Derivation of the Time Dependent Gross–Pitaevskii Equation in Two Dimensions

We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting from an interacting N-particle system of bosons. We consider the interaction potential to be given either by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_\beta (x)=N^{-1+2 \beta }W(N^\beta x)$$\end{document}Wβ(x)=N-1+2βW(Nβx), for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta >0$$\end{document}β>0, or to be given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_N(x)=e^{2N} V(e^N x)$$\end{document}VN(x)=e2NV(eNx), for some spherical symmetric, nonnegative and compactly supported \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W,V \in L^\infty ({\mathbb {R}}^2,{\mathbb {R}})$$\end{document}W,V∈L∞(R2,R). In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_N$$\end{document}VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.


Introduction
We are interested in the time evolution of bosonic quantum systems of N particles in two dimensions that interact with each other by a two-particle interaction potential. At a given time t, the state of the system is described by a wave function t ∈ L 2 s (R 2N , C), where L 2 s (R 2N , C) denotes the Hilbert space of all ∈ L 2 (R 2N , C) which are symmetric under permutations of the variables x 1 , . . . , x N ∈ R 2 . The Hamiltonian of the system is given by with A · : R 2 × R → R being a time-dependent external potential and U : R 2 → R modeling the interaction between the particles. The time evolution of the system is described by the Schrödinger equation with initial datum 0 ∈ L 2 s (R 2N , C). In general, even for small particle numbers N , it is not possible to solve the Schrödinger equation exactly or numerically. The time evolution of the system, however, can approximately be determined if one studies special classes of initial conditions and certain types of interaction potentials. In this paper, we are concerned with the dynamical evolution of a Bose-Einstein condensate. This state of matter appears if one cools bosons in an external trapping potential near absolute zero temperature such that almost all particles occupy the same quantum state (see e.g. [38] for a comprehensive discussion). After the trapping potential has been changed or completely switched off, the condensate is no longer in equilibrium and one would like to study its evolution in space.
Mathematically, the appearance of a Bose-Einstein condensate is described by means of the one-particle reduced density matrix γ (1) of the state . γ (1) is a non-negative trace class operator on L 2 (R 2 , C) with an integral kernel given by γ (1) A state is said to exhibit complete Bose-Einstein condensation, if there exists a oneparticle wave function ϕ ∈ L 2 (R 2 , C) with ϕ = 1 such that γ (1) → |ϕ ϕ| in trace norm as N → ∞. 1 Initially, we consider a complete condensed state 0 and then show 1 We like to remark that it is well known that the convergence of γ (1) to |ϕ ϕ| in trace norm is equivalent to the respective convergence in operator norm since |ϕ ϕ| is a rank-1-projection, see Remark 1.4. in [51]. For other indicators of condensation and their relation we refer to [41].
that γ (1) t → |ϕ t ϕ t | as N → ∞, where ϕ t solves a nonlinear Schrödinger equation. This statement shows that the condensate is stable during the time evolution. Moreover, it proves that the time-evolution of the one-particle reduced density matrix which is given by the many-body Schrödinger equation can approximately be described by a much simpler nonlinear one-particle equation.
To state the exact form of the one-particle equation, we specify the potentials U we are interested in.
• For β > 0, we consider the so called nonlinear Schrödinger (NLS) scaling U (x) = W β,N (x) = N −1+2β W (N β x), for a compactly supported, spherically symmetric and nonnegative potential W ∈ L ∞ c (R 2 , R). In the case of β > 1/2, such a scaling models strong but short range repulsive interactions. The origin of the scaling can heuristically be motivated by the fact that for a completely factorized wave function = ϕ ⊗N with ϕ ∈ H 2 (R 2 , C) the kinetic energy per particle 2 is of the same order as the potential energy per particle 1 with V ∈ L ∞ c (R 2 , R) being spherically symmetric and nonnegative. This scaling will be denoted Gross-Pitaevskii scaling in the following. The motivation to consider an exponential scaling is similar to the Gross Pitaevskii scaling V N (x) = N 2 V (N x) in three space dimensions. Namely, the kinetic and interaction energy are of the same order for a gas of fixed volume. This will be shown below, when discussing the scattering process of two particles, see (4). Furthermore, the interaction originates from a N -independent potential by rescaling space and time coordinates [see (7)]. Our results can be generalized to a wider class of N -dependent interactions covering most of the relevant cases discussed in the literature on two dimensional Bose gases [39].
For these scalings the condensate wave function ϕ t satisfies the cubic nonlinear Schrödinger equation with initial datum ϕ 0 . The precise definition of b U will be given in Definition 2.1. At the moment however if suffices to note that for the potentials from above we have b W β,N = N W β,N 1 = W 1 if U = W β,N and b V N = 4π for U = V N . In case that the coupling constant is given by b V N = 4π Eq. (3) is also referred to as Gross-Pitaevskii equation.
We are going to explain on a heuristic level why the coupling constants differ in the NLS and Gross-Pitaevskii scaling. We first consider the exponential scaling and assume that the energy of the many-body state t is comparable to the ground state energy of the system. In this case, the wave function develops a short scale correlation structure which prevents the particles from being too close to each other [39]. If we neglect for the moment all but two particle correlations, one may heuristically think of t to be of Jastrow-type [38, p. 15 and p. 28], i.e. t (x 1 , . . . , x N ) ≈ i< j F(x i − x j ) N k=1 ϕ t (x k ). The function F accounts for pair correlations between the particles at scales of order O(e −N ). These correlations determine the time evolution of the condensate in a crucial manner and must therefore explicitly be taken into account. Since V N is a strong, short range potential, the interaction between the particles can in first order be described as a two-body scattering process. That is, the correlation function F should approximately be given by the zero energy scattering state j N ,R ∈ C 1 (R 2 , R) which is defined by for some R ∈ (0, ∞) used to normalize j N ,R via the second line of (4). Note, that it is a peculiarity of two dimensional scattering states that lim x→∞ | j N ,R (x)| does not exist for short range potentials and can not be used for normalization. A particle at location x then experiences the effective interaction see e.g. [19] for a nice derivation. It will be shown in Sect. 5 that where a denotes the scattering length of the potential V . Since 4π ln R ae −N ≈ 4π N holds for a > 0, the effective coupling b V N will be given by 4π . This shows that the scaling we used gives us a system where the kinetic energy and the interaction energy are of the same order.
Let us now turn to the NLS scaling and consider for β > 0 the scattering equation of the potential W β,N − x + N −1+2β W (N β x) F N ,β,R (x) = 0, F N ,β,R (x) = 1 for |x| = R.
Due to the factor N −1 , the zero energy scattering state is almost constant for large N , F N ,β,R (x) ≈ 1 ∀ |x| ≤ R. It can therefore be concluded that the microscopic structure has a negligible effect on the effective interaction on each particle which is approximated by 3 3 Even if the effect of the microscopic structure is negligible in the interaction, one should note that t is not close to a full product state N k=1 ϕ t (x k ) in norm. For certain types of interactions, it has rigorously been shown that t can be approximated by a quasifree state satisfying a Bogoliubov-type dynamics. We refer to [5,12,[23][24][25]31,35,[43][44][45][46] for the precise statements. This yields to the correct coupling in the effective equation (3) in the case of U (x) = W β,N (x).
Let us briefly compare the phenomenon of Bose-Einstein condensation in two and three dimensions. In three dimension the NLS scaling is defined by N −1+3β W (N β x) only for 0 < β < 1 while in the case of β = 1 the microscopic structure must be taken into account. This difference originates from the different form of the scattering state in two and three dimension, see Appendix C of [38]. In the case that the time evolution of t is generated by H V N it is interesting to note that the effective evolution equation of ϕ t does not depend on the scattering length a. Also this contrasts the three-dimensional case, where the correct mean field coupling is given by 8πa 3D , a 3D denoting the scattering length of the potential in three dimensions. The universal coupling 4π in the case of a two-dimensional setup is known within the physical literature, see e.g. (30) and (A3) in [18] (note thath = 1, m = 1 2 in our choice of coordinates). Actually, our dynamical result complements a more general theory describing the ground state properties of dilute, two-dimensional Bose gases. It was shown in [39] that for a gas with repulsive interaction V ≥ 0, the ground state energy per particle is to leading order given by either the Gross-Pitaevskii energy functional with coupling parameter 8π/| ln(ρa 2 )| or a Thomas-Fermi type functional, depending on the diluteness of the gas, i.e. the mean-particle distance compared to the scattering length of the interaction. Here, ρ denotes the mean density of the gas and a is the scattering length which must decrease exponentially with N in the Gross-Pitaevskii limit [39, p. 20].
It should be pointed out that there has been some debate about the question whether two-dimensional Bose-Einstein condensation can be observed experimentally. This amounts to the question whether condensation takes place for temperatures T > 0. For an ideal, noninteracting gas in a box, the standard grand canonical computation for the critical temperature T c of a Bose-Einstein condensate shows that there is no condensation for T > 0. For trapped, noninteracting bosons in a confining power-law potential, the findings in [3] however show that in that case T c > 0 holds. Finally, it was proven in [37] that γ (1) converges to |ϕ ϕ| in trace norm if is the ground state of H V N and ϕ is the ground state of the Gross-Pitaevskii energy functional, see (8). It was furthermore proven that one does not observe 100% condensation in the ground state of an interacting homogenous system. The emergence of 100% Bose-Einstein condensation as a ground state phenomenon thus highly depends on the particular physical system. Our approach is the following: Initially, we assume the convergence of γ (1) 0 to |ϕ 0 ϕ 0 |. We then show the persistence of condensation for time scales of order one. Our assumption is thus in agreement with the findings in [37].
The rigorous derivation of effective evolution equations is well known in the literature, see e.g. [2,5,[9][10][11][19][20][21][22]30,43,44,[48][49][50][51] and references therein. For the twodimensional case we consider, it has been proven, for 0 < β < 3/4 and W nonnegative, that γ (1) t converges to |ϕ t ϕ t | as N → ∞ [27]. For 0 < β < 1/6, it has been established in [14] that the reduced density matrices converge, assuming that the potential W is attractive, i.e. W ≤ 0. This result was later extended to a larger class of scaling parameters β, under some assumptions on the negative part of the potential W [26,34]. In [45] a norm approximation to the two-dimensional focusing Schrödinger equation in the NLS scaling with 0 < β < 1 was considered. Here, the evolution of the condensate is effectively described by the nonlinear Schrödinger equation while the evolution of the fluctuations around the condensate is governed by a quadratic Hamiltonian, resulting from Bogoliubov approximation. Another approach which relates more closely to the experimental setup is to consider a three-dimensional gas of bosons which is strongly confined in one spatial dimension. Then, one obtains an effective two-dimensional system in the unconfined directions. We remark that in this dimensional reduction two limits appear, the length scale in the confined direction and the scaling of the interaction in the unconfined directions. A derivation of the two-dimensional Gross-Pitaevskii equation from the three-dimensional quantum many-body dynamics of strongly confined bosons was just recently given in [7]. Further results in this direction can be found in [4,6,8,[15][16][17]28,29]. For known results regarding the ground state properties of dilute Bose gases, we refer to the monograph [38], which also summarizes the papers [37,39,40].
Our proof is based on [49], which covers the derivation of the time dependent Gross-Pitaevskii equation in three dimensions. In particular, the exponential scaling of the interaction forces us to adapt crucial ideas and refine many estimates. Additional difficulties arise amongst others from the logarithmic behaviour of the scattering state and the fact that We shortly discuss the physical relevance of the Gross-Pitaevskii scaling. It is possible to rescale space-and time-coordinates in such a way that in the new coordinates the interaction is not N -dependent. Choosing y = e N x and τ = e 2N t the Schrödinger equation reads The latter equation thus corresponds to an extremely dilute gas of bosons with density ∼ e −2N . In order to observe a nontrivial dynamics, this condensate is then monitored over time scales of order τ ∼ e 2N . Since the trapping potential is adjusted according to the density of the gas in the experiment, the N dependence of A e −2N τ (e −N ·) is reasonable.

