Another proof of BEC in the GP-limit

We present a fresh look at the methods introduced by Boccato, Brennecke, Cenatiempo, and Schlein concerning the trapped Bose gas and give a conceptually very simple and concise proof of BEC in the Gross-Pitaevskii limit for small interaction potentials.


Introduction
One of the major achievements in mathematical quantum mechanics within the last 25 years was the proof of Bose-Einstein condensation (BEC) for trapped Bose gases by Lieb and Seiringer in 2002 [13], see also [14,16]. Their work was based on preceding works of Dyson [9] and Lieb and Yngvason [17] on the ground state energy of dilute Bose gases. Since then several different proofs of BEC in the Gross-Pitaevskii regime, and beyond, have been carried out. Nam, Rougerie and Seiringer [19] gave a proof using the quantum de Finetti theorem. Nam, Napiorkowski, Ricaud and Triaud [18] used ideas of [7,8], while more recently Fournais [10], by means of techniques developed in his joint work with Solovej [11] on the Lee-Huang-Yang conjecture, gave a relatively short proof, valid well beyond the GP-regime. While the above approaches are based on localization techniques in configuration space, Boccato, Brennecke, Cenatiempo and Schlein (BBCS), prior to [18,10,7,11], developed a different approach, more in the spirit of Bogolubov's original work. They use unitary rotations to encode the expected ground state. On the one hand they achieved optimal error bounds in their proof of BEC [2,4], on the other hand this approach culminated in the rigorous establishment of Bogolubov theory on a periodic box [3]. Although their works are voluminous, the methods are conceptually quite accessible and rather straight forward.
The aim of the present work is to take a fresh look at the approach of BBCS, with the obvious difference that we treat the system in a grand-canonical way, inspired by Brietzke and Solovej [7]. This allows us to conceptually simplify the approach of BBCS and additionally streamline the error estimates. A further advantage of this approach is that the emergence of the scattering length in the final result comes out automatically and does not have to be put in from the beginning. Thanks to the smallness assumption on the interaction potential we achieve to present a concise proof of BEC in the GP-limit with optimal error bounds. It should also be remembered that another advantage of the use of unitary rotations is the fact that one simultaneously produces precise upper and lower bounds. On the downside it is fair to mention that one needs regularity assumptions on the interaction V , excluding the hardcore potential which is included in the results [13,19,10,18]. The smallness assumption of the interaction potential simplifies our approach significantly. In [1] Adhikari, Brennecke and Schlein managed to overcome the smallness condition by an independent argument, without relying on previous results [13,19], using additional cubic and quartic transformations, which makes the proof technically even harder than [2,4]. This suggests that getting rid of this smallness condition, however, seems to be a major task within this approach. We consider a system of bosons in the Gross-Pitaevskii regime by means of the grand canonical Hamiltonian with H N = ∞ n=0 H n N , where the n-particle Hamiltonian is given by H µ is acting on the bosonic Fock space Further the GP-regime is reflected by the scaling of the potential where V (x) is assumed to be positive and compactly supported and We impose periodic boundary conditions on the box Λ = [−1/2, 1/2] 3 . In that sense the Hamiltoinan H µ should actually contain the periodized potential. However, we will work mainly with the variant in momentum space, where the periodization is automatic. Notice that N in the Hamiltonian H µ acts as a parameter. However, we choose the chemical potential so that the expected number of particles in the ground state, to the leading order, is N . More precisely, we follow [7] and choose the chemical potential as µ = 8πa, where a is the scattering length of κV .
In contrast to previous works we define the scattering length via its Born series, in the form for the simple reason that this is exactly the way how the scattering length appears in our approach. In a more concise way, see [12], the scattering length (1) can also be expressed as

Main results
For convenience we rewrite the Hamiltonian H µ in momentum space via with p ∈ Λ * := 2πZ 3 , and a † p := a † (φ p ), a p := a(φ p ) the usual creation and annihilation operators on the Fock space over the periodic box, i.e., φ p (x) = e ix·p . Notice, be the ground stat energy on the Fock space. The first theorem concerns the grand-canonical ground state energy. The statement resembles well known results in the literature, e.g. [15,2,3]. Its proof forms the basis for the subsequent establishment of BEC.
