Bose-Einstein Condensation Beyond the Gross-Pitaevskii Regime

We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order $N^{-1+\kappa}$, for $\kappa>0$. Assuming that $\kappa\in (0;1/43)$, we show that low-energy states of the system exhibit complete Bose-Einstein condensation by providing explicit bounds on the expectation and on higher moments of the number of excitations.


Introduction
We consider systems of N ∈ N bosons trapped in the box Λ = [0; 1] 3 with periodic boundary conditions (the three-dimensional torus with volume one) and interacting through a repulsive potential with scattering length of the order N −1+κ , for κ ∈ (0; 1/43). We are interested in the limit of large N . The Hamilton operator has the form 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 ∈ N particles. Here, we assume the interaction potential V ∈ L 3 (R 3 ) to have compact support and to be nonnegative, ie. V (x) ≥ 0 for almost all x ∈ R 3 .
For κ = 0, the Hamilton operator (1.1) describes bosons in the so-called Gross-Pitaevskii limit. This regime is frequently used to model trapped Bose gases observed in recent experiments. Another important regime is the thermodynamic limit, where N bosons interacting through a fixed potential V (independent of N ) are trapped in the box Λ L = [0; L] 3 and where the limits N, L → ∞ are taken, keeping the density ρ = N/L 3 fixed. After rescaling lengths (introducing new coordinates x = x/L), the Hamilton operator of the Bose gas in the thermodynamic limit is given (up to a multiplicative constant) by (1.1), with κ = 2/3. Choosing 0 < κ < 2/3, we are interpolating therefore between the Gross-Pitaevskii and the thermodynamic limits.
The goal of this paper is to show that low-energy states of (1.1) exhibit Bose-Einstein condensation in the zero-momentum mode ϕ 0 ∈ L 2 (Λ) defined by ϕ 0 (x) = 1 for all x ∈ Λ and to give bounds on the number of excitations of the condensate. To achieve this goal, it is convenient to switch to an equivalent representation of the bosonic system, removing the condensate and focusing instead on its orthogonal excitations. To this end, we notice that every ψ N ∈ L 2 s (Λ N ) can be uniquely decomposed as where ⊗ s denotes the symmetric tensor product and α j ∈ L 2 ⊥ (Λ) ⊗sj for all j = 0, . . . , N, with L 2 ⊥ (Λ) the orthogonal complement in L 2 (Λ) of ϕ 0 . This observation allows us to define a unitary map U N : The truncated Fock space F ≤N + = N j=0 L 2 ⊥ (Λ) ⊗sj is used to describe orthogonal excitations of the condensate (some properties of the map U N will be discussed in Sect. 2 below). On F ≤N + , we introduce the number of particles operator, defining (N + ξ) (n) = nξ (n) for every ξ = {ξ (0) , . . . ξ (N ) } ∈ F ≤N + . We are now ready to state our main theorem, which provides estimates of the expectation and on higher moments of the number of orthogonal excitations of the Bose-Einstein condensate for low-energy states of (1.1). Theorem 1.1. Let V ∈ L 3 (R 3 ) be pointwise nonnegative and spherically symmetric. Let a 0 > 0 denote the scattering length of V . Let H N be defined as in (1.1) with 0 < κ < 1/43. Then, for every ε > 0, there exists a constant C > 0 such that Let ψ N ∈ L 2 s (Λ N ) with ψ N = 1 and

4)
for a ζ > 0. Then, for every ε > 0 there exists a constant C > 0 such that for all N ∈ N large enough. If moreover ψ N = χ(H N ≤ E N + ζ)ψ N , then for all k ∈ N and all ε > 0 there exists C > 0 such that (1.6) for all N ∈ N large enough.
The convergence E N /4πa 0 N 1+κ → 1, as N → ∞, has been first established, for Bose gases trapped by an external potential, in [19] (the choice κ > 0 corresponds, in the terminology of [19], to the Thomas-Fermi limit).
It follows from (1.5) that the one-particle density matrix γ N = tr 2,...,N |ψ N ψ N | associated with a normalized ψ N ∈ L 2 s (Λ N ) satisfying (1.4) is such that as N → ∞. Here, we used the formula U N a * (ϕ 0 )a(ϕ 0 )U N = N − N + ; see (2.5). Equation (1.7) implies that low-energy states of (1.1) exhibit complete Bose-Einstein condensation, if κ < 1/43. We remark that the estimate (1.6) follows, in our analysis, from a stronger bound controlling not only the number but also the energy of the excitations of the condensate. As we will explain in Sect. 3, in order to estimate the energy of excitations in low-energy states, we first need to remove (at least part of) their correlations. If we choose, as we do in (1.6), ψ N ∈ L 2 s (Λ N ) with ψ N = 1 and ψ N = χ(H N ≤ E N + ζ)ψ N , we can introduce the corresponding renormalized excitation vector ξ N = e B U N ψ N ∈ F ≤N + , with the antisymmetric operator B defined as in (3.21) (the unitary operator e B will be referred to as a generalized Bogoliubov transformation). We will show in Sect. 6 that for every k ∈ N, there exists C > 0 such that p,q∈Λ * + ,r∈Λ * : r =−p,−q N κ V (r/N 1−κ )a * p+r a * q a q+r a p (1.9) are the kinetic and potential energy operators, restricted to F ≤N + . (Here, V is the Fourier transform of the potential V , defined as in (2.4).) Equation (1.6) follows then from (1.8), because N + commutes with H N , N + ≤ K ≤ H N and because conjugation with the generalized Bogoliubov transformation e B does not change the number of particles substantially; see Lemma 3.2 (for k ∈ N even, we also use simple interpolation).