Main Result
Our main theorem consists of two parts, which consider potentials in the NLS and Gross-Pitaevskii scaling, respectively. For the proof of the theorem it is useful to enlarge the class of potentials in the NLS regime because it allows us in the derivation of the Gross-Pitaevskii equation to refer to various estimates that appear in first part of the proof.
(c) Define the set of potentials W β by To ease the notation, we often omit to display the dependence on N and denote both the sequence W β,N N ∈N and the element W β,N by W β .
For notational convenience, it is in addition helpful to define a class of potentials with Gross-Pitaevskii scaling.
With a slight abuse of notation we use V N to denote the sequence V N N ∈N and its N th element.
where ·, · denotes the scalar product on L 2 (R 2N , C). Furthermore, define the Gross-Pitaevskii energy functional where ·, · denotes the scalar product on L 2 (R 2 , C). Note that both E U ( ) and E G P b U (ϕ) depend on t, due to the time varying external potential A t . For the sake of readability, we will not indicate this time dependence explicitly. Our main theorem is the following.
(a) Let β > 0, W β ∈ W β and let t the unique solution to i∂ t t = H W β t with initial datum 0 . Let ϕ t the unique solution to i∂ t ϕ t = h G P b W β ϕ t with initial datum ϕ 0 and Then, for any t > 0 there exists a constant 0 < C t < ∞, which depends on t but not on N , such that Tr γ (1) t − |ϕ t ϕ t | ≤ e C t 4 Tr γ (1) where γ = β for 0 < β < 1/12 and γ = 1/20 for β ≥ 1/12.
Then, for any t > 0 there exists a constant 0 < C t < ∞, which depends on t but not on N , such that Remarks. (a) If one considers initial many-body states which exhibit condensation and whose energy per particle converges to the corresponding Gross-Pitaevskii energy, i.e.