Theorem 1. Let µ = 8πa. Then for κ small enough asymptotically as N tends to infinity.
In the proof of Theorem 1 we use the fact, that the number operator N commutates with the Hamiltonian, i.e. [H µ , N ] = 0, which allows us to restrict the determination of the ground state energy as well as the proof of BEC to eigenfunctions ψ of the number operator, N ψ = nψ.
Remark 2. As a consequence of the proof of Theorem 1 we see that any approximate ground state ψ n , with fixed particle number n, N ψ n = nψ n , whose energy is , meaning that a-priori n equals the external parameter N only up to an error of √ N .
Theorem 3. Let µ = 8πa, and let ψ ∈ F be normalized, with N ψ = nψ and satisfy Then, for κ small enough, Since, More precisely, the highest eigenvalue of the one particle density matrix γ ψ of any approximate ground state ψ, with N ψ = nψ is macroscopically occupied, implying BEC with optimal error bounds in terms of n. Let us recall that in terms of creation and annihilation operators the one particle density matrix γ ψ can be expressed via the matrix elements ψ, a † p a q ψ , with p, q ∈ 2πZ 3 , i.e.,γ ψ (p, q) = ψ, a † p a q ψ .
Remark 4. As easy consequence of the proof of Theorem 1 we will see that one can find a state Ψ N , with N ψ N = N ψ N , and such that the corresponding one-particle density matrix γ ψ N satisfies with optimal rate in the parameter N .
As corollary of Theorem 1 and Remark 4 we immediately obtain the energy asymptotic of the ground state energy of an N -particle system. Recall, Corollary 5. Let E N = inf spec H N N , then Proof. The upper bound is provided in the proof of Theorem 1, cf. Remark 4, by an explicit trial state with particle number N . The lower bound follows from a simple variational argument [7]. Let ψ N be the ground state of H N N . Then, (µ = 8πa), using Theorem 1.
Remark 6. The proof of Theorem 1 can easily be extended to general values of the chemical potential µ > 0. Indeed, for any µ > 0 and κ small enough one gets as N tends to infinity. For approximate ground states ψ ∈ F, with N ψ = nψ, and one obtains again complete condensation ψ, N + ψ ≤ O(1), however, the expectation number of particles is now n = N µ In the following section 3 we present the main steps of the proof of Theorem 1 and complete the proof of BEC in section 4. The rest of the paper is concerned with technical estimates which are not important for the understanding of the main ideas of the proof.

Strategy and main steps of the proofs
Notice that the Hamiltonian H µ and the number operator N commute, which tells us that we can restrict to states with fixed quantum number N ψ = nψ. Following ideas from Brietzke and Solovej [7] we can restrict our attention to the case where N ≤ 10N , with 10 being chosen for aesthetic reasons (anything larger than 4 would do).
The key observation from [7] is that whenever there are more than 10N particles one can combine them in groups with each group consisting of a number of particles between 5N and 10N . Since the interaction is positive, one can simply drop the interaction between different groups for a lower bound. Since we will further show that the energy of a system with more than 5N particles is actually nonnegative, this tells us that we can restrict from the very beginning to N ≤ 10N . Notice that in the grand canonical case with positive chemical potential it is easy to see that the ground state must necessarily be negative.
Under the assumption of N ≤ 10N we are now in the position to apply the strategy developed by Boccato, Brennecke, Cenatiempo, and Schlein [2,4,3], based on ideas of [6].
We will look for an appropriate unitary rotation e B , with B = B −B * a number conserving operator on the Fock space, which encodes the ground state, in the sense that e −B H µ e B has, to leading order, Π N i=1 φ 0 as approximate ground state. This further implies that ψ ≃ e B Π N i=1 φ 0 is an approximate minimizer for H µ .
