DERIVATIO OF THE CUBIC LS AD GROSS-PITAEVSKII HIERARCHY FROM MAYBODY DYAMICS I d = 2, 3 BASED O SPACETIME ORMS THOMAS CHE AD ATAŠA PAVLOVIĆ Abstract. We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimensions d = 2, 3, from an -body Schrödinger equation describing a gas of interacting bosons in the GP scaling, in the limit. The main result of this paper is the proof of convergence of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work on the well-posedness of the Cauchy problem for GP hierarchies, [6, 7, 8], which are inspired by the solutions spaces based on space-time norms introduced by Klainerman and Machedon in [23]. We note that in d = 3, this has been a wellknown open problem in the field. While our results do not assume factorization of the solutions, consideration of factorized solutions yields a new derivation of the cubic, defocusing nonlinear Schrödinger equation (LS) in d = 2, 3. 1. Introduction We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy from an -body Schrödinger equation in dimensions 2 and 3 describing a gas of interacting bosons in the Gross-Pitaevskii (GP) scaling, as. The main result of this paper is the proof of convergence in the spaces introduced in our previous work on the wellposedness of the Cauchy problem for GP hierarchies, [6, 7, 8], which are inspired by the solutions spaces based on space-time norms introduced by Klainerman and Machedon in [23]. In dimension 3, this problem has so far remained a key open problem, while in dimensions 1 and 2, it was solved in [24, 5] for the cubic and quintic case. The derivation of nonlinear dispersive PDEs, such as the nonlinear Schrödinger (LS) or nonlinear Hartree (LH) equations, from many body quantum dynamics is a central topic in mathematical physics, and has been approached by many authors in a variety of ways; see [14, 15, 16, 24, 23, 29] and the references therein, and also [1, 3, 1, 11, 13, 17, 18, 19, 21, 2, 22, 28, 31]. This problem is closely related to the mathematical study of Bose-Einstein condensation in systems of interacting bosons, where we refer to the important works [2, 25, 26, 27] and the references therein. 1.1. The Gross-Pitaevkii limit for Bose gases. As a preparation for our analysis in the present paper, we will outline some main ingredients of the approach due to L. Erdös, B. Schlein, and H.-T. Yau. In an important series of works, 1
2 T. CHE AD. PAVLOVIĆ [14, 15, 16], these authors developed a powerful method to derive the cubic nonlinear Schrödiner equation (LS) from the dynamics of an interacting Bose gas in the Gross-Pitaevskii limit. We remark that the defocusing quintic LS can be derived from a system of bosons with repelling three body interactions, see [5]. 1.1.1. From -body Schrödinger to BBGKY hierarchy. We consider a quantum mechanical system consisting of bosons in R d with wave function Φ L 2 (R d ). According to Bose-Einstein statistics, Φ is invariant under the permutation of particle variables, Φ (x π(1), x π(2),..., x π() ) = Φ (x 1, x 2,..., x ) π S, (1.1) where S is the -th symmetric group. We denote by L 2 sym(r d ) the subspace of L 2 (R d ) of elements obeying (1.1). The dynamics of the system is determined by the -body Schrödinger equation i t Φ = H Φ. (1.2) The Hamiltonian H is given by a self-adjoint operator acting on the Hilbert space L 2 sym(r d ), of the form H = ( xj ) + 1 1i<j V (x i x j ), (1.3) where V (x) = dβ V ( β x) with V spherically symmetric, sufficiently regular, and for < β < 1 d+1. Since the Schrödinger equation (1.2) is linear and H self-adjoint, the global well-posedness of solutions is evident. To perform the infinite particle number limit, we outline the strategy developed in [14, 15] as follows. One introduces the density matrix γ Φ (t, x, x ) = Φ (t, x ) Φ (t, x ) := Φ (t, x )Φ (t, x ) where x = (x 1, x 2,..., x ) and x = (x 1, x 2,..., x ). Furthermore, one considers the associated sequence of k-particle marginal density matrices γ (k) Φ (t), for k = 1,...,, as the partial traces of γ Φ over the degrees of freedom associated to the last ( k) particle variables, γ (k) Φ = Tr k+1,k+2,..., Φ Φ. Here, Tr k+1,k+2,..., denotes the partial trace with respect to the particles indexed by k + 1, k + 2,...,. Accordingly, γ (k) is explicitly given by γ (k) Φ (x k, x k) = = Φ dx k γ Φ (x k, x k ; x k, x k ) dx k Φ (x k, x k )Φ (x k, x k ). (1.4) It follows immediately from the definitions that the property of admissibility holds, γ (k) Φ = Tr k+1 (γ (k+1) Φ ), k = 1,..., 1, (1.5) for 1 k 1, and that Trγ (k) Φ = Φ 2 L 2 s (Rd ) k = 1, 2,...,. = 1 for all, and all