lim N →∞
Tr γ (1) Our result consequently shows the stability of the condensate during the time evolution. (b) It has been shown that in the limit N → ∞ the energy-difference E V N ( gs ) − E G P 4π (ϕ gs ) → 0, where gs is the ground state of a trapped Bose gas and ϕ gs the ground state of the respective Gross-Pitaevskii energy functional, see [39,40].
(c) The necessity to require ϕ t ∈ H 3 (R 2 , C) stems from the fact that the constant C t in (9) and (11) depends on ϕ t H 3 , see the discussion before Lemma 4.7. For regular enough external potentials A t we expect the assumption ϕ t ∈ H 3 (R 2 , C) to follow from regularity assumptions on the initial datum ϕ 0 .
x f < ∞} holds, the bound ϕ t H 3 < ∞ has been proven for external potentials which are at most quadratic in space, see [13] and Lemma 4.7. In particular, for ϕ 0 ∈ 3 (R 2 , C), the bound ϕ t H 3 ≤ C with C > 0 uniformly bounded in t holds if the external potential is not present, i.e. A t = 0 [see [13] above (1.3.)]. (d) One can relax the conditions on the initial condition and only require 0 ∈ L 2 s (R 2N , C) using a standard density argument.

Organization of the Proof
The method we are applying to prove Theorem 2.4 was originally introduced in [50] and later generalized to derive various mean-field equations [1,8,30,32,33,42,[46][47][48][49]. Our proof is primarily based on [49] which covers the three-dimensional counterpart of our system. The key idea of the method is to show the existence of Bose-Einstein condensation not in terms of reduced density matrices but to consider an equivalent measure of condensation. Heuristically speaking, we count for each time t the relative number of those particles which are not in the state of the condensate wave function ϕ t . It is then possible to show that the rate of the particles which leave the condensate is small, if initially almost all particles were in the state ϕ 0 . The counting of the particles will be performed with the help of a functional. In order to define it, we introduce the following operators.
and q ϕ j = 1 − p ϕ j . We shall also use, with a slight abuse of notation, the bra-ket notation p For negative k and k > N we set P ϕ k = 0.
(c) For any function m: N 0 → R + 0 we define the operator m ϕ : We also need the shifted operators m ϕ d : Following a general strategy 4 we will define a functional α: If t is a solution of (2) and ϕ t a solution of (3), α( t , ϕ t ) can be estimated by α( 0 , ϕ 0 )+ t 0 ds C s α( s , ϕ s )+O(1) for some time dependent constant C s . Using a Grönwall type estimate, it then follows that α( t , ϕ t ) ≤ e 2 t 0 dτ C τ α( 0 , ϕ 0 ) + O (1) . (c) α( t , ϕ t ) → 0 implies the convergence of the one-particle reduced density matrix of t to |ϕ t ϕ t | in trace norm as well as the convergence of the energy per particle of the many-body system to the energy of the condensate wave function.
In [30,50] the mean field scaling W 0 (x) = N −1 W (x) and a condensate wave function which evolves according to the Hartree equation i∂ t ϕ t = − + A t ϕ t + W * |ϕ t | 2 ϕ t were considered in the three-dimensional setting. In these works it was shown that the persistence of condensation can be proven if one chooses where n(k) = √ k/N , j > 0 and t is a solution of (2) with U = W 0 . The choice j = 2 corresponds to the functional t , N k=0 k N P ϕ t k t , whose action on t can be viewed as "counting the relative number of particles which are not in the state ϕ t ". Other values of j or a different choice of m ϕ t should be understood as a weighted measure of counting the number of particles which are not in the condensate state. We will therefore sometimes call m the weight function of the functional α.
In this work we are interest in interaction potentials which get peaked as N → ∞. As explained in Sect. 6.1, it is then no longer possible to obtain a Grönwall estimate with the previous choice of the functional and we have to adjust it in accordance with the scaling of the interaction. The precise definition of the functional and the proof of Theorem 2.4 are given in Sect. 6. In the preceding chapters we introduce the necessary preliminaries.
The rest of the paper is organized as follows: (a) In Sect. 4 we start by fixing the notation. Afterwards, we recall important properties of the operator m and explain the required regularity conditions on the solutions of the nonlinear Schrödinger equation.
(b) In case of the exponential scaling, the interaction is so strong such that the many-body state develops a short scale correlation structure. This correlation structure affects the time evolution of the condensate and must therefore also be regarded in the definition of the functional. In Sect. 5, we explain the correlations structure in greater detail, provide certain estimates on the zero-energy scattering state and explain how the effective coupling parameter b V N can be inferred from the microscopic structure. (c) In Sect. 6 we prove Theorem 2.4. We first consider the potential W β and define a counting measure which allows us to establish a Grönwall estimate for all β > 0. We will explain in detail how one arrives at this Grönwall estimate. Afterwards, the counting measure is adjusted to the case V N , taking the microscopic structure j N ,R of the wave function into account. We then establish a Grönwall estimate and finally prove the second part of the main theorem. (d) In order to improve the readability of the paper we only state the estimates which are needed for the proof of Theorem 2.4 in Sect. 6. Their derivation is provided afterwards in Sect. 7.