First we follow Bogolubov's way and decompose the interaction potential in different terms depending on the number of a 0 's and a † 0 's: Let us denote the number operator counting the number of particles in the state φ 0 as The rest of the interaction then has the form In the following we will assume that a p or a † p automatically means that p = 0 which allows us to skip the distinctions in the sums. E. g., instead of p =0 a † p a p we simply write p a † p a p . Let us recall Duhamel's formula Applying the second equality of the formula to H 1 + Q 4 and the first to Q 2 we obtain Let us explain the main idea of the strategy. The term H 0 (µ) clearly contributes to the leading term. In Bogolubov's original approach Q 3 and Q 4 was omitted and Bogolubov [5] diagonalized the quadratic part H 1 + H 2 + Q 2 . But he did not get the leading term correctly, since he missed the contribution coming from Q 4 . We perform an "almost" diagonalization by choosing B in such a way that We will treat H 2 and Q 3 as error terms, since they do not contribute to the leading order. The requirement that [H 1 + Q 4 , B] + Q 2 vanishes apart from higher order terms, suggests a choice of B, of the form with ϕ p appropriately chosen. In fact, to leading order, ϕ p will satisfy the scattering equation.
Lemma 7. Let B be defined as in (5). If ϕ p satisfies the equation then with Proof. Straightforward calculations yield where the two terms in the last line are actually equal, which can be seen by changing variables, p → p − r, q → q − r and then r → −r. Collecting all terms involving a † p a † −p a 0 a 0 and recalling the form of Q 2 we see that Remark 8. Let us apply the discrete inverse Fourier transform where P 0 is the projection on the constant function φ 0 , and the orthogonal projection comes about because all sums run over p = 0. Applying this transformation to equation (6) and assume that N is large enough thatV (x) = V (x) on [−N/2, N/2], then we obtain the equation which can be inverted by Among others this shows that equation (6) has a unique solution ϕ p . Useful properties of this function ϕ p are provided in Lemma 14.
We continue with equation (4). We plug [H 1 + Q 4 , B] = −Q 2 + Γ into the last two terms in (4) and obtain 1 0ˆs 0 Rewriting (4) accordingly we arrive at The idea now is rather simple. The leading order is contained in the first two terms on the right hand side, The positive term H 1 + Q 4 is used to dominate the error terms coming from the rest. Here the smallness of the potential κ will be used. Thanks to the gap, all errors of the form κCN + will be absorbed by H 1 for small enough κ. In the following we extract the leading contribution of the term´1 0´1 s e −tB [Q 2 , B]e tB dtds. with Proof. We calculate The two terms in the bracket are hermitian conjugates. Hence it suffices to calculate the second one, (15) and integrating over s, t implies the statement. Observe that for the leading term the integral over s, t gives a factor 1/2.
Let us now define With this definition and the previous Lemma we rewrite (12) as where E = Γ +ˆ1 Let us recall that we are able to restrict to wavefunctions with fixed particle number, N ψ = nψ, (and additionally assume n ≤ 10N ). Further, The following lemma, which was proven in [2], tells us that the error terms E + Ξ can be absorbed by H 1 + Q 4 .
Then for κ small enough Furthermore, we now point out that a N converges to the scattering length, see [6,2].
Lemma 11. Let a be the scattering length defined in (1). Then we obtain for a N , defined in (16), We postpone the proof of Lemma 11. Applying these two lemmata to (18) we conclude that there is a constant C, such that Lemma 12. With µ = 8πa and N ψ = nψ, and n ∈ [5N, 10N ], then for N large enough.
Lemma 12 and equation (21) imply for µ = 8πa that This implies the lower bound in the statement of Theorem 1. The upper bound is obtained using the simple trial stateψ = e B Π N i=1 φ 0 plugged into (18). This implies where we used the simple fact
Equation (21) implies for such ψ and µ = 8πa that Hence, in order to deduce condensation it suffices to show that N + is invariant under unitary transformation of e B , at least for N ≤ 10N .
By Lemma 13 which finally allows us to conclude from (23) that which implies complete BEC condensation. Further

Proof of Lemma 10
The proof of Lemma 10 was carried out in detail in [3]. The estimates of some terms are tedious, however, straightforward. The goal of this section is to outline and streamline the strategy of [3]. Let us start with collecting some information about ϕ p .

Lemma 14.
For small enough κ one has Proof. The estimates (29), (28), (31) are a consequence of the inequality To see this, recalling (6), we estimate which can be seen by treating the latter expression as the Riemann sum of the convolution |V | * 1/p 2 which is a uniformly bounded function. Hence one can bound the absolute value of the left hand side of (6) from below by sup p |p 2 ϕ p |(1 − Cκ) which implies (32) using the boundedness ofV . The inequality |ϕ p | κ/p 2 immediately implies (29), (28), (31). In order to see (30) we need more decay for large p. In configuration space this corresponds to more smoothness. This is usually implied by a bootstrap argument. This can be done here as well. Plugging |ϕ p | κ/p 2 into equation (6) yields with G(p/N ) at least bounded by Continuing the bootstrap argument allows to improve the fall off properties of ϕ p further, which however, is not necessary for our purpose. The bound (33) implies (30) by considering the sum as Riemann sum.