In the Gross-Pitaevskii regime corresponding to κ = 0 the convergence γ N → |ϕ 0 ϕ 0 | has been first established in [16][17][18] and later, using a different approach, in [21]. 1 In this case (ie. κ = 0), the bounds (1.3), (1.5) and (1.6) with ε = 0 (which are optimal in their N -dependence) have been shown in [4]. Previously, they have been established in [2], under the additional assumption of small potential. A simpler proof of the results of [2], extended also to systems of bosons trapped by an external potential, has been recently given in [20]. The result of [4] was used in [3] to determine the second order corrections to the ground state energy and the low-energy excitation spectrum of the Bose gas in the Gross-Pitaevskii regime. Note that our approach in the present paper could be easily extended to the case κ = 0, leading to the same bounds obtained in [4]. We exclude the case κ = 0 because we would have to modify certain definitions, making the notation more complicated (for example, the sets P H in (3.14) and P L in (4.2) would have to be defined in terms of cutoffs independent of N ).
The methods of [16][17][18] can also be extended to show Bose-Einstein condensation for low-energy states of (1.1), for some κ > 0. In fact, following the proof of [18,Theorem 5.1], it is possible to show that for a normalized ψ N ∈ L 2 s (Λ N ) with ψ N = 1 and such that ψ N , H N ψ N ≤ E N + ζ, the expectation of the number of excitations is bounded by which implies complete Bose-Einstein condensation for low-energy states, for all κ < 1/10. For sufficiently small κ > 0, Theorem 1.1 improves (1.10) because it gives a better rate 2 (if κ < 15/711) and because, through (1.6), it also provides (under stronger conditions on ψ N ) bounds for higher moments of the number of excitations N + .
In [10], in a slightly different setting, the authors obtain a bound of the form (1.6) for k = 1, for the choice κ = 1/(55 + 1/3) (for normalized ψ N ∈ L 2 s (Λ N ) that satisfy ψ N , H N ψ N ≤ E N + ζ). They use this result to show a lower bound on the ground state energy of the dilute Bose gas in the thermodynamic limit matching the prediction of Lee-Yang and Lee-Huang-Yang [13,14].
After completion of our work, two more papers have appeared whose results are related with Theorem 1.1. Based on localization arguments from [8,10], a bound for the expectation of N + in low-energy states has been shown in [9], establishing Bose-Einstein condensation for all κ < 2/5 (as pointed out there, using a refined analysis similar to that of [10], the range of κ can be slightly improved). On the other hand, following an approach similar to [2], but with substantial simplifications (partly due to the fact that the author works in the grand canonical, rather than the canonical, ensemble), a new proof of Bose-Einstein condensation was obtained in [11], in the Gross-Pitaevskii regime, under the assumption of small potential. There is hope that the approach of [11] can be extended beyond the Gross-Pitaevskii regime, providing a simplified proof of Theorem 1.1, potentially allowing for larger values of κ.
The derivation of the bounds (1.5), (1.6), (1.8) is crucial to resolve the low-energy spectrum of the Hamiltonian (1.1). The extension of estimates on the ground state energy and on the excitation spectrum obtained in [3] for the Gross-Pitaevskii limit, to regimes with κ > 0 small enough will be addressed in a separate paper [6], using the results of Theorem 1.1. With our techniques, it does not seem possible to obtain such precise information on the spectrum of (1.1) using only previously available bounds like (1.10).
Let us now briefly explain the strategy we use to prove Theorem 1.1. The first part of our analysis follows closely [4]. We start in Sect. 2 by introducing the excitation Hamiltonian L N = U N H N U * N , acting on the truncated Fock space F ≤N + ; the result is given in (2.6), (2.7). The vacuum expectation Ω, L N Ω = N 1+κ V (0)/2 is still very far from the correct ground state energy of L N (and thus of H N ); the difference is of order N 1+κ . This is a consequence of the definition (1.2) of the unitary map U N , whose action removes products of the condensate wave function ϕ 0 , leaving however all correlations among particles in the wave functions α j ∈ L 2 ⊥ (Λ) ⊗sj , j = 1, . . . , N. To factor out correlations, we introduce in Sect. 3 a renormalized excitation Hamiltonian G N = e −B L N e B , defined through unitary conjugation of L N with a generalized Bogoliubov transformation e B . The antisymmetric operator B : F ≤N is quadratic in the modified creation and annihilation operators b p , b * p defined, for every momentum p ∈ Λ * + = 2πZ 3 \{0}, in (2.8) (b * p creates a particle with momentum p annihilating, at the same time, a particle with momentum zero; in other words, b * p creates an excitation, moving a particle out of the condensate). The properties of G N are listed in Prop. 3.3. In particular, Proposition 3.3 implies that to leading order, Ω, Unfortunately, G N is not coercive enough to prove directly that lowenergy states exhibit condensation (in the sense that it is not clear how to estimate the difference between G N and its vacuum expectation from below by the number of particle operator N + ). For this reason, in Sect. 4, we define yet another renormalized excitation Hamiltonian J N = e −A G N e A , where now A is the antisymmetric operator (4.1), cubic in (modified) creation and annihilation operators (to be more precise, we only conjugate the main part of G N with e A ; see (4.3)). Important properties of J N are stated in Proposition 4.1. Up to negligible errors, the conjugation with e A completes the renormalization of quadratic and cubic terms; in (4.5), these terms have the same form they would have for particles interacting through a mean-field potential with Fourier transform 8πa 0 N κ 1(|p| < N α ), with a parameter α > 0 that will be chosen small enough, depending on κ (in other words, the renormalization procedure allows us to replace, in all quadratic and cubic terms, the original interaction with Fourier transform N −1+κ V (p/N 1−κ ) decaying only for momenta |p| > N 1−κ , with a potential whose Fourier transform already decays on scales N α N 1−κ ).