Preliminaries
We will first fix the notation we are going to employ during the rest of the paper. Notation 4.1. (a) Throughout the paper hats · will always be used in the sense of Definition 3.1(c). The label n will always be used for the function n(k) = √ k/N . (b) For better readability, we will often omit the upper index ϕ on p j , q j , P j and ·. It will be placed exclusively in formulas where the ϕ-dependence is crucial. (c) The operator norm, defined for any linear operator f : will be denoted by f .
(d) We will bound expressions which are uniformly bounded in N and t by some constant C. Constants appearing in a sequence of estimates will not be distinguished, i.e. in X ≤ CY ≤ C Z the constants may differ. (e) We will denote by K(ϕ t , A t ) a generic polynomial with finite degree in ϕ t ∞ , ∇ϕ t ∞ , ∇ϕ t , ϕ t , A t ∞ , t 0 ds Ȧ s ∞ and Ȧ t ∞ . Note, in particular, that for a generic constant C the inequality C ≤ K(ϕ t , A t ) holds. The exact form of K(ϕ t , A t ) which appears in the final bounds can be reconstructed, collecting all contributions from the different estimates. (f) We will denote for any multiplication operator F: L 2 (R 2 , C) → L 2 (R 2 , C) the corresponding operator acting on the N -particle Hilbert space by F(x k ). In particular, we will use, for any , ∈ L 2 (R 2N , C) the notation In analogy, for any two-particle multiplication operator K : L 2 (R 2 , C) ⊗2 → L 2 (R 2 , C) ⊗2 , we denote the operator acting on any ∈ L 2 (R 2N , C) by multiplication in the variable x i and x j by K (x i , x j ). In particular, we denote Next, we prove some properties of the projectors p j and q j , which are defined in Definition 3.1.
(c) For any weight m: N 0 → R + 0 and any function f ∈ L ∞ R 4 , C and any j, k = 0, 1, 2 (d) For any weight m : N 0 → R + 0 and any functions f ∈ L ∞ R 4 , C , g ∈ L ∞ R 2 , C the commutation relations Proof. (a) follows immediately from Definition 3.1, using that p j and q j are orthogonal projectors.
Using also (q j ) 2 = q j and q j p j = 0 we get (c) Using the definitions above we have The number of projectors q j in P l Q j in the coordinates j = 3, . . . , N is equal to l − j. The p j and q j with j = 3, . . . , Similarly one gets the second and third formula.
We will show that the right hand side is zero. Multiplying the right hand side with p 1 p 2 from the left and using (c) one gets Multiplying (18) with p 1 q 2 from the left one gets Using (c) the latter is zero. Also multiplying with q 1 p 2 yields zero due to symmetry in interchanging x 1 with x 2 . Multiplying (18) with q 1 q 2 from the left one gets which is again zero and so is (18). By means of the identity 1 = p 1 + q 1 one has The second relation from part (d) then follows from (a) and (c).
(e) To show (15), note that It follows that For (16) we write With (15) we get (16). For (17) we use The Lemma then follows from the fact that, for bounded operators A, A op = A * op holds, where A * is the adjoint operator of A.
Within our estimates we will encounter wave functions where some of the symmetry is broken (at this point the reader should exemplarily think of the wave function V β (x 1 − x 2 ) which is not symmetric under exchange of the variables x 1 and x 3 ). This leads to the following definition Based on the combinatorics of the p j and q j , we obtain the following Proof. Let ∈ H M a for some finite set 1 ∈ M a ⊂ {1, 2, . . . , N }. By Lemma 4.2 (b), (20) can be estimated as Similarly, we obtain for which concludes the Lemma.
Proof. Using p 2 + q 2 = 1 and triangle inequality, With Lemma 4.2 (c) we get Note that the wave function p 2 ∇ 2 q 2 is symmetric under the exchange of any two variables but x 2 . Thus we can use Lemma 4.4 to get Since √ k ≤ √ k + 1 for k ≥ 0 it follows that the latter is bounded by Using that r op = sup 0≤k≤N {r (k)} = r d op for any d ∈ N and any weight r , the Corollary follows.
Proof. Using symmetry and Cauchy Schwarz For the second factor we can write Since ab ≤ 1/2a 2 + 1/2b 2 and (a + b) 2 ≤ 2a 2 + 2b 2 holds for any real numbers a and b, the Lemma follows.
In our estimates, we need the regularity conditions . Then, |ϕ t | 2 , |ϕ t | 2 1 and ϕ 2 t , which also appear in our estimates, can be bounded by Recall the Sobolev embedding Theorem, which implies in particular follows since both ϕ and ∇ϕ have to decay at infinity. Thus, , which suffices for our estimates. Since ϕ t obeys a defocusing nonlinear Schrödinger equation, we expect the regularity of the solution ϕ t to follow from the regularity of the initial datum ϕ 0 . For a certain class of external potentials A t this has been proven in [13]: x , C) real valued and smooth with respect to the space variable: Proof. Part (a) is Corollary 1.4. in [13]. We like to remark that ϕ t H k ≤ C holds, if A t = 0, see Section 1.2. in [13]. The conditions on A t are for example satisfied if can be verified directly, using the existence of global in time solutions. Part (c) follows from (a) and the embedding

The scattering state.
In this section we analyze the microscopic structure which is induced by V N . In particular, we explain why the dynamical properties of the system are determined by the low energy scattering regime.
Next, we want to recall some important properties of the scattering state j N ,R , see also Appendix C of [38].
For the scattering state defined previously the following relations hold: (a) There exists a nonnegative number a, called scattering length of the potential V , such that (in the case a = 0 we have I R = 0). The scattering length a does not depend on R and fulfills a ≤ diam(supp(V )). Furthermore, Proof.
(a)+(b) Rescaling x → e N x = y, we obtain, settingR = e N R and sR(y) = j 0,e N R (y), the unscaled scattering equation Since we assume V to be nonnegative, one can define the scattering state sR by a variational principle. Theorem C.1 in [38] then implies that sR is a nonnegative, spherically symmetric function in |y|. It is then easy to verify that for diam (supp(V )) ≤ |y| there exists a number A ∈ R such that Next, we show that A = R 2 d 2 yV (y)sR(y). This can be seen by noting that, for r > diam (supp(V )), By Theorem C.1 in [38], there exists a number a ≥ 0, not depending onR, such that for all |y| ≥ diam (supp(V )) Comparing this with (28), we obtain Since sR is nonnegative, it furthermore follows that a ≤ diam (supp(V )). This directly implies A ≥ 0. By scaling, we obtain Assuming that the energy per particle E V N ( ) is of order one, the wave function will have a microscopic structure near the interactions V N , given by j N ,R . The interaction among two particles is then determined by N . Keeping in mind that each particle interacts with all other N − 1 particles, we obtain the effective Gross-Pitaevskii Thus, choosing V N (x) = e 2N V (e N x) leads in our setting to an effective one-particle equation which is determined by the low energy scattering behavior of the particles. We remark that, for any s > 0, the potential e 2N s V (e N s x) yields to the coupling 4π/s.