Let us remark that we perform all our estimates on states ξ ∈ F, with N ξ = nξ, n ≤ 10N.
Equivalently we will frequently use the operator estimates We start with looking at the terms Ξ + E. The strategy is rather straightforward. Whenever the terms inside the bracket of e −sB (...)e sB can be estimated by C(N + + 1) m , then Lemma 13 can be used to bound the total expressions by κ(N + + 1).
Let us demonstrate this in the case of the first two terms of Ξ in (13) as well as e −B H 2 e B . For convenience denote Using N + ≤ N N and Lemma 13 we derive where in the last step we used (31). The estimate for e −B H 2 e B works in an analogous way.
Hence we obtain Next, let us look at the term Γ. This cannot simply be estimated by N + . We additionally need the interaction Q 4 . This is no problem as long as the term is not in between e −sB ...e sB , due to the fact that e −sB Q 4 e sB cannot be dominated by Q 4 and N + , since e −sB Q 4 e sB produces an terms of order N .
Remark 15. In the following it is important to absorb some error terms by the interaction term Q 4 . To this aim let us recall first that the bosonic Fock space can written as where F 0 is the Fock space spanned by the one dimensional space {φ 0 } and F ⊥ = F(H ⊥ ), the Fock space built by all states orthogonal to φ 0 . In order to see the positivity of Q 4 one has to rewrite the term in configuration space. Witȟ where the last term is fundamentally positive for any positive interaction. For that reason it turns out to be convenient to estimate some of the error terms in configuration space. Hence, whenever we use the interaction˜dxdyκV N (x − y)ǎ † xǎ † yǎyǎx it has to be remembered that it only acts on F ⊥ . For sake of convenience we omit the corresponding symbols indicating the restrictions on F ⊥ .
Lets come back to the term Γ. Denoting Λ = [−1/2, 1/2] 3 we calculate V N (r)ϕ p a † p+r a † q a † −p a q+r a 0 a 0 = r,p,qV where we used from the second to the third line that q e iq·(y−w) = δ(y − w). In terms of expectation values we thus obtain for Γ By means of Cauchy-Schwarz, and the fact that V N 1 = V 1 /N , we conclude that for any δ there is a C δ such that ξ, Γξ ≤ δ ξ, Q 4 ξ + C δ κ 3 ξ, (N + + 1)ξ .
Next, consider the termˆ1 For convenience, we neglect the terms a 0 a 0 /N , which are bounded by a constant anyway at the end. To this aim we first calculate the commutator [Γ, B] Evaluating these commutators leads to three types of terms. First second, −N 0 (N 0 − 1)/N 2 times the expression κ p,q,rV N (r)ϕ p a † p+r a † q a p a q+r ϕ p + a † p+r a † −p a −q a q+r ϕ q + a † q a † −p a −p−r a q+r ϕ p+r The third term stems from the commutator [a 0 a 0 , a † 0 a † 0 ] = 2(2N 0 + 1), i.e., Recall that all terms have to be sandwiched between e −tB ...e tB . This complicates the estimates whenever it is not possible to bound the terms solely by the number operator N + , but instead we are forced to use the potential Q 4 . For that reason we postpone the estimation of the terms (38) and (40). For the moment we only concentrate on (39). Let us start with the quadratic expressions in the last line in (39). Since (N 0 +1) N 1, the corresponding first two terms in the second line of (39) are simply bounded by using the L 2 -bound of ϕ p . For the third quadratic term we use that with Lemma 14, such that κ r,pV N (r)ϕ p ϕ p+r a † p a p κ 3 N + .