The main problem with J N is that its quartic terms (the restriction of the initial potential energy on the orthogonal complement of the condensate wave function) are still proportional to the local interaction with Fourier transform One possibility to solve this problem is to neglect the original quartic terms (they are positive) and insert instead quartic terms proportional to the renormalized mean-field potential 8πa 0 N κ 1(|p| < N α ), so that Bose-Einstein condensation follows as it does for mean-field systems (see [22]). Since (with the notationχ for the inverse Fourier transform of the characteristic function on the ball of radius one) and since we know from (1.10) that N + N 15+20κ 17 in low-energy states, the insertion of the renormalized quartic terms produces an error that can be controlled by localization in the number of particles, if This strategy was used in [4] to prove Bose-Einstein condensation with optimal rate in the Gross-Pitaevskii regime κ = 0 (in this case, one can choose α = 0). Here, we follow a different approach. We perform a last renormalization step, conjugating J N through a unitary operator e D , with D quartic in creation and annihilation operators. This leads to a new Hamiltonian M N = e −D J N e D (in fact, it is more convenient to conjugate only the main part of J N , ignoring small contributions that can be controlled by other means; see (5.5)), where the original interaction N −1+κ V (p/N 1−κ ) is replaced by the mean-field potential 8πa 0 N κ 1(|p| < N α ) in all relevant terms. 3 Condensation can then be shown as it is done for mean-field systems, with no need for localization. This is the main novelty of our analysis, compared with [4]. In Sect. 5, we define the final Hamiltonian M N and in Proposition 5.1 we bound it from below. The proof of Proposition 5.1, which is technically the main part of our paper, is deferred to Sect. 7. In Sect. 6, we combine the results of the previous sections to conclude the proof of Theorem 1.1.
The results we prove with our new technique are stronger than what we would obtain using the approach of [4] in the sense that they allow for larger values of κ and better rates. More importantly, we believe that the approach we propose here is more natural and that it leaves more space for extensions. In particular, with the final quartic renormalization step, we map the original Hamilton operator (1.1), with an interaction varying on momenta of order N 1−κ , into a new Hamiltonian having the same form, but now with an interaction restricted to momenta smaller than N α . If α < 1 − κ, this leads to an effective regularization of the potential and it suggests that further improvements may be achieved by iteration; we plan to follow this strategy, which bears some similarities to the renormalization group analysis developed in [1], in future work.
In order to control errors arising from the quartic conjugation, it is important to use observables that were not employed in [4]. In particular, the expectation of the number of excitations with large momenta and of its powers N 2 ≥N γ , N 3 ≥N γ , as well as the expectation of products of the form K L N ≥N γ and K L N 2 ≥N γ , involving the kinetic energy operator restricted to low momenta K L , will play a crucial role in our analysis. It will therefore be important to establish bounds for the growth of these observables through all steps of the renormalization procedure (Lemmas 4.2, 4.3, 7.1, 7.2). In Sect. 6, an important step in the proof of Theorem 1.1 will consist in controlling the expectation of these observables on low-energy states of the renormalized Hamiltonian G N .

The Excitation Hamiltonian
We denote by F = n≥0 L 2 (Λ) ⊗sn the bosonic Fock space over the oneparticle space L 2 (Λ) and by Ω = {1, 0, . . .} the vacuum vector. We can define the number of particles operator N by setting (N ψ) (n) = nψ (n) for all ψ = {ψ (0) , ψ (1) , . . .} in a dense subspace of F. For every one-particle wave function g ∈ L 2 (Λ), we define the creation operator a * (g) and its hermitian conjugate, the annihilation operator a(g), through Creation and annihilation operators are defined on the domain of N 1/2 , where they satisfy the bounds for all g, h ∈ L 2 (Λ) ( ., . denotes here the inner product on L 2 (Λ)). For p ∈ Λ * = 2πZ 3 , we define the plane wave ϕ p ∈ L 2 (Λ) through ϕ p (x) = e −ip·x for all x ∈ Λ, and the operators a * p = a(ϕ p ) and a p = a(ϕ p ) creating and, respectively, annihilating a particle with momentum p. It is sometimes convenient to switch to position space, introducing operator valued distributionsǎ x ,ǎ * x such that In terms of creation and annihilation operators, the number of particles operator can be written as We will describe excitations of the Bose-Einstein condensate on the truncated Fock space constructed over the orthogonal complement L 2 ⊥ (Λ) of the condensate wave function ϕ 0 . On F ≤N + , we denote the number of particles operator by N + . It is given by is the momentum space for excitations. Given Θ ≥ 0, we also introduce the restricted number of particles operators measuring the number of excitations with momentum larger or equal to Θ, and we can also express U N in terms of creation and annihilation operators; we obtain It is then easy to check that U * N : F ≤N and that U * N U N = 1, ie. U N is unitary. Using 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 L N , we first write the Hamiltonian (1.1) in momentum space, in terms of creation and annihilation operators. We find is the Fourier transform of V , defined for all k ∈ R 3 (in fact, (1.1) is the restriction of (2.3) to the N ∈ N-particle sector of the Fock space F). We can now determine the excitation Hamiltonian 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 3 \{0}, we find (see [15]): where we introduced generalized creation and annihilation operators for all p ∈ Λ * + . Observe that by (2.5), In other words, b * p creates a particle with momentum p ∈ Λ * + but, at the same time, it annihilates a particle from the condensate; it creates an excitation, preserving the total number of particles in the system. 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 . Modified creation and annihilation operators satisfy the commutation relations Furthermore, we find for all p, q, r ∈ Λ * + ; this implies in particular that [b p , 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 The commutation relations (2.9) take the form