DERIVATIO OF THE CUBIC GP HIERARCHY 3 Moreover, γ (k) Φ is positive semidefinite as an operator S(R kd ) S(R kd ) C, (f, g) dxdx f(x)γ(x; x )g(x ). The time evolution of the density matrix γ Φ is determined by the Heisenberg equation i t γ Φ (t) = [H, γ Φ (t)], (1.6) which has the explicit form i t γ Φ (t, x, x ) = ( x x )γ Φ (t, x, x ) (1.7) + 1 [V (x i x j ) V (x i x j)]γ Φ (t, x, x ). 1i<j Accordingly, the k-particle marginals satisfy the BBGKY hierarchy i t γ (k) Φ (t, x k ; x k) = ( xk x k )γ (k) Φ (t, x k, x k) + 1 [V (x i x j ) V (x i x j)]γ (k) Φ (t, x k ; x k ) (1.8) + k 1i<jk k dx k+1 [V (x i x k+1 ) V (x i x k+1 )] (1.9) i=1 γ (k+1) Φ (t, x k, x k+1 ; x k, x k+1) where xk := k x j, and similarly for x k. We note that the number of terms in (1.8) is k2 k( k), and the number of terms in (1.9) is k as. Accordingly, for fixed k, (1.8) disappears in the limit described below, while (1.9) survives. 1.1.2. From BBGKY hierarchy to GP hierarchy. It is proven in [14, 15, 16] that, for asymptotically factorized initial data, and in the weak topology on the space of marginal density matrices, one can extract convergent subsequences γ (k) Φ γ (k) as, for k, which satisfy the the infinite limiting hierarchy i t γ (k) (t, x k ; x k) = ( xk x k )γ (k) (t, x k ; x k) (1.1) k ( + κ Bj,k+1 γ k+1) (t, x k ; x k ), which is referred to as the Gross-Pitaevskii (GP) hierarchy. Here, (B j,k+1 γ k+1 )(t, x k ; x k) := dx k+1 dx k+1[δ(x j x k+1 )δ(x j x k+1) δ(x j x k+1 )δ(x j x k+1)] γ (k+1) (t, x k, x k+1 ; x k, x k+1). The coefficient κ is the scattering length if β = 1 (see [14, 26] for the definition), and κ = V (x)dx if β < 1 (corresponding to the Born approximation of the scattering length). For β < 1, the interaction term is obtained from the weak limit V (x) κ δ(x) in (1.9) as. The proof for the case β = 1 is much more difficult, and the derivation of the scattering length in this context is a breakthrough
4 T. CHE AD. PAVLOVIĆ result obtained in [14, 15]. For notational convenience, we will mostly set κ = 1 in the sequel. Some key properties satisfied by the solutions of the GP hierarchy are: The solution of the GP hierarchy obtained in [14, 15] exists globally in t. It satisfies the property of admissibility, γ (k) = Tr k+1 (γ (k+1) ), k, (1.11) which is inherited from the system at finite. There exists a constant C depending on the initial data only, such that the a priori energy bound Tr( S (k,1) γ (k) (t) ) < C k (1.12) is satisfied for all k, and for all t R, where S (k,α) := k xj α x j α. (1.13) This is obtained from energy conservation in the original -body Schrödinger system. Solutions of the GP hierarchy are studied in spaces of k-particle marginals {γ (k) γ (k) h 1 < } with norms γ (k) h α := Tr( S (k,α) γ (k) ). (1.14) This is in agreement with the a priori bounds (1.12). 1.1.3. Factorized solutions of GP and LS. The LS emerges as the mean field dynamics of the Bose gas for the very special subclass of solutions of the GP hierarchy that are factorized. Factorized k particle marginals at time t = have the form k (x k ; x k ) = φ (x j )φ (x j ), γ (k) where we assume that φ H 1 (R d ). One can easily verify that γ (k) (t, x k ; x k) = k φ(t, x j )φ(t, x j ), is a solution (usually referred to as a factorized solution) of the GP hierarchy (1.1) with κ = 1, if φ(t) H 1 (R d ) solves the defocusing cubic LS, for t I R, and φ() = φ H 1 (R d ). i t φ = x φ + φ 2 φ, (1.15)
DERIVATIO OF THE CUBIC GP HIERARCHY 5 1.1.4. Uniqueness of solutions of GP hierarchies. While the existence of factorized solutions can be easily verified in the manner outlined above, the proof of the uniqueness of solutions of the GP hierarchy (which encompass non-factorized solutions) is the most difficult part in this analysis. The proof of uniqueness of solutions to the GP hierarchy was originally achieved by Erdös, Schlein and Yau in [14, 15, 16] in the space {γ (k) γ (k) h 1 < }, for which the authors developed highly sophisticated Feynman graph expansion methods. In [23], Klainerman and Machedon introduced an alternative method for proving uniqueness in a space of density matrices defined by the Hilbert-Schmidt type Sobolev norms γ (k) H α k := S (k,α) γ (k) L 2 (R dk R dk ) <. (1.16) While this is a different (strictly larger) space of marginal density matrices than the one considered by Erdös, Schlein, and Yau, [14, 15], the authors