Properties of the scattering state.
Note that the potential V N is strongly peaked within an exponentially small region. In order to control the short scale structure of t , we define a potential M μ with softer scaling behaviour in such a way that the potential V N − M μ has scattering length zero. This allows us to "replace" V N by M μ , which has better scaling behavior and is easier to control. In particular, M μ ≤ CN −1+μ can be controlled for μ sufficiently small.
Furthermore, we define the zero energy scattering state Note that M μ and f μ depend on R μ .
Remark 5.4. In the following, we choose R μ to be the smallest value such that the scattering length of the potential The existence of such R μ < ∞ will be proven in Lemma 5.5. Note, that choosing R μ to be the minimal value such that (V N − M μ ) has scattering length zero excludes the possibility for bound states for the potential. This will be shown in Lemma 7.10 (a). Heuristically speaking, the absence of bound states can be seen in the following way: The attractive part of the potential, i.e. −M μ , is chosen to be as small as possible, i.e. just to compensates the repulsive part. Then, there is not enough attractiveness left to form a bound state.

Lemma 5.5.
For the scattering state f μ , defined by (30), the following relations hold: For the rest of the paper we assume that R μ is the minimum we get in (a). (f) For any fixed 0 < μ, N sufficiently large such that V N and M μ do not overlap, we obtain Then, Proof. (a) In the following, we will sometimes denote, with a slight abuse of notation, for r = |x| (for this, recall that f μ and j N ,R are radially symmetric). We further denote by f μ (r ) the derivative of f μ with respect to r . We first show by contradiction that there exists a Using (30) and Gauss'-theorem, we further obtain (33) and (34) then imply for r > r 0 Taking the supreme over the interval [r 0 , r ], the inequality above then implies that there exists a constant C(r, r 0 ) = 0, lim r →r 0 C(r, r 0 ) = 0 such that Thus, for r close enough to r 0 , the inequality above can only hold if f μ (s) = 0 for s ∈ [r 0 , r ], yielding a contradiction to the choice of r 0 .
and satisfies the zero energy scattering equation (26) for x ∈ B N −μ (0). Note that the scattering equations (26) and (30) have a unique solution on any compact set. It then follows that for the rest of the proof of part (a). For μ fixed, consider the function We show by contradiction that the function s has at least one zero. Assume s = 0 were to hold. We can assume w.l.o.g. s > 0. It then follows from Gauss'-theorem that f By uniqueness of the solution of the scattering equation (30), forR μ < R μ there exists a constant KR μ ,R μ = 0, such that for all |x| ≤R μ we have f and s are continuous, we can further conclude KR μ ,R μ > 0. From s = 0, it then follows that, Remark 5.6. As mentioned, we will from now on fix R μ ∈ [N −μ , ∞[ as the minimal value such that s(R μ ) = 0. Furthermore, we may assume a > 0 and R μ > N −μ in the following. For a = 0, we can choose R μ = N −μ , such that f μ (x) = j N ,R (x). It is then easy to verify that the Lemma stated is valid.
(b) From (a), we can conclude that Next, we show that the constant K μ is positive. Since j N ,R μ (N −μ ) is positive, it follows from Eq. (35) that K μ and f μ (N −μ ) have equal sign. By (a), the sign of f μ is constant for |x| ≤ R μ . Since j N ,R μ and V N are nonnegative functions, we obtain by Gauss-theorem and the scattering equation (30) sgn Recall that R μ is the smallest value such that If it were now that K μ is negative, we could conclude from (35) and (36) that we were able to conclude from the continuity of the derivative that ∂ f μ ∂r < 0 for all r < R μ and hence f (R μ ) < 0. However, this were in contradiction to the boundary condition of the zero energy scattering state [see (30)] and thus K μ > 0 follows. (c) This directly follows from e −N < CN −μ for N sufficiently large. (d) From the proof of property (b), we see that f μ and its derivative is positive at N −μ .
From (34), we obtain f μ (r ) = 0 for all r > R μ . Further (34) gives that R μ is the smallest value such that f μ (R μ ) = 0. This and continuity imply that f μ (r ) > 0 for all r < R μ . Since f μ is continuous, positive at N −μ , and its derivative is a nonnegative function, it follows that f μ is a positive, monotone nondecreasing function in |x|.
(f) Since f μ is a positive monotone nondecreasing function in |x|, we obtain We obtain the lower bound For the upper bound we first prove that Using the scatting equations (26) and (30) we obtain Using the minimum principle, we obtain, using that f μ − j N ,R μ is spherically symmetric If it were now that min is superharmonic for all |x| ≤ R μ . Using the minimum principle once again, we then obtain As a consequence, we obtain the desired bound Since f μ is a nonegative, monotone nondecreasing function in |x|, Using as well as Since g μ (x) is a nonnegative, monotone nonincreasing function, it follows with and (j) follows.

Proof of the Theorem
In this section, we present the proof of Theorem 2.4. We start with the NLS regime and then pursue with the exponential scaling. In both cases we follow the same strategy: After giving the precise definition of the functional we explain its connection to the notion of Bose-Einstein condensation in terms of reduced density matrices. Thereupon, we differentiate the functional with respect to its time variable, perform a Grönwall estimate and finally prove the respective part of the theorem.

Definition of the functional
The goal of this section is to define a functional α: which is adapted to potentials with NLS scaling and which meets all the requirements stated in Sect. 3. In short, we demand the functional to converge to zero for properly chosen initial states and its time derivative to be controllable by means of a Grönwall estimate. Additionally, the functional should allow to prove both Bose-Einstein condensation and the convergence of the energy per particle of the manybody system to the effective energy functional.
While interactions in the mean-field scaling (W β with β = 0) become weak for large particle numbers, potentials W β with β > 1/2 are getting peaked as N → ∞. This fact needs to be taken into account when defining a suitable counting functional. For small β and a large class of different choices of the weight m ϕ t with ϕ t being a solution of (3), it is possible to show that This enables us to perform an integral type Grönwall estimate if we choose Here, the smallness of the distance between the energies is used to control the kinetic energy per particle of the many-body system (Lemma 7.6). This prevents the wave function from being strongly localized in the support of the potential and in this way softens the effect of the interaction. Moreover, it allows us to bound the kinetic energy of the particles which are not in the condensate state ϕ t by α( t , ϕ t ), see Lemma 7.9. For large β, the interaction is harder to control and several estimates break down, if one defines α as above. It is therefore necessary to redefine the functional α( t , ϕ t ) and to carefully choose a new weight function m. Let us explain why this is necessary. To obtain an integral type Grönwall estimate, we will calculate the time derivative of In total, this suggests the following form of the functional Definition 6.1. For 0 < ξ < 1 3 define Remark 6.2. It should be noted, that α < depends on the parameter ξ which will be chosen later. For better readability, we disregard the ξ dependence in the notation. The counting measure can be related to the trace norm distance of the one-particle reduced density matrix. Lemma 6.3. Let 0 < ξ < 1/3, ∈ L 2 s (R 2N , C), ϕ ∈ L 2 (R 2 , C) and α < ( , ϕ) be defined as in Definition 6.1. Then, Proof. We would like to mention, that this Lemma has been proven in [ If one then uses that n(k) 2 ≤ n(k) ≤ m(k) and m(k) ≤ n(k) + 1 2 N −ξ imply the relations the Lemma follows.