Renormalized Excitation Hamiltonian
Conjugation with U N extracts, from the original quartic interaction in (2.3), some constant and some quadratic contributions, collected in L To extract the missing energy, we have to take into account correlations. To this end, we consider the ground state solution f of the Neumann problem on the ball |x| ≤ N 1−κ (we omit the N ∈ N-dependence in the notation for f and for λ ; notice that λ scales as N 3κ−3 ), with the normalization f (x) = 1 if |x| = N 1−κ . By scaling, we observe that f (N 1−κ .) satisfies the equation on the ball |x| ≤ . From now on, we fix some 0 < < 1/2, so that the ball of radius is contained in the box Λ = [−1/2; 1/2] 3 . We then extend f (N 1−κ .) if |x| ≤ and f N (x) = 1 for x ∈ Λ, with |x| > . As a consequence, where χ denotes the characteristic function of the ball of radius . The Fourier coefficients of the function f N are given by The Fourier coefficients of w N are given by denotes the Fourier transform of the (compactly supported) function w . We The next lemma summarizes important properties of the functions w and f . Its proof can be found in [4, Appendix A] (replacing N ∈ N by N 1−κ and noting that still N 1−κ 1 for N ∈ N sufficiently large and fixed ∈ (0; 1/2)).
Lemma 3.1. Let V ∈ L 3 (R 3 ) be nonnegative, compactly supported and spherically symmetric. Fix > 0 and let f denote the solution of (3.1). For N ∈ N large enough, the following properties hold true.
(iii) There exists a constant C > 0 such that for all x ∈ R 3 and all N ∈ N large enough.
(iv) There exists a constant C > 0 such that for all p ∈ R 3 and all N ∈ N large enough (such that N 1−κ ≥ −1 ).
We define η : Λ * → R through In position space, this means that for x ∈ Λ, we havě so that we have in particular the L ∞ -bound for all p ∈ Λ * + = 2πZ 3 \{0}, and for some constant C > 0 independent of N ∈ N (for N ∈ N large enough). From (3.4), we find the relation (3.12) or equivalently, expressing the r.h.s. through the coefficients η p , (3.13) In our analysis, it is useful to restrict η to high momenta. To this end, let α > 0 and P H = {p ∈ Λ * + : |p| ≥ N α }. (3.14) We define η H ∈ 2 (Λ *  (3.18) for all N ∈ N large enough. 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. (3.13) involves η 0 ). From Lemma 3.1, part iii), we obtain It will also be useful to have bounds for the functionη H : Λ → R, having Fourier coefficients η H (p) as defined in (3.15).
With the coefficients (3.15), we define the antisymmetric operator and the generalized Bogoliubov transformation e B : F ≤N A first important observation is that conjugation with this unitary operator does not change the number of particles by too much. The proof of the following Lemma can be found in [7, Lemma 3.1] (a similar result has been previously established in [22]). (3.21), with the coefficients η p as in (3.8), satisfying (3.17). For every n ∈ N, there exists a constant C > 0 such that

Lemma 3.2. Assume B is defined as in
as an operator inequality on F ≤N + . (The constant depends only on η H and on n ∈ N.) With the generalized Bogoliubov transformation e B , we can now define the renormalized excitation Hamiltonian G N : F ≤N In the next propositions, we collect important properties of G N . Recall the notation H N = K + V N , introduced in (1.9). Proposition 3.3. Let V ∈ L 3 (R 3 ) be compactly supported, pointwise nonnegative and spherically symmetric. Let G N be defined as in (3.23). Assume that the exponent α introduced in (3.14) is such that Then, and there exists C > 0 such that, for all δ > 0 and all N ∈ N large enough, we have and the improved lower bound for all N ∈ N sufficiently large. Furthermore, there exists a constant C > 0 such that Finally, for every k ∈ N, there exists a constant C > 0 such that The

Cubic Renormalization
From Eq. (3.28), we observe that the cubic terms in G eff N still depend on the original interaction, which decays slowly in momentum (in contrast to the quadratic terms in the second line of (3.28), where the sum is now restricted To renormalize the cubic terms in (3.28), we are going to conjugate G eff N with a unitary operator e A , where the antisymmetric operator A : F ≤N The high-momentum set P H = {p ∈ Λ * + : |p| ≥ N α } is as in (3.14). The low-momentum set P L is defined by with exponent β > 0, that will be chosen as in (3.28).
Using the unitary operator e A , we define J N : F ≤N Observe here that we only conjugate the main part G eff N of the renormalized excitation Hamiltonian G N ; this makes the analysis a bit simpler (the difference G N −G eff N is small and can be estimated before applying the cubic conjugation). The next proposition summarizes important properties of J N ; it can be shown very similarly to [4,Prop. 5.2], of course with the appropriate changes of the scaling of the interaction. In the version of this paper that is posted on the arXiv, we give a complete proof of Proposition 4.