of [23] impose an additional a priori condition on space-time norms of the form B j;k+1 γ (k+1) L 2 t H 1 k < Ck, (1.17) for some arbitrary but finite C independent of k. The strategy in [23] developed to prove the uniqueness of solutions of the GP hierarchy (1.1) in d = 3 involves the use of certain space-time bounds on density matrices (of generalized Strichartz type), and crucially employs the reformulation of a combinatorial result in [14, 15] into a board game argument. The latter is used to organize the Duhamel expansion of solutions of the GP hierarchy into equivalence classes of terms which leads to a significant reduction of the complexity of the problem. Subsequently, Kirkpatrick, Schlein, and Staffilani proved in [24] that the a priori spacetime bound (1.17) is satisfied for the cubic GP hierarchy in d = 2, locally in time. Their argument is based on the conservation of energy in the original -body Schrödinger system, and a related a priori H 1 -bounds for the BBGKY hierarchy in the limit derived in [14, 15], combined with a generalized Sobolev inequality for density matrices. 1.2. Cauchy problem for GP hierarchies. In [6], we began investigating the well-posedness of the Cauchy problem for GP hierarchies, with both focusing and defocusing interactions. We do so independently of the fact that it is currently not known how to rigorously derive a GP hierarchy from the limit of a BBGKY hierarchy with L 2 -supercritical, attractive interactions. In [6], we introduced the notions of cubic, quintic, focusing, or defocusing GP hierarchies, according to the type of LS obtained from factorized solutions. In [6], we introduced the following topology on the Banach space of sequences of k-particle marginal density matrices G = { Γ = ( γ (k) (x 1,..., x k ; x 1,..., x k) ) k Trγ (k) < }. (1.18) Given >, we defined the space H α = {Γ Γ H α < } (1.19)
6 T. CHE AD. PAVLOVIĆ with the norm Γ H α := k k γ (k) H α, (1.2) where γ (k) H α k := S (k,α) γ (k) L 2 (1.21) is the norm (1.16) considered in [23]. If Γ H α, then 1 an upper bound on the typical H α -energy per particle; this notion is made precise in [6]. We note that small energy results are characterized by large > 1, while results valid without any upper bound on the size of the energy can be proven for arbitrarily small values of > ; in the latter case, one can assume < < 1 without any loss of generality. The GP hierarchy can then be written in the form i t Γ + ± Γ = BΓ, (1.22) with Γ() = Γ, where the components of Γ and BΓ can be read off from (1.1). Here we have set κ = 1. In [6], we prove the local well-posedness of solutions for energy subcritical focusing and defocusing cubic and quintic GP hierarchies in a subspace of H α defined by a condition related to (1.17). The parameter α determines the regularity of the solution, ( 1 2, ) if d = 1 α A(d, p) := ( d 2 1 2(p 1), ) if d 2 and (d, p) (3, 2) [ (1.23) 1, ) if (d, p) = (3, 2), where p = 2 for the cubic, and p = 4 for the quintic GP hierarchy. Our result is obtained from a Picard fixed point argument, and holds for various dimensions d, without any requirement on factorization. The parameter > is determined by the initial condition, and it sets the energy scale of the given Cauchy problem. In addition, we prove lower bounds on the blowup rate for blowup solutions of focusing GP-hierarchies in [6]. The Cauchy problem for GP hierarchies was also analyzed by the authors of [12], and the cubic GP hierarchy was derived in [11] with the presence of an external trapping potential in 2D. In the joint work [9] with. Tzirakis, we identify a conserved energy functional E 1 (Γ(t)) = E 1 (Γ ) describing the average energy per particle, and we prove virial identities for solutions of GP hierarchies. In particular, we use these ingredients to prove that for L 2 -critical and supercritical focusing GP hierarchies, blowup occurs whenever E 1 (Γ ) < and the variance is finite. We note that prior to [9], no exact conserved energy functional on the level of the GP hierarchy was identified in any of the previous works, including [24] and [14, 15]. In [7], we discovered an infinite family of multiplicative energy functionals and prove that they are conserved under time evolution; their existence is a consequence of the mean field character of GP hierarchies. Those conserved energy functionals allow us to prove global wellposedness for H 1 subcritical defocusing GP hierarchies, and for L 2 subcritical focusing GP hierarchies.