Preliminaries for the Grönwall estimate
Subsequently, we will perform a Grönwall estimate for α < and prove part (a) of Theorem 2.4. For this, we define Definition 6.4. Let 0 < ξ < 1/3 and W β ∈ W β . Define and we define for l ∈ {a, b, c} the functionals γ < l : The value of the functional α < ( t , ϕ t ) at time t is then bounded by Proof. For the proof of the Lemma we restore the upper index ϕ t in order to pay respect to the time dependence of m ϕ t . The time derivative of ϕ t is given by (3) acting on the j th coordinate x j .
We then obtain where we used the symmetry of t . Using Lemma 4.2 (d), it follows that (dropping the explicit dependence on ϕ t from now on) Since Z ϕ t β and p 1 p 2 ( m − m 2 ) as well as p 1 q 2 ( m − m 1 ) are selfadjoint, we obtain Note that in view of Lemma 4.2 (c) r Q j Z x 2 )Q j r for any j ∈ {0, 1, 2} and any weight r . Therefore, Using Symmetry and Lemma 4.2 (c), we obtain the first line (43). Furthermore, (1)) P 0 and we get By explicit estimates, one can show that the functions γ <

The Grönwall estimate
In order to establish a Grönwall estimate for α < , we have to find a suitable bound for the right hand side of (46).
(a) Let β < 1/12. Moreover, let α < ( t , ϕ t ), γ < a ( t , ϕ t ) and γ < b ( t , ϕ t ) be defined as in Definitions 6.1 and 6.4 with ξ = 1/6. Then (b) Let β ≥ 1/12. Moreover, let α < ( t , ϕ t ), γ < a ( t , ϕ t ) and γ < b ( t , ϕ t ) be defined as in Definitions 6.1 and 6.4 with ξ = 1/10. Then The proof of Lemma 6.6 is given in Sect. 7.3. At this point, we only consider the most relevant term γ < b ( t , ϕ t ) and explain on a heuristic level why it is small. The principle argument follows the ideas and estimates of [49]. The first line in (44) is the most important one. This expression is only small if the correct coupling parameter b W β ≈ N W β 1 is used in the mean-field equation (3). Then, converges against the mean-field potential, and hence the first expression of (44) is small.
In order to estimate the second and third line of (44), one tries to bound With the help of Lemma 6.5, Lemma 6.6 and Grönwall's Lemma, we obtain (a) Let β < 1/12 and α < ( t , ϕ t ) be defined as in Definition 6.1 with ξ = 1/6. Then, (b) Let β ≥ 1/12 and α < ( t , ϕ t ) be defined as in Definition 6.1 with ξ = 1/10. Then, Proof. From Lemmas 6.5 and 6.6, we have in the case of β < 1/12. Thus if we apply Grönwall's Lemma, we get With the help of the relation |x| ≤ e |x| this can be further simplified and one obtains (50). Part (b) of the Lemma is shown in complete analogy.

6.2.
Proof for the exponential scaling V N .

Definition of the functional
In case of the exponential scaling, the interaction is so strong such that the many-body wave function develops a non-negligible short scale correlation structure which prevents the particles from being localized too close to each other. These correlations determine the statical and dynamical properties of the condensate in a crucial manner and need to be taken into account explicitly. It is therefore reasonable to expect that the counting measure needs to be modified, too. In order to motivate how the correlation structure will appear in the definition of the functional we think for the moment of the most simple counting measure, namely t , q t . This functional counts the relative number of particles which are not in the state ϕ t and consequently measures if the many-body state is approximately given by the product state ϕ ⊗N t , in the sense of reduced density matrices. However, in the face of the exponential scaling, one should picture the many-body state not as the product of one-particle states but rather as a wave function of Jastrow-type, i.e.
with j N ,R being the zero energy scattering state as defined in (26).
In the following, we will consider the correlation structure to be induced by f μ (see Definition 5.3) rather than by j N ,R . This replacement does not change the heuristic discussion above, since f μ (x) ≈ j N ,R μ (x) ∀|x| ≤ N −μ for N large (see Lemma 5.5), but will allow us to smoothen the singular interaction, as we will explain in the following.
Instead of projecting onto the state ϕ t , the previous discussion suggests to replace p The counting measure would then be given by This expression can be further simplified, if we use g μ = 1 − f μ and only keep the terms which are at most linear in g μ With the help of the symmetry of the many-body wave function and the identity q

Using (30) and neglecting the mixed derivatives we get that [H
. Further one can show that the leading order of g μ (x 1 − 1 . This is due to the smallness of the support of g μ and V N which significantly overlap only for this term. Hence the leading order of d dt (52) is given by Summarizing we can say that due to this adjustment and by means of the scattering equation (30), the interaction V N got replaced by the less singular potential M μ f μ in in the first line of the equation above. It is this less singular potential that can be controlled using the results from the previous chapter. M μ f μ has the properties of the W β considered above [see Lemma 5.5 (k)], making (53) controllable. This explains why we chose to use f μ in the definition of the modified counting functional instead of j N ,R . In return we obtain additional error terms, which, however, can be estimated sufficiently well, see Lemma 6.13. Making use of Lemma 4.2 (c) and (d) this idea can also be used for weight functions different from k N . Note that due to symmetry the correction term in (52) can be written as Moreover, N −1 can be viewed as the discrete time derivative of the weight k N , in other words We will use this insight to modify the functional α < ( t , ϕ t ) from Definition 6.1. We first compute the time derivative of t , m ϕ t t and then add an additional term to the counting measure in a way such that the interaction V N gets replaced by the potential M μ f μ .
Pursuing this approach results in the following definition.
Then, α: Remark 6.9. It should be noted that r depends on ξ and the functional α depends on ξ and μ. Both parameters are later chosen in a way such that we can establish an integral type Grönwall estimate.
If one recalls Definition 6.3, one sees that α is obtained from α < by adding an additional correction term. It is important to note that (see proof of Lemma 6.10) For μ chosen large enough this allows us to show that the convergence of α to zero can be related to the notion of complete Bose-Einstein condensation in terms of reduced density matrices. Lemma 6.10. Let 0 < ξ < 1/3, μ > 0, ∈ L 2 s (R 2N , C), ϕ ∈ L 2 (R 2 , C)∩ L ∞ (R 2 , C) and α( , ϕ) be defined as in Definition 6.1. Then, there exists a constant C ∈ (0, ∞) such that Tr γ (1) Proof. Using m a op + m b op ≤ CN −1+ξ , see (76), together with Eq. (16) and Lemma 5.5 (i), we obtain . By means of Lemma 6.3 the Lemma follows.