The bounds for J N given in Proposition 4.1 are still not enough to show Theorem 1.1. As we will discuss in the next section, the main problem is the quartic interaction term, contained in H N , which still depends on the singular interaction potential (in all other terms on the r.h.s. of (4.5), the singular potential has been replaced by the regular mean-field type potential, with Fourier transform 8πa 0 N κ 1 P c H (p), supported on momenta |p| < N α ). To renormalize the quartic interaction, we will have to conjugate J eff N with yet another unitary operator, this time quartic in creation and annihilation operators. This last conjugation (which will be performed in the next section) will produce error terms. These errors will controlled in terms of the observables N + , K and V N (as in (4.6)) but also, as we stressed at the end of Sect. 1, in terms of observables having the form N ≥N γ (the number of excitations having momentum larger or equal to N γ ), For this reason, we need to control the action of e A on all these observables.
Proof. The case m = 0 follows from m = 1. We start therefore with the case where A 1 as in (4.1). By the assumptions on γ and c, we have N α ≥ N α −N β ≥ cN γ for N ∈ N large enough. This implies in particular that for r ∈ P H and p ∈ P L , by (2.1) and (2.10). We then obtain (4.10) as well as (4.11) for some function Θ : N → (0; 1) by the mean value theorem. Using the pullthrough formula N + a * p = a * p (N + + 1) and Cauchy-Schwarz, we estimate With the operator inequality N ≥cN γ ≥ N ≥N α and with (4.7), we find that (4.12) The same arguments show that (4.13) Finally, we have that (4.14) Recalling (4.9), (4.10) and that α ≥ 2κ, the bounds (4.12) to (4.14) show that Since the bounds are independent of ξ ∈ F ≤N + and the same bounds hold true replacing A by −A in the definition of ϕ ξ , the first inequality in (4.8) follows by Gronwall's Lemma.
To prove (4.8) with m = 2, we proceed similarly. Given ξ ∈ F ≤N + , we define the function ψ ξ : R → R by Its derivative is equal to Comparing the contribution containing the double commutator in the last line on the r.h.s. of the last equation with (4.10) and using once again that (4.16) Hence, the bounds (4.12) and (4.13) prove that To bound the second contribution on the r.h.s. in (4.15), we recall (4.10) and we estimate Finally, the last contribution in (4.15) can be bounded as in (4.14), using (4.11). We have where, in the last step, we used that N ≥cN γ ≤ N + . In conclusion, we have proved that Since the bounds are independent of ξ ∈ F ≤N + and the same bounds hold true replacing −A by A in the definition ψ ξ , Gronwall's lemma implies the last inequality in (4.8).
We denote the kinetic energy restricted to low momenta by We will need the following estimates for the growth of the restricted kinetic energy.
Next, let us prove the second inequality in (4.18). We define ψ ξ : R → R by and we compute First, we proceed as in (4.20) and obtain with (4.7) that (4.21) Equation (4.21) and Lemma 4.2 then imply Next, we recall the identity in (4.10) and that by assumption on c 1 and γ 1 . We then estimate (4.23) Hence, putting (4.22) and (4.23) together, we have proved that . This implies the second bound in (4.18), by Gronwall's lemma.
Next, we seek a bound for the growth of the potential energy operator. To this end, we first compute the commutator of V N with the antisymmetric operator A. We introduce here the shorthand notation for the low-momentum part of the kinetic energy There exists a constant C > 0 such that (4.28) Here and in the following, the notation * indicates that we only sum over those momenta for which the arguments of the creation and annihilation operators are nonzero. The first term on the r.h.s. of (4.27) appears explicitly in (4.25), so let us estimate next the size of the operators Θ 1 to Θ 4 , defined in (4.28). The bounds can be obtained similarly as in the proof of [4, Prop. 8.1]. Consider first Θ 1 . For ξ ∈ F ≤N + , we switch to position space and find (4.29) The term Θ 2 on the r.h.s. of (4.28) can be controlled by In the last step, we used (4.7) to estimate for any ξ ∈ F ≤N + . The contributions Θ 3 and Θ 4 can be bounded similarly. We find Summarizing (using α > 3β + 2κ) we proved that   Proof. We apply Gronwall's lemma. Given ξ ∈ F ≤N + , we define ϕ ξ (s) = ξ, e −sA V N e sA ξ and compute its derivative s.t.
Hence, we can apply (4.25) and estimate Here, we used (3.10), which shows that η ∞ ≤ CN . Using Lemma 4.2, this simplifies to

Quartic Renormalization
To explain why the bounds for J N obtained in Prop. 4.1 are not enough to show Theorem 1.1, we introduce, for r ∈ Λ * + , the operators We denote the adjoints of c * r and e * r by c r and e r , respectively. Notice in particular that e * r = e −r for all r ∈ Λ * + . A straightforward computation shows that (5.2) Together with (4.5), this suggests to bound the Hamiltonian J N from below by completing the square in the operators g * r := b * r + c * r + e * r and g r := b r + c r + e r , for r ∈ P c H ⊂ Λ * + . A better look at (4.5) reveals, however, that several terms that are needed to complete the square are still hidden in the energy H N . Since these terms are not small, we need to extract them from H N by conjugation with a unitary operator e D , with The next proposition provides an important lower bound for M N . Its proof is given in Sect. 7. (3.14)) and β (in the definition of the set P L in (4.2)) are such that

Proposition 5.1. Suppose the exponents α (in the definition of the set P H in
be compactly supported, pointwise nonnegative and spherically symmetric. Then, M N , as defined as in (5.5), is bounded from below by for a self-adjoint operator E MN satisfying for all N ∈ N sufficiently large.