DERIVATIO OF THE CUBIC GP HIERARCHY 7 In the paper [8], we prove the existence of solutions to the GP hierarchy, without the assumption of the Klainerman-Machedon condition. This is achieved via considering a truncated version of the GP hierarchy (for which existence of solutions can be easily obtained) and showing that the limit of solutions to the truncated GP hierarchy exists as the truncation parameter goes to infinity, and that this limit is a solution to the GP hierarchy. Such a truncation-based proof of existence of solutions to the GP hierarchy motivated us to try to implement a similar approach at the level of the BBGKY hierarchy, which is what we do in this paper. 1.3. Main results of this paper. As noted above, our results in [6] prove the local well-posedness of solutions for spaces W α (I) := { Γ L t IH α BΓ L 2 t IH α }, α A(d, p), (1.24) where the condition on the boundedness of the L 2 t I Hα spacetime norm corresponds to the condition (1.17) used by Klainerman and Machedon, [23]. This is a different solution space than that considered by Erdös, Schlein and Yau, [14, 15]. As a matter of fact, it has so far not been known if the limiting solution to the GP hierarchy constructed by Erdös, Schlein, and Yau is an element of (1.24) or not in dimension d 3 (for d 2, it is known to be the case, [5, 24]). This is a central open question surrounding the well-posedness theory for GP hierarchies in the context of our approach developed in [6, 7, 8, 23]. In this paper, we answer this question in the affirmative. We give a derivation of the cubic GP hierarchy from the BBGKY hierarchy in dimensions d = 2, 3 based on the spacetime norms used in [6, 23]. The main result can be formulated as follows: Let d = 2, 3, and < β < 1 d + 1. (1.25) Moreover, let δ > be an arbitrary, small, fixed number. Suppose that the pair potential V (x) = dβ V ( β x), for V W 2, (R d ) L 1 (R d ), is spherically symmetric, positive, and V C δ (R d ) L (R d ) decays rapidly outside the unit ball. Let (Φ ) denote a sequence of solutions to the -body Schrödinger equation (1.2) for which we have that for some < < 1, and every, Γ Φ () = (γ (1) Φ (),..., γ () Φ (),,,... ) H 1+δ holds at initial time t =, and moreover, that the strong limit Γ = lim ΓΦ () H 1+δ (1.26) exists. We emphasize that Γ does not need to be of factorized form. The additonal δ amount of regularity is introduced to control the convergence of certain terms, see section 5. We denote by Γ Φ (t) := (γ (1) Φ (t),..., γ () Φ (t),,,...,,... ) (1.27) the solution to the associated BBGKY hierarchy (1.8) (1.9), trivially extended by γ (n) Φ for n >.
8 T. CHE AD. PAVLOVIĆ We define the truncation operator P K by P K Γ = (γ (1),..., γ (K),,,... ), (1.28) and let K() := b log (1.29) for some sufficiently large constant b >. Then, the following hold for sufficiently small < < 1: (1) There exists Γ L t [,T ] H1 such that the limit holds strongly in L t [,T ] H1. (2) Moreover, the limit holds strongly in L 2 t [,T ] H1. s lim P K()Γ Φ = Γ (1.3) s lim B P K() Γ Φ = BΓ (1.31) (3) The limit point Γ L t [,T ] H1 is a mild solution to the cubic GP hierarchy with initial data Γ, satisfying with Γ() = Γ. Γ(t) = U(t) Γ + i t An outline of our proof is given in Section 3 below. U(t s) B Γ(s) ds, (1.32) Remark 1.1. We emphasize the following: The results stated above imply that the -BBGKY hierarchy (truncated by P K() with a suitable choice of K()) has a limit in the space introduced in [6], which is based on the space considered by Klainerman and Machedon in [23]. For factorized solutions, this provides the derivation of the cubic defocusing LS in those spaces. In [14, 15, 24], the limit γ (k) Φ γ (k) of solutions to the BBGKY hierarchy to solutions to the GP hierarchy holds in the weak, subsequential sense, for an arbirary but fixed k. In our approach, we prove strong convergence for a sequence of suitably truncated solutions to the BBGKY hierarchy, in an entirely different space of solutions. An important ingredient for our construction is that this convergence is in part controlled by use of the parameter >, which is not available in [14, 15, 24]. Moreover, we assume initial data that are slightly more regular than of class H 1.
DERIVATIO OF THE CUBIC GP HIERARCHY 9 We assume that the initial data has a limit, Γ φ () Γ H 1+δ as, which does not need to be factorized. We note that in [14, 15], the initial data is assumed to be asymptotically factorizing. The method based on spacetime norms developed in this paper works for the cubic case in d = 2, 3, and is expected to have a straightforward generalization for the quintic case in d = 2. Our result is completely new for the cubic case in d = 3; the other cases (of cubic and quintic GP in d 2) were covered in [5, 24]; however the mode of convergence proven here is different and the initial data in this paper do not need to be of factorized form. A main obstacle in treating the quintic GP hierarchy in d = 3 is the fact that the currently available Strichartz estimates are not good enough for the quintic GP hierarchy, [5].