Preliminaries for the Grönwall estimate
Definition 6.11. Let 0 < ξ < 1/3, μ > 0 and r be defined as in Definition 6.8. Then, γ : where the different summands are: (a) The change in the energy-difference where, using M μ from Definition 5.3, (c) The mixed derivative term (e) Interaction terms of the correction (f) Correction terms of the mean field The value of α( t , ϕ t ) at time t is then bounded by (57) and (61). Then

ϕ t ) and γ ( t , ϕ t ) be defined as in
Using symmetry and (i z) = − (z), we obtain The third and fourth lines equal γ d (recall that is symmetric), the fifth line equals γ e and the seventh line equals γ f . Using that The first, second and the fourth line give γ b + γ d + γ e + γ f . Using Definition (5.3) the commutator in the fifth line equals Using symmetry the third and fifth line in (63) give By means of and the fundamental theorem of calculus the result follows.

The Grönwall estimate
Again, we will bound the time derivative of α( t , ϕ t ) such that we can employ a Grönwall estimate. ϕ t ), i ∈ {a, b, c, d, e, f } be defined as in Definitions 6.8 and 6.11 with ξ = 1/10 and μ = 10. Then, The proof of the Lemma can be found in Sect. 7.4. By means of Lemma 5.5 (h) and (i), the terms γ a and γ b can be estimated in the same way as γ < a and γ < b . The estimates for γ c , γ d , γ e and γ f are based on the smallness of the L p -norms of g μ , see Lemma 5.5 (i).
Thus, combining Lemmas 6.12 and 6.13, we obtain the following estimate for α( t , ϕ t ) by means of Grönwall's Lemma Lemma 6.14. Let V N ∈ V N . Let t the unique solution to i∂ t t = H V N t with initial Let α( t , ϕ t ) be defined as in Definition 6.8 with ξ = 1/10 and μ = 10. Then, Proof. This is proven in the same way as Lemma 6.7.

Moreover, one obtains
Finally, this shows part (b) of Theorem 2.4.

Smearing out the potential W β .
To control the potential W β for β large, we use a technique which allows us to replace the potential W β by some potential The function h β 1 ,β can be thought as an electrostatic potential which is caused by the charge W β − U β 1 ,β . It is then possible to rewrite for χ, ω ∈ L 2 s (R 2N , C). We will verify that the L p -norms of h β 1 ,β and ∇h β 1 ,β are better to control than the respective L p -norm of W β . With additional control of ∇ 1 and ∇ 1 χ , it is therefore possible to obtain a sufficient bound for χ, for large β.

Estimates on the cutoff.
In order to smear out singular potentials as explained in the previous section and to obtain sufficient bounds, it seems at first necessary to show that ∇ 1 q 1 t decays in N . However, this term will in fact not be small for the dynamic generated by V N . There, we rather expect that ∇ 1 q 1 t = O(1) holds. It has been shown in [18,37] that the interaction energy is purely kinetic in the Gross-Pitaevskii regime, which implies that a relevant part of the kinetic energy is concentrated around the scattering centers. We must thus cutoff the part which is used to form the microscopic structure. For this, we define the set A (d) j which includes all configurations where the distance between particle x i and particle x j , j = i is smaller than N −d . It is then possible to prove that the kinetic energy concentrated on the complement of A   For any j, k = 1, . . . , N and Proof. (a) First note that the volume of the sets a (d) j,k introduced in Definition 7.3 are |a where we defined 1,k as well as 1 , we obtain This implies The second statement of (a) can be proven similarly. Analogously, we obtain (b) Without loss of generality, we can set j = 1. Recall the two-dimensional Sobolev inequality, for ∈ H 1 (R 2 , C) and for any 2 < m < ∞, there exists a positive constant C m , such that m ≤ C m ∇ m−2 m 2 m holds. Using Hölder and Sobolev for the x 1 -integration, we get, for p > 1 where C p denotes a positive constant, depending on p.
Using Hölder for the x 2 , . . . x N -integration with the conjugate pair r = p p−1 and s = p, we then obtain Since the sets C k are pairwise disjoint, the 1 C k are pairwise orthogonal and we get (d) 7.3. Proof of Lemma 6.6. The goal of this section is to prove Lemma 6.6. To this end, we bound each of the functionals γ < a , γ < b and γ < c separately and then collect the estimates. In view of the conditions required in Lemma 6.6, the following is assumed in the rest of this section: Let Proof. Using 1 = p 1 + q 1 , where we used Lemma 4.2 (c). Since p 1 Using Lemma 7.5, n − m op ≤ CN −ξ and setting B =Ȧ t , we get Control of γ < b To control γ < b we will first prove that ∇ 1 t is uniformly bounded in N , if initially the energy per particle E U ( 0 ) is of order one. Then This yields Next, we control m a and m b which were defined in Definition 6.4. The difference m(k) − m(k + 1) and m(k) − m(k + 2) is of leading order given by the derivative of the function m(k)-k understood as real variable-with respect to k. The k-derivative of m(k) equals It is then easy to show that, for any j ∈ Z, there exists a C j < ∞ such that Now, we prove some general bounds, which will allow us to estimate the different terms of γ < b in (44). In order to facilitate the notation, let w ∈ {N m a −1 , N m b −2 }. Then w(k) < Cn(k) −1 and w 1 op ≤ C w op ≤ CN ξ follows.
In addition, we have the slightly improved bound , n + N −1/6 ln(N ) Proof. Since the left hand sides of all these statements are bounded, it follows that all these estimates hold uniformly in N being in any finite subset of N. Hence it suffices to prove the validity of (a), (b) and (c) for sufficiently large N ∈ N.
(a) In view of Lemma 4.4, we obtain x 2 )q 1 p 2 op can be estimated using p 1 q 1 = 0 and (19): Let h be given by As above (see Lemma 7.2), we obtain h(x) = 0 for x / ∈ B RN −β (0), where R N −β is the radius of the support of W β . Thus, Integration by parts and Young's inequality give that Thus, we obtain the bound which then proves (a). (b) We will first consider β < 1/2.