Proof of Theorem 1.1
For ε > 0 sufficiently small, we define With (6.2) and setting G N = G N − E N , we deduce that Next, we prove (1.5). From (3.29) and (6.3) we arrive at Writing G eff = e A J N e −A and recalling that κ < 1/43 (and that ε > 0 is small enough), Prop. 4.1 and (6.3) imply that Inserting J eff = e D M N e −D and applying Prop. 5.1, we obtain With K ≥ (2π) 2 N + and Lemma 4.2 (with m = 0 and k = 1) we have for a constant c > 0 small enough (but independent of N ). If N is large enough, we conclude (using also the upper bound (6.2)), that To bound the error term e A e D E MN e −D e −A , we need (according to (5.8)) to control observables of the form N −1 KN ≥cN γ . To this end, we observe, first of all, that, by Cauchy-Schwarz and by (6.3), (6.7) Choosing δ > 0 sufficiently small, we thus have Using (6.3) (similarly as we did in (6.7)) and N ≥cN γ ≤ N , N ≥cN γ ≤ CN −2γ K, we can bound the expectation of the first term on the r.h.s. of the last equation, for an arbitrary ξ ∈ F ≤N + , by On the other hand, to estimate the commutator term in Eq. (6.9), we notice that A : by (3.30). Setting μ = max(α, 3γ), this implies, with (6.3), for all ξ ∈ F ≤N + . Plugging (6.10) and (6.11) into (6.9), we find that, for sufficiently small δ > 0, Inserting into (6.8) and choosing δ > 0 small enough, we obtain 13) Applying (6.13) to the r.h.s. of (5.8) we find, using also (6.3), (6.1), and the choice κ < 1/43, (6.14) Inserting the last equation into (6.6) and using (6.2), we conclude that for N large enough, which proves (1.5), using Lemma 3.2. From (6.3), we obtain also an estimate that will be needed to arrive at (1.6). Evaluating (6.14) on a normalized ground state ξ N of G N and inserting the result in (6.4) we also deduce that Together with the upper bound (6.2), this concludes the proof of (1.3).
We still have to show (1.6) for k > 0. To this end, we will prove the stronger bound (1.8); Eq. (1.6) follows then immediately from N + ≤ H N and by Lemma 3.2. We denote by Q ζ the spectral subspace of G N associated with energies below E N + ζ. We use induction to show that for all k ∈ N, there exists a constant C > 0 (depending on k) such that for all k ∈ N. This proves (1.8) and thus, with the bound N + ≤ H N and with Lemma 3.2, also (1.6). The case k = 0 follows from (6.15). From now on, we assume (6.16) to hold true, and we prove the same bound, with k replaced by (k + 1) (and with a new constant C). To this end, we start by observing that combining (6.3) and (6.6), Hence, We estimate the first term on the r.h.s. by By Cauchy-Schwarz, we find for every ξ ∈ Q ζ . Hence, for any δ > 0, we have To bound the contribution proportional to e A e D E MN e −D e −A on the r.h.s. of (6.17), we have to control, according to (6.8), terms of the form For an arbitrary ξ ∈ Q ζ , we can bound the expectation of A by Cauchy-Schwarz as As for the term B, we can write From (6.18) and using (3.30) to estimate Applying the bounds N + ≤ N , N ≥cN γ ≤ CN −2γ K and (6.3) yields on the one hand for any δ > 0. Since 8κ + 2ε − γ ≤ 1 + κ/2 − γ and κ + γ/2 ≤ 1 + κ/2 − γ for all γ ≤ α if κ < 1/43, this implies with the choice δ = 1 (6.21) On the other hand, we can estimate (6.22) Expressing V N in position space, we find, with φ = N ≥cN γ (N + + 1) k+1 ξ, Ann We haveǎ is such that χ x = χ ≤ CN 3γ/2 . Hence, we find Inserting in (6.23), we find From (6.22), we conclude that for all γ ≤ α = 14κ + 4ε, if κ < 1/43. Using now similar arguments as before (6.21), we conclude that together with (6.21), we have Combining this with (6.20), we arrive at for all ξ ∈ Q z . With (6.8), we obtain Applying this bound to (5.8) and recalling that κ < 1/43, we conclude that Therefore, for any δ > 0, we find (if N is large enough) From the last bound, (6.19) and (6.17), we obtain for any ξ ∈ Q ζ . Taking the supremum over all ξ ∈ Q ζ , and choosing δ > 0 small enough, we arrive at by the induction assumption.

Growth of Number and Energy of Excitations
The first lemma of this section controls the growth of the number of excitations with high momentum. with D 1 as in (7.1). By assumption, N α ≥ N α − N β ≥ cN γ for sufficiently large N ∈ N. This implies that for r ∈ P H and p, q ∈ P L , by (2.1) and (2.10). We then compute (7.4) and apply Cauchy-Schwarz to obtain We have The contribution of the first term on the r.h.s. of (7.6) can be controlled as in (7.5) (replacing e sD ξ with (N ≥cN γ + 1)e sD ξ). With (7.4) and using again that All these contributions can be controlled like those in (7.4). We conclude that This proves (7.2) with m = 3. The case m = 2 follows by operator monotonicity of the function x → x 2/3 .