1 T. CHE AD. PAVLOVIĆ 2. Definition of the model In this section, we introduce the mathematical model that will be studied in this paper. Most notations and definitions are adopted from [6], and we refer to [6] for additional motivations and details. 2.1. The -body Schrödinger system. We consider the -boson Schrödinger equation ( i t Φ = xj + 1 ) V (x j x l ) Φ (2.1) 1j<l on L 2 Sym (Rd ), with initial data Φ () = Φ, L 2 Sym (Rd ). Here, V (x) = dβ V ( β x) for V W 2, (R d ) L 1 (R d ) spherically symmetric, and positive. Moreover, we assume that V C 1 (R d ) with rapid decay outside the unit ball. The parameter < β < 1 is assumed to satisfy the smallness condition (3.1). Let γ (k) Φ := Tr k+1,..., ( Φ Φ ). (2.2) It is proved in [14, 15, 24] that for V satisfying the above assumptions, Φ, ( + H ) K Φ C K K Tr(S (1,K) γ (K) Φ ) (2.3) for some positive constant C < independent of, K. This a priori bound makes use of energy conservation in the -body Schrödinger equation satisfied by Φ, and will be used in the proof of our main results. 2.2. The solution spaces. We recall the space introduced in [6] G := L 2 (R dk R dk ) k=1 of sequences of marginal density matrices Γ := ( γ (k) ) k where γ (k), Trγ (k) = 1, and where every γ (k) (x k, x k ) is symmetric in all components of x k, and in all components of x k, respectively, i.e. γ (k) (x π(1),..., x π(k) ; x π (1),..., x π (k) ) = γ(k) (x 1,..., x k ; x 1,..., x k) (2.4) holds for all π, π S k. For brevity, we will write x k := (x 1,, x k ), and similarly, x k := (x 1,, x k ). The k-particle marginals are assumed to be hermitean, γ (k) (x k ; x k) = γ (k) (x k ; x k). (2.5) We call Γ = (γ (k) ) k admissible if γ (k) = Tr k+1 γ (k+1), that is, γ (k) (x k ; x k ) = dx k+1 γ (k+1) (x k, x k+1 ; x k, x k+1)
DERIVATIO OF THE CUBIC GP HIERARCHY 11 for all k. Let < < 1. We define { H α := Γ G Γ H α where with Γ H α = } < k γ (k) H α k (R dk R dk ), k=1 (2.6) γ (k) H α k := S (k,α) γ (k) L 2 (2.7) where S (k,α) := k α α. xj x j 2.3. The GP hierarchy. The main objective of the paper at hand will be to prove that, in the limit, solutions of the BBGKY hierarchy converge to solutions of an infinite hierarchy, referred to as the Gross-Pitaevskii (GP) hierarchy. In this section, we introduce the necessary notations and definitions, adopting them from [6]. The cubic GP (Gross-Pitaevskii) hierarchy is given by k i t γ (k) = [ xj, γ (k) ] + κ B k+1 γ (k+1) (2.8) in d dimensions, for k. Here, where and B k+1 γ (k+1) = B + k+1 γ(k+1) B k+1 γ(k+1), (2.9) B + k+1 γ(k+1) = B k+1 γ(k+1) = k B + j;k+1 γ(k+1), (2.1) k B j;k+1 γ(k+1), (2.11) with ( B + j;k+1 γ(k+1)) (t, x 1,..., x k ; x 1,..., x k) = dx k+1 dx k+1 and δ(x j x k+1 )δ(x j x k+1)γ (k+1) (t, x 1,..., x k+1 ; x 1,..., x k+1), ( B j;k+1 γ(k+1)) (t, x 1,..., x k ; x 1,..., x k) = dx k+1 dx k+1 δ(x j x k+1 )δ(x j x k+1)γ (k+1) (t, x 1,..., x k+1 ; x 1,..., x k+1).
12 T. CHE AD. PAVLOVIĆ We remark that for factorized initial data, γ (k) (; x k ; x k ) = k φ (x j ) φ (x j ), (2.12) the corresponding solutions of the GP hierarchy remain factorized, k γ (k) (t, x 1,..., x k ; x 1,..., x k) = φ(t, x j ) φ(t, x j). (2.13) if the corresponding 1-particle wave function satisfies the defocusing cubic LS i t φ = φ + κ φ 2 φ. The GP hierarchy can be rewritten in the following compact manner: where and i t Γ + ± Γ = κ BΓ ± Γ := ( (k) ± γ (k) ) k, with (k) ± = We will also use the notation Γ() = Γ, (2.14) k ) ( xj x, j BΓ := ( B k+1 γ (k+1) ) k. (2.15) B + Γ := ( B + k+1 γ(k+1) ) k, B Γ := ( B k+1 γ(k+1) ) k. 2.4. The BBGKY hierarchy. In analogy to the compact notation for the GP hierarchy described above, we introduce a similar notation for the cubic defocusing BBGKY hierarchy. We consider the cubic defocusing BBGKY hierarchy for the marginal density matrices, given by k i t γ (k) (t) = [ xj, γ (k) (t)] + 1 ( k) + 1jk 1j<k [V (x j x k ), γ (k) (t)] Tr k+1 [V (x j x k+1 ), γ (k+1) (t)], (2.16) for k = 1,..., n. We extend this finite hierarchy trivially to an infinite hierarchy by adding the terms γ (k) = for k >. This will allow us to treat solutions of the BBGKY hierarchy on the same footing as solutions to the GP hierarchy. We next introduce the following compact notation for the BBGKY hierarchy. k i t γ (k) = [ xj, γ (k) ] + µb ;k+1γ (k+1) (2.17)