Complete estimate:
In total, we obtain To estimate γ < b we recall that w ∈ {N m a −1 , N m b −2 } with w(k) < n(k) −1 and w 1 op ≤ C w op ≤ CN ξ . Lemma 7.7, n − m op ≤ CN −ξ and ξ < 1/3 imply In addition, we have the improved bound for al β < 1/2. Control of γ < c With Definition 2.1 and (76) we estimate Collecting all the estimates for γ < a , γ < b and γ < c then proves Lemma 6.6.

7.4.
Proof of Lemma 6.13. Next, we prove Lemma 6.13. We will proceed in a similar way as in the previous section and consecutively estimate the functionals γ i with i ∈ {a, b, c, d, e, f }. In the rest of this section we assume that For the most involved scaling which is induced by V N , we need to control Proof. We estimate With Lemma 4.2 (e) we get as well as we obtain

Control of γ a
In total analogy to (73) we get With Definition 6.8 and (58) we have

Control of γ b
Recall that Estimate (102) yields to the bound The first line of γ b can be bounded with (62) and f μ = 1 − g μ by Since M μ f μ ∈ W μ , (103) is of the same form as γ < b ( , ϕ). By means of Lemma 7.7, n − m op ≤ C N −ξ and (58), we obtain Using Lemma 5.5 (h), the second term is controlled by The last term is controlled by which implies the bound

Control of γ d
To control γ d and γ e we will use the notation Since the second k-derivative of m is given by (see (74) for the first derivative) it is easy to verify that Recall that Since p j + q j = 1, we can rewrite r as Thus, Using Lemma 4.2 (d), we obtain the following estimate: Both lines are bounded by In view of Lemmas 4.2 (e) and 5.5 (i), g μ ( We continue with Next, we estimate (116). The support of the function g μ ( In the fourth line, we applied Sobolev inequality as in the proof of Lemma 7.4, then setting p = 2. Furthermore, we used ∇ 1 m a 1/2 m a 1/2 ≤ CN −1+ξ , as well as g μ ∞ ≤ C, see Lemma 5.5. Using Lemma 4.2 (d), (117) can be bounded by In total, we obtain

Control of γ e
Recall that Using symmetry, Lemma 4.2 (d) and notation (110), γ e is bounded by We get with (111), Lemma 5.5 and Lemma 4.2 that

Energy estimates. In this section we show that 1 A (d)
1 ∇ 1 q 1 2 can be controlled sufficiently well in terms of the counting functionals α < and α. If t is evolving according to W β , one could actually show that ∇ 1 q 1 t 2 is small already without cutoff. While such a proof would be less involved, we chose a unified presentation which both covers the Gross-Pitaevskii scaling and the NLS scaling.
1 as in Definition 7.3. Then, for N large enough and d ≥ 3, Proof. We start with expanding E W β ( ) − E G P N W β 1 (ϕ). This yields where we have defined Notice that the first two terms in Q β ( , ϕ) are nonnegative. This yields to the bound We therefore obtain Thus if we use that Definition 2.1 implies the estimate we get the bound: Next, we split up the energy difference E V N ( ) − E G P 4π (ϕ), In order to better estimate the terms corresponding to the two-particle interactions, we introduce, for ν > d, the potential M ν (x), defined in Definition 5.3. Note, that ν > d assures that that part of the interaction M ν which lies within the set A (d) 1 will be negligible. We continue with After reordering, the identity q 1 = 1 − p 1 , together with the symmetry of The first three terms inQ ν ( , ϕ) are nonnegative. For ν > d and N large enough, Lemma 7.10 implies that (124) is also nonnegative. Thus, for ν > d, we obtain the boundS In total, we obtain It is therefore left to estimate M( , ϕ), S β ( , ϕ) andS ν ( , ϕ).
Let now Z n be a subset Z n ⊂ R 2 with |Z n | = n which is such that the supports of the potentials M ν (· − z k ) lie within the Ball around zero with radius R and are pairwise disjoint for any two z k ∈ Z n . Since we are in two dimensions we can choose a n which is of order R 2 . Let now χ R (x) = ξ R (x) z k ∈Z nψ (x −z k ). By construction, there exists a D = O (1) such that χ R (x) =ψ(x − z k ) for |x − z k | ≤ D. From this, we obtain Choosing R and hence n large enough and small, we can find a Z n such that χ R , H Z n χ R is negative, contradicting nonnegativity of H Z n . Now, we can prove that holds for any ∈ H 2 (R 2N , C). Using the coordinate transformationx 1 = x 1 − x 2 ,x i = x i ∀i ≥ 2, we have ∇ x 1 = ∇x 1 . Thus (132) is equivalent toQ( ) := 1 |x 1 |≤R ν ∇ 1 2 + 1 2 , (V N − M ν )(x 1 ) ≥ 0 ∀ ∈ H 2 (R 2N , C). If it were now thatQ( ) is not nonnegative, then there exists a ∈ H 2 (R 2N , C) such thatQ( ) < 0. By the Schmidt decomposition theorem, there exist two orthonormal bases { k } k∈N ⊂ H 2 (R 2N −2 , C), {ϕ l } l∈N ⊂ H 2 (R 2 , C) and nonnegative numbers {λ k } k∈N such that = k∈N λ k ϕ k ⊗ k .
Note that 1 B (d) 1 depends only on x 2 , . . . , x N . By this This yields where the last inequality follows from (a), using 1 B (d) , , n + N −1/6 ln(N ) . (c) Proof. (a) We will only consider the first inequality of (a). The second inequality of (a) can be proven analogously. Let first = 1 B (d) 1 . Then, Using Lemma 7.4 (c) with = 1, together with p 2 W β (x 1 − x 2 ) p 2 op ≤ ϕ 2 ∞ W β 1 , the first line can be bounded by The second term is bounded by This yields (a) in the case = 1 B (d) 1 . The inequality (a) can be proven analogously for = .
We obtain (note that 1 B (d) 1 does not depend on x 1 ) since both ∇q 1 and q 1 are bounded uniformly in N . Since q 2 is a projector it follows that where we used Lemma 7.2 in the last step. Next, we estimate The first term can be estimated as for any d ≥ 3. In the last line we used Lemma 7.4 (c) with = 1. The last term can be estimated as Combining both estimates we obtain, for any β > 1/4, where the last inequality comes from choosing η = 1/3 and β 1 = 1/4. For = , (b) can be estimated the same way, yielding the same bound. (c) This follows from (a) and (b), using that 1 − p 1 p 2 = q 1 q 2 + p 1 q 2 + q 1 p 2 .
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