with ±E 2 ≤ C for all α ≥ 1/2. On the other hand, using (3.18) we have With the first bound in (3.27) we conclude that where ±E 3 ≤ C, if α ≥ 1/2. Using (3.17), we can also handle the fourth line of (A.59); we find The last two terms on the right hand side of (A.61) are error terms. With (3.18) and (A.51) we have The second term on the right hand side of (A.61) can be bounded in position space: The term in parenthesis can be bounded similarly as in (6.6). Namely, for any q > 2 and 1 < q ′ < 2 with 1/q + 1/q ′ = 1. Choosing q = log N , we get and, from (A.61), we conclude that if α > 1. Combining (A.59) with (A.60) and (A.62), and using the definition (3.25) we conclude that for any α > 1. Observing that | V (p/e N ) − V (0)| ≤ C|p|e −N in the second line on the r.h.s. of (A.63), we arrive at G N,α = G eff N,α + E G , with G eff N,α defined as in (3.28) and with E G that satisfies (3.29).

B Properties of the Scattering Function
Let V be a potential with finite range R 0 > 0 and scattering length a. For a fixed R > R 0 , we study properties of the ground state f R of the Neumann problem on the ball |x| ≤ R, normalized so that f R (x) = 1 for |x| = R. Lemma 3.1, parts i)-iv), follows by setting R = e N ℓ in the following lemma.
Lemma B.1. Let V ∈ L 3 (R 2 ) be non-negative, compactly supported and spherically symmetric, and denote its scattering length by a. Fix R > 0 sufficiently large and denote by f R the Neumann ground state of (B.1). Set w R = 1 − f R . Then we have Moreover, for R large enough there is a constant C > 0 independent of R such that Finally, there exists a constant C > 0 such that for R large enough.
To show Lemma B.1 we adapt to the two dimensional case the strategy used in [6, Lemma A.1] for the three dimensional problem. We will use some well known properties of the zero energy scattering equation in two dimensions, summarized in the following lemma.
Lemma B.2. Let V ∈ L 3 (R 2 ) non-negative, with supp V ⊂ B R0 (0) for an R 0 > 0. Let a ≤ R 0 denote the scattering length of V . For R > R 0 , let φ R : R 2 → R be the radial solution of the zero energy scattering equation normalized such that φ R (x) = 1 for |x| = R. Then for all |x| > R 0 . Moreover, |x| → φ R (x) is monotonically increasing and there exists a constant C > 0 (depending only on V ) such that for all x ∈ R 2 . Furthermore, there exists a constant C > 0 such that for all x ∈ R 2 .
On the other hand, Eq.(B.18), together with m R (x) = |x| for |x| ≥ R 0 , implies the lower bound Hence, with (B.9), we conclude that To prove the lower bound for λ R it is convenient to show some upper and lower bounds for f R . We start by considering f R outside the range of the potential. We denote ε R = √ λ R R. Keeping into account the boundary conditions at |x| = R, we find, for R 0 ≤ |x| ≤ R, From (B.24), we have |ε R | ≤ C | log(R/a)| −1/2 . Thus, we can expand f R for large R, using (B.11) and, for Y 0 , the improved bound Y 0 (r) − 2 π log(re γ /2) 1 − 1 4 r 2 ≤ C r 2 , we find We can also compute the radial derivative With the expansions (B.11) and (B.25) we conclude that for all R 0 ≤ |x| < R we have The bound (B.27) shows that ∂ r f R (x) is positive, for, say, R 0 < |x| < R/2. Since ∂ r f R (x) must have its first zero at |x| = R, we conclude that f R is increasing in |x|, on R 0 ≤ |x| ≤ R. From the normalization f R (x) = 1, for |x| = R, we conclude therefore that f R (x) ≤ 1, for all R 0 ≤ |x| ≤ R. for all R * < |x| ≤ R.
Finally, we show that f R (x) ≤ 1 also for |x| ≤ R 0 . First of all, we observe that, by elliptic regularity, as stated for example in [9, Theorem 11.7, part iv)], there exists 0 < α < 1 and C > 0 such that we conclude that 0 ≤ f R (x) ≤ 1 + C f 2 for all |x| ≤ R 0 (because we know that f R (x) ≤ 1 for R 0 ≤ |x| ≤ R). To improve this bound, we go back to the differential equation (B.1), to estimate This implies that f R (x) + λ R (1 + C f 2 )x 2 /2 is subharmonic. Using (B.26), we find f R (x) ≤ 1 − Cε 2 R for |x| = R 0 . From the maximum principle, we obtain therefore that f R (x) ≤ 1 − Cε 2 R + Cλ R (1 + C f R 2 ) (B.31) for all |x| ≤ R 0 . In particular, this implies that f R 1 |x|≤R0 2 ≤ C + Cλ R f R 2 , and therefore that With f R (x) ≤ 1 for R 0 ≤ |x| ≤ R, we find, on the other hand, that f R 1 R0≤|x|≤R 2 ≤ CR. We conclude therefore that f R 2 ≤ CR and, from (B.31), that f R (x) ≤ 1 − Cε 2 R + C/R ≤ 1, for all |x| ≤ R 0 , if R is large enough. We are now ready to prove the lower bound for λ R . We use now that any function Φ satisfying Neumann boundary conditions at |x| = R can be written as Φ(x) = q(x)Ψ R (x), with Ψ R (x) the trial function used for the upper bound and q > 0 a function that satisfies Neumann boundary condition at |x| = R as well. This is in particular true for the solution f R (x) of (B.1). In the following we write where q R satisfies Neumann boundary conditions at |x| = R. From (B.18), we find |Ψ R (x)| ≥ C/ log(ka). The bound f R (x) ≤ 1 implies therefore that there exists c > 0 such that q R (x) ≤ C log(ka) ∀ |x| ≤ R 0 . (B.32) From the identity From (B.22) and (B.23), we have With (B.32), we obtain With (B.29) (recalling that R * = max{R 0 , ea}), we bound f R 2 ≥ R * ≤|x|≤R |f R (x)| 2 dx ≥ CR 2 log 2 (R/a) and, inserting in (B.33), we conclude that where in the last inequality we used (B.9).
To prove (B.3) we use the scattering equation (B.1) to write Passing to polar coordinates, and using that ∆f R (x) = |x| −1 ∂ r |x|∂ r f R (x), we find that the first term vanishes. Hence With the upper bound f R (r) ≤ 1 and with (B.2), we find dx V (x)f R (x) ≤ 2πR 2 λ R ≤ 4π log(R/a) 1 + C log(R/a) .
To obtain a lower bound for the same integral we use that f R (r) ≥ 0 inside the range of the potential. Outside the range of V , we use (B.26). We find dr r (1 − Cε 2 R log(R/r)) ≥ Moreover ∂ r f R (x) = 0 if |x| = R, by construction.
On the other hand, if |x| ≤ R 0 , we have w R (x) = 1 − f R (x) ≤ 1. As for the derivative, we define f R on R + through f R (r) = f R (x), if |x| = r, and we use the representation