DERIVATIO OF THE CUBIC GP HIERARCHY 13 for k. Here, we have γ (k) = for k >. Moreover, we set B ;k to be given by multiplication with zero for k >. For k = 1,..., 1, we define where with and with ( B +,main ;j;k+1 γ(k+1) = B ;k+1 γ (k+1) B ± ;k+1 γ(k+1) B ±,main ;k+1 γ(k+1) = B + ;k+1 γ(k+1) = B ±,main ;k+1 γ(k+1) := k B ±,error ;k+1 γ(k+1) := 1 k i<j k ) (t, x 1,..., x k ; x 1,..., x k) B ;k+1 γ(k+1), (2.18) + B ±,error ;k+1 γ(k+1) (2.19) B ±,main ;j;k+1 γ(k+1), (2.2) B ±,error ;i,j;k+1 γ(k+1), (2.21) dx k+1 V (x j x k+1 )γ (k+1) (t, x 1,..., x k, x k+1 ; x 1,..., x k, x k+1 ) (2.22) and ( ) B +,error ;i,j;k+1 γ(k) (t, x 1,..., x k ; x 1,..., x k) = V (x i x j )γ (k) (t, x 1,..., x k ; x 1,..., x k) = dx k+1 V (x i x j )γ (k+1) (t, x 1,..., x k, x k+1 ; x 1,..., x k, x k+1 ), (2.23) where the last line follows thanks to admissibility of γ (k) ( ). Moreover, B,main ;j;k+1 γ(k+1) (t, x 1,..., x k ; x 1,..., x k) = dx k+1 V (x j x k+1 )γ (k+1) (t, x 1,..., x k, x k+1 ; x 1,..., x k, x k+1 ). and ( ) B,error ;i,j;k+1 γ(k+1) (t, x 1,..., x k ; x 1,..., x k) = V (x i x j)γ (k) (t, x 1,..., x k ; x 1,..., x k) = dx k+1 V (x i x j)γ (k+1) (t, x 1,..., x k, x k+1 ; x 1,..., x k, x k+1 ). The advantage of this notation will be that we can treat the BBGKY hierarchy and the GP hierarchy on the same footing. We remark that in all of the above definitions, we have that B ±,main ;k, B ±,error ;k, etc. are defined to be given by multiplication with zero for k >. As a consequence, we can write the BBGKY hierarchy compactly in the form i t Γ + ± Γ = B Γ Γ () H α, (2.24)
14 T. CHE AD. PAVLOVIĆ where and ± Γ := ( (k) ± γ (k) ) k, with (k) ± = k ) ( xj x, j B Γ := ( B ;k+1 γ (k+1) ) k. (2.25) In addition, we introduce the notation which will be convenient. B + Γ := ( B + ;k+1 γ(k+1) ) k B Γ := ( B ;k+1 γ(k+1) ) k
DERIVATIO OF THE CUBIC GP HIERARCHY 15 3. Statement of main results and outline of proof strategy The main result proven in this paper is the following theorem. Theorem 3.1. Let d = 2, 3 and let β < 1 d + 1. (3.1) Moreover, let δ > be an arbitrary small, fixed number. Assume that Φ solves the -body Schrödinger equation (2.1) with initial condition Φ (t = ) = Φ, L 2 (R d ), where the pair potential V (x) = dβ V ( β x), for V W 2, (R d ) L 1 (R d ), is spherically symmetric, positive, and V C δ (R d ) L (R d ) with rapid decay outside the unit ball. Let Γ Φ = (γ (n) Φ,..., γ (n) Φ,,,... ) (3.2) denote the associated sequence of marginal density matrices (trivially extended by zeros), which solves the -BBGKY hierarchy, Γ Φ (t) = U(t) Γ Φ () + i U(t s) B Γ Φ (s) ds. (3.3) Furthermore, we assume that Γ Φ, H 1+δ exists for some < < 1. Γ = Define the truncation operator P K by for all, and that lim ΓΦ, H 1+δ (3.4) P K Γ = (γ (1),..., γ (K),,,... ), (3.5) and observe that P K Γ Φ (t) = U(t) P K Γ Φ () + i U(t s) P K B Γ Φ (s) ds. (3.6) Let K() := b log (3.7) for a sufficiently large constant b >. Then, there exists Γ L t I H1 with BΓ such that the limits L 2 t I H1 and lim P K()Γ Φ Γ L t I H 1 = (3.8) lim B P K() Γ Φ BΓ L 2 t I H 1 = (3.9) hold, for I = [, T ] with < T < T (), and for > sufficiently small (it is sufficient that < < η with η specified in Lemma B.3 below). In particular, Γ solves the cubic GP hierarchy, Γ(t) = U(t) Γ + i U(t s) B Γ(s) ds, (3.1) with initial data Γ.