Next, we prove bounds for the growth of the low-momentum part of the kinetic energy, defined as in (4.17).
Next, let us prove the second inequality in (7.7). We define ψ ξ : R → R by ψ ξ (s) = ξ, e −sD K ≤c1N γ 1 (N ≥c2N γ 2 + 1)e sD ξ , and we compute ≤c1N γ 1 N ≥c2N γ 2 , D 1 e sD ξ . First, we proceed as in (7.9) and obtain with (4.7) that Here, we used in the last step that [a q−r , N ≥c2N γ 2 ] = a q−r for r ∈ P H , q ∈ P L and that N c2N γ 2 ≥ N N α −N β for N ∈ N large enough. The last bound and Lemma 7.1 imply that (7.10) Next, we recall the identity (7.4) and that K ≤c1N γ 1 , a * p+r = K ≤c1N γ 1 , a * q−r = 0 whenever r ∈ P H , p, q ∈ P L and N ∈ N is sufficiently large. We then obtain Hence, putting (7.10) and (7.11) together, we have proved that , which implies the second bound in (7.7), by Gronwall's lemma.
It will also be important to control the potential energy operator, restricted to low momenta. We define u∈Λ * ,p,q∈Λ * + : p+u,q+u,p,q∈PL p+u a * q a p a q+u . (7.12) Notice that V N,L = V * N,L by symmetry of the momentum restrictions. To calculate e D V N,L e −D , we will use the next lemma, which will also be useful in the next subsections. Lemma 7.3. Assume the exponents α, β satisfy (5.6). Let F = (F p ) p∈Λ * + ∈ ∞ (Λ * + ) and define Then, there exists a constant C > 0 such that 14) for all s ∈ [−1; 1], and for all N ∈ N sufficiently large.
which has derivative By assumption, we have α > 3β+2κ so that |r|, |v+r|, |w−r| ≥ N α −N β > N β if r ∈ P H and v, w ∈ P L , for sufficiently large N ∈ N. This implies in particular that [a p a q+u , a * v+r a * w−r ] = 0 whenever q + u, p ∈ P L and r ∈ P H , v, w ∈ P L . As a consequence, we find With (4.7) and N α − N β > 1 2 N α for N ∈ N large enough, we can bound , and for all N ∈ N sufficiently large.
We also need rough bounds for the conjugation of the full potential energy operator V N . To this end, we will make use of the following estimate for the commutator of V N with D = D 1 − D * 1 , with D 1 defined in (7.1).
Proof. We have To compute the commutator [V N , D 1 ], we compute first of all that Putting the terms in the first and last line on the r.h.s. into normal order, we obtain The first term on the r.h.s. in (7.18) appears explicitly in (7.16). Hence, let us estimate the size of the operators Φ 1 to Φ 4 , defined in (7.19).
Starting with Φ 1 , we switch to position space and find (7.20) The term Φ 2 on the r.h.s. of (7.19) can be controlled by Finally, the contributions Φ 3 and Φ 4 can be bounded as follows. We obtain In conclusion, the previous bounds imply with the assumption (5.6) (in particular, since α > 3β + 2κ and 3β − 2 < 0) that for any δ > 0. This concludes the proof. With Proposition 7.5, we obtain a bound for the growth of V N . Corollary 7.6. Assume the exponents α, β satisfy (5.6). Then, there exists a constant C > 0 such that the operator inequality for all s ∈ [−1; 1] and for all N ∈ N sufficiently large.
Finally, we need control for the growth of the full kinetic energy operator K. To this end, we need to estimate its commutator with D.
a w−r a v+r ξ avawξ By Cauchy-Schwarz, the first term on the r.h.s. of (7.27) can be controlled by The second contribution on the r.h.s. of (7.27) can be bounded by Similarly, we find that (7.29) In summary, the previous three bounds imply that 30) for some constant C > 0 and all δ > 0.

Action of Quartic Renormalization on Excitation Hamiltonian
We compute now the main contributions to M N = e −D J eff N e D . From (4.5) and recalling that [N + , D] = 0, we can decompose (7.41) Here, χ {p∈S} denotes as usual the characteristic function for the set S ⊂ Λ * + , evaluated at p ∈ Λ * + . Let us briefly explain how to bound the different contributions V 1 to V 5 , defined in (7.41). Using Cauchy-Schwarz, the first two contributions are bounded by where, for V 2 , we used that v + r ∈ P c H implies that |r| ≤ N α + N β and furthermore that N α ≤|r|≤N α +N β |η r | ≤ N κ+β . The contributions V 3 to V 5 , on the other hand, can be controlled by for any ξ ∈ F ≤N + . In conclusion (since 2κ + 3β − α/2 − 1 < κ from (5.6)), we have proved that and there exists a constant C > 0 such that (7.44) for all N ∈ N sufficiently large.