Concerning the quartic terms in (39), the term where both functions ϕ * have the same index cannot be estimated solely by N + either, but also needs the interaction Q 4 . The other terms, however, can simply be bounded by N + . Thanks to Lemma 13 the application of e −B ...e B lets the bounds unchanged. Let us demonstrate such an estimate on the term including ϕ p ϕ q . Using Cauchy-Schwarz in p, q, r we obtain κ N p,q,rV (r/N )ϕ p ϕ q ξ, a † p+r a † −p a −q a q+r ξ κ N p,q,r |ϕ q | a −p a p+r ξ a −q a q+r ξ |ϕ p | κ 3 N + ξ 2 N κ 3 N whose estimates are more elaborate. Brennecke and Schlein realized in [6] that these terms can be expressed via a convergent geometric sum where the bounds of each terms can be classified in a straight forward way. Thereby the convergence is guaranteed by the smallness of κ. In the following we present an alternative way of estimating these terms. The method is essentially the same for all terms in (42). We will demonstrate the method on the first and the last two terms. The others work analogously. We start with the term e −B Q 3 e B . To this aim we rewrite it as Via Duhamel's formula we have The idea behind this is the simple fact that the corresponding term in (43) involving a † q+r a † −r will be estimated by Q 4 . The remaining terms, however, coming from [a † q+r a † −r , B] can be bounded by N + , which is stable under application of e −B ..e B . In order to recover Q 4 the term κ N q,rV (r/N )a † q+r a † −r e −B a q a 0 e B has to be estimated in configuration space, where the term reads whose expectation value of ξ is bounded by The remaining term has the form Since [a † q+r a † −r , B] = − 2 N ϕ r a † q+r a r + ϕ r+q a † −r a −q−r + ϕ r δ q,0 a † 0 a † 0 , and the fact that the sum q,r , by assumption, does only include indices different from 0, only the first two terms need to be estimated. Since they are similar we only consider Using Cauchy-Schwarz for the expectation value of ξ we deduce 0 ds a q+r a 0 a 0 e sB ξ a r e (s−1)B a q a 0 e B ξ κ 2 (N + + 1)ξ 2 , where we used |a 0 a 0 | N N , Lemma 13, and Next we look at the second to last term in (42). To this aim, notice that the operator norm of Φ = p ϕ p a −p a p can be estimated by which can be seen by applying Cauchy-Schwarz Further we write and Hence, the expression coming from the second term on the right hand side of (50) gives The first term on the right hand side of (50), including a † p a † −p , is again evaluated by rewriting it in configuration spacê which can be bounded by Finally, consider the last term in (42), which we conveniently rewrite as 0ˆ1 s e −tB a † p+r a † q e tB e −tB a † −p a q+r Φ(N 0 + 1)e tB dtds, using again the notation Φ = k ϕ k a −k a k . Next we apply Duhamel again, similar to (44), to e −tB a † p+r a † q e tB and obtain two terms, where the first one including a † p+r a † q has to be estimated by Q 4 . More precisely, in configuration space that term has the form N 2 e tB dtds δQ 4 + κ 2 (N + + 1). is again estimated by κ 3 (N + + 1).

Proof of Lemma 11
Using (9) we can write where we used (11) to obtain the last equality. This implies now for a N 4πa N = κV (0) 2 On a formal level this converges for N → ∞ as with v = κ 2 V . The right hand side is 4πa, with a being the scattering length. In order to obtain the bound |a − a N | ≤ O(1/N ), lets denote |e p e p |, |e p = e −i p N ·x /N 3/2 , such that 1 N + |e 0 e 0 | is the identity on L 2 ([−N/2, N/2] 3 ). Then we can write v, P ⊥ with δ N = 1−1 N , where 1 denotes the identity on L 2 (R 3 ). Notice, we implicitly assume that the application of 1 N means that one only integrates over [−N/2, N/2] 3 . We also assume that N is large enough such that the support of v is in [−N/2, N/2] 3 . Hence, we can write the difference of a − a N as Observe that ϕ = − 1 −∆+v v is the solution of the scattering equation on the whole space, which is smooth function with falloff 1/|x| in configuration space, due to the properties of V . The first term of the right hand side of (55) is which is the difference of the Riemann sum and its integral. A second order Taylor expanding ofφ(p) shows that this error is of order 1/N due to the 1/p 2 behavior for small p. Notice that a quick first order expansion gives an error of log N/N . For the second term in (55), observe which is again the difference of a specific integral and its Riemann approximation.