16 T. CHE AD. PAVLOVIĆ We note that the limits K and are taken simultaneously, and that the smallness of the parameter > is used (since small > corresponds to large energy per particle, this does not lead to any loss of generality). In our proof, we will significantly make use of our work [8] which proves the unconditional existence of solutions Γ L t I Hα of GP hierarchies, without assuming Γ L 2 t I Hα <. 3.1. Outline of the proof strategy. The proof contains the following main steps: Step 1: In a first step, we construct a solution to the -BBGKY hierarchy with truncated initial data. First, we recall that the -BBGKY hierarchy is given by for all k. k i t γ (k) = [ xj, γ (k) ] + B,k+1γ (k+1) (3.11) Given K, we let P K denote the projection operator P K : G G Γ = (γ (1), γ(2),..., γ(),,,... ) (γ(1),..., γ(k),,,... ), (3.12) and P >K = 1 P K, as well as P K := P K P K 1. Instead of considering the solution obtained from Φ, we consider (3.11) with truncated initial data Γ K, := P KΓ,, for some fixed K. We will refer to solutions of this system as the K-truncated -BBGKY hierarchy, or (K, )-BBGKY hierarchy in short. We note that in contrast, Γ Φ solves (3.11) with un-truncated initial data Γ,. ext, we prove via a fixed point argument that there exists a unique solution of the (K, )-BBGKY hierarchy for every initial condition Γ K, H1+δ in the space { Γ K L t I K H 1+δ B Γ K L 2 t I K H 1+δ }. (3.13) To this end, we re-interpret Γ K, as an infinite sequence, extended by zeros for elements (Γ K, )(k) = with 1 k > K. Hence, we have obtained solutions Γ K (t) of the BBGKY hierarchy, for the truncated initial data i t Γ K = ± Γ K + B Γ K, (3.15) Γ K () = P K Γ () = (γ (1) (),..., γ(k) (),,,... ) (3.16) 1 We observe that then, (3.11) determines a closed, infinite sub-hierarchy, for initial data γ (k) () =, for k > K, which has the trivial solution (γ K )(k) (t) =, t I = [, T ], k > K. (3.14)
DERIVATIO OF THE CUBIC GP HIERARCHY 17 for an arbitrary, large, fixed K, and where component (Γ K )(m) (t) = for the m-th component, for all m > K. By the Duhamel formula, the solution of (3.15) is given by Γ K (t) = U(t)Γ K () + i for initial data Γ K () = P KΓ (). t U(t s) B Γ K (s) ds (3.17) For a fixed scale < < 1, it is sufficient to iterate the Duhamel formula (3.17) for Γ K only finitely many times, in order to obtain a fully explicit solution to (3.15) for fixed K that satisfies Γ K L t I H 1+δ, B Γ K L 2 t I H 1+δ C(T, ) Γ H 1+δ. (3.18) Step 2: In this step, we let K() = b log for a sufficiently large constant b >, and take the limit of the solution Γ K() to (3.15) which was obtained in Step 1. To this end, we invoke the solution Γ K of the GP hierarchy with truncated initial data, Γ K (t = ) = P K Γ H 1. In [8], we proved the existence of a solution ΓK that satisfies the K-truncated GP-hierarchy in integral form, Γ K (t) = U(t)Γ K () + i t U(t s) BΓ K (s) ds (3.19) where (Γ K ) (k) (t) = for all k > K. Moreover, it is shown in [8] that this solution satisfies BΓ K L 2 t I H1. We then prove the following convergence: (a) In the limit, Γ K() satisfies lim ΓK() Γ K() L t H 1 =. (3.2) (b) In the limit, B Γ K() satisfies lim B Γ K() BΓ K() L 2 t H 1 =. (3.21) The proof of these limits makes use of the δ amount of extra regularity of the initial data Γ, Γ, H 1+δ beyond H 1. Step 3: We compare the solution Γ K of the K-truncated -BBGKY hierarchy to the the truncated solution P K() Γ Φ of the -BBGKY hierarchy. otably, both have the same value at t =, given by P K() Γ,. Letting K() = b log for a sufficiently large constant b > (the same as in the previous step), we prove that and lim ΓK() P K() Γ Φ L t I H 1 =. (3.22) lim B Γ K() B P K() Γ Φ L 2 t I H 1 =. (3.23)
18 T. CHE AD. PAVLOVIĆ The proof of this limit involves the a priori energy bounds for the -body Schrödinger system (2.3) established in [14, 15, 24]. The freedom to choose the parameter > to be sufficiently small is used in this step. Step 4: Finally, we determine the limit of Γ K() from Step 2, obtaining that: (i) The strong limit lim Γ K() exists in L t H 1, and satisfies lim ΓK() = Γ L t H, 1 (3.24) where Γ is a solution to the full GP hierarchy (2.8) with initial data Γ. (ii) In addition, the strong limit lim BΓ K() exists in L 2 t H 1, and satisfies lim BΓK() = BΓ L 2 t H. 1 (3.25) The results of Step 4 were proven in our earlier work [8].
DERIVATIO OF THE CUBIC GP HIERARCHY 19 4. Local well-posedness for the (K, )-BBGKY hierarchy In this section, we prove the local well-posedness of the Cauchy problem for the K-truncated -BBGKY hierarchy, which we refer to as the (K, )-BBGKY hierarchy for brevity. In the sequel, we will have d = 2, 3. Lemma 4.1. Assume that Γ K, = P KΓ, H 1+δ for some < < 1 and δ. Then, there exists a unique solution Γ K L t I H1+δ of (3.17) for I = [, T ] with T > sufficiently small, and independent of K,. In particular, B Γ K L 2 t I H1+δ and. Moreover, Γ K L t I H 1+δ B Γ K L 2 t I H 1+δ C (T,, ) Γ K, H 1+δ (4.1) C (T,, ) Γ K, H 1+δ (4.2) hold for < < sufficiently small (it is sufficient that < < η with η specified in Lemma B.3 below). The constant C = C (T,, ) is independent of K,. Furthermore, (Γ K (t))(k) = for all K < k, and all t I. Proof. To obtain local well-posedness of the Cauchy problem for the (K, )-BBGKY hierarchy, we consider the map M K ( Θ K 1 ) := B U(t)Γ K, + i t B U(t s) Θ K 1 (s), (4.3) where P ΘK 1 K 1 = Θ K 1 on the subspace Ran(P K ) L 2 t I H1+δ L 2 t I H1+δ. Using the K-truncated Strichartz estimate in Proposition A.2, we find that M K ( Θ K 1 1 ) M K ( Θ K 1 2 ) L 2 t I H 1+δ t ds B U(t s)( Θ K 1 T Thus, for (T (K)) 1 2 < ds B U(t s)( Θ K 1 T C (K) 1 ds ( Θ K 1 C (K) 1 T 1 2 Θ K 1 1 K 1 1 Θ K 1 1 Θ K 1 1 Θ K 1 Θ 2 2C (K), we find that MK 2 )(s) 2 )(s) L 2 t I H 1+δ H 1+δ L 2 t I H 1+δ By the fixed point principle, we obtain a unique solution Θ K 1 = P K 1 Θ K 1 satisfying Θ K 1 In particular, Θ K 1 L 2 t I H1+δ Θ K 1 (t) = B U(t)Γ K, + i t B U(t)Γ K, L 2 t I H 1+δ L 2 t I 2 )(s) H 1+δ. (4.4) is a contraction on L2 t I H1+δ L 2 t I H1+δ. with B U(t s)θ K 1 (s). (4.5) + C (K) 1 T 1 2 Θ K 1 L 2 t I H1+δ (4.6)
2 T. CHE AD. PAVLOVIĆ and use of Proposition A.2 implies that holds. Θ K 1 L 2 t I H1+δ C (K) 1 1 C (K) 1 T 1 2 Γ K, H 1+δ (4.7) ext, we let Clearly, Γ K (t) := U(t)Γ K, + i Γ K L t I H 1+δ t Γ K, L t I H 1+δ 1 1 C (K) 1 T 1 2 from (4.7). Comparing the right hand sides of B Γ K holds, and that B Γ K Γ K (t) = U(t)Γ K, + i U(t s)θ K 1 (s). (4.8) + T 1 2 Θ K 1 L 2 t I H1+δ Γ K, H 1+δ and ΘK 1 (4.9), we conclude that = Θ K 1 (4.1) t U(t s)b Γ K (s) ds (4.11) is satisfied, with B Γ K L2 t I H1+δ. So far, we have established well-posedness of solutions of the (K, )-BBGKY hierarchy for t [, T ] with T < T (K, ). We can piece those together, in order to extend the solution to longer time intervals. As a matter of fact, we can prove that (4.9) can be enhanced to an estimate with both C and T independent of K. To this end, we observe that applying B to (4.11), we find B Γ K (t) = B U(t)Γ K, + i t B U(t s)b Γ K (s) ds. (4.12) It is easy to verify that the assumptions of Lemma B.3 in the Appendix are satisfied for We assume that Θ K := B Γ K, Ξ K := B U(t)Γ K,. (4.13) < η < η 2 (4.14) where < η < 1 is as in Lemma B.3. Then, Lemma B.3 implies that B Γ K L 2 t I H 1+δ C(T,, η) B U(t)Γ K, L 2 t I H 1+δ C (T,, η) Γ K, H 1+δ (4.15) holds for a constant C = C (T,, η) independent of K,, and for T < T (, η). It remains to prove that (Γ K (t))(k) = for all K < k, and all t I. To this end, we first note that (B P K P K 1 B ) Γ K =, (4.16)
DERIVATIO OF THE CUBIC GP HIERARCHY 21 as one easily verifies based on the componentwise definition of B in (2.22) and (2.23). Hence, in