Proof. Let us define the operator Y : F ≤N This implies that it is enough to control the commutator [Y, D 1 ] after conjugation with e tD , for any t ∈ [−1; 1]. Note that, if p ∈ P c H , q ∈ P L , r ∈ P H and v, w ∈ P L , we have |v+r| ≥ N α −N β > 1 2 N α > N β s.t. [a * −p a q , a * v+r a * w−r ] = 0, for N ∈ N large enough. Then, a lengthy but straightforward calculation shows that As a consequence, we conclude that where Let us explain how to control the operators Ψ 1 to Ψ 6 , defined in (7.48). We start with Ψ 1 . Given ξ ∈ F ≤N + , we find that The contribution Ψ 2 can be bounded by Notice here, that we used that |r| ≤ N α + N β if r + v ∈ P c H and v ∈ P L . Next, we apply as usual Cauchy-Schwarz to estimate the terms Ψ 3 to Ψ 5 by for all α > 3β + 2κ. Finally, the term Ψ 6 can be controlled by In conclusion, the previous estimates show that so that, together with (7.46) and (7.47), an application of the Lemmas 4.2, 4.3, 7.1, 7.2 and the operator bound N ≥ 1 2 N α ≤ 4N −2α K proves the claim.
(7.52) and there exists a constant C > 0 s.t. E 1 (s) and E 2 (s) satisfy  for all p, q ∈ P L and r ∈ P H , v, w ∈ P L and N ∈ N sufficiently large. Then, proceeding as in the proof of Proposition 7.5, we obtain [a * p+u a * q−u a p a q , a * v+r a * w−r a v a w ] = −a * v+r a * w−r a * q−u a w a p a q δ p+u,v − a * v+r a * w−r a * p+u a w a p a q δ q−u,v − a * v+r a * w−r a v a * q−u a p a q δ p+u,w − a * v+r a * w−r a v a * p+u a p a q δ q−u,w . (7.57) Combining the last two identities and putting non-normally ordered contributions into normal order, we find that (7.59) Let us briefly explain how to control the operators ζ 1 to ζ 6 , defined in (7.2.3).
Noting that v + u ∈ P L implies |u| ≤ 2N β whenever v ∈ P L , the first two contributions ζ 1 and ζ 2 in (7.2.3) can be controlled by (7.60) By switching to position space, the term ζ 3 can be bounded by We proceed similarly as above for the terms ζ 4 and ζ 5 which yields where, for ζ 5 , we used that v + r + u ∈ P L implies that |u| ≥ 3 4 N α , and thus |q + u| ≥ 1 2 N α , whenever v, q ∈ P L , r ∈ P H and N ∈ N sufficiently large (otherwise |v + r + u| ≥ 1 4 N α − N β > N β for large enough N ∈ N). Finally, ζ 6 can be controlled by In summary, the previous estimates show that On the other hand, by Lemma 7.3, we also know that 64) where the self-adjoint operators E 1 (s) and E 2 (s) are bounded by as well as for all δ > 0 and uniformly in s ∈ [−1; 1]. Defining E 2 (s) = E 2 (s, N −β−κ ), this concludes the proof.
Choosing C > 0 sufficiently large (but independently of N ∈ N) and arguing as right before (7.66), we deduce that for all α, β satisfying (5.6) and N ∈ N sufficiently large. This follows through another application of Corollaries 4.5, 7.4 and 7.6, together with Lemmas 4.2, 4.3, 7.1 and 7.2. We summarize these bounds in the following corollary.
Corollary 7.13. Let m 0 ∈ R be such that m 0 β = α and let M (4) N be defined as in (7.35). For every C > 0, there exists a constant C > 0 such that for all exponents α, β satisfying (5.6) and for all N ∈ N sufficiently large.

Proof of Proposition 5.1
Recall from (7.34) the decomposition Collecting the results of Propositions 7.9, 7.10 and Corollary 7.13, we deduce that  Arguing in the same way for the contribution on the fifth line in (7.3), using that ( f N −η/N )(p) = δ p,0 for all p ∈ Λ * + , and using that v ∈ P L and v +r ∈ P L implies in particular that r ∈ P c H , we therefore obtain that such that, after switching to position space, the pointwise positivity V ≥ 0 implies (7.84) Here, we used that Λ * + = P L ∪ P c L and we denote byχ S the distribution which has Fourier transform χ S , the characteristic function of the set S ⊂ Λ * + . Combining (7.3), (7.3), (7.3) and (7.84), it follows that To express also the first term in the third line of (7.89) in terms of the modified creation and annihilation fields defined in (5.1), we first observe that Then, for a fixed r ∈ P c H , we have that where S 1 = (v, w) ∈ Λ * + × Λ * + : v ∈ P L , w ∈ P L , v + r ∈ P L , w − r ∈ P L , S 2 = (v, w) ∈ Λ * + × Λ * + : v ∈ P L , w ∈ P L , v + r ∈ P L , w − r ∈ P c L , S 3 = (v, w) ∈ Λ * + × Λ * + : v ∈ P L , w ∈ P L , v + r ∈ P c L , w − r ∈ P L , S 4 = (v, w) ∈ Λ * + × Λ * + : v ∈ P L , w ∈ P L , v + r ∈ P c L , w − r ∈ P c L , S 5 = (v, w) ∈ Λ * + × Λ * + : v ∈ P c L , w ∈ P L , v + r ∈ P L , w − r ∈ P L , S 6 = (v, w) ∈ Λ * + × Λ * + : v ∈ P L , w ∈ P c L , v + r ∈ P L , w − r ∈ P L , S 7 = (v, w) ∈ Λ * + × Λ * + : v ∈ P c L , w ∈ P c L , v + r ∈ P L , w − r ∈ P L .
In particular, the union 7 j=1 S j is a disjoint union. As a consequence, we find that This concludes the proof of Proposition 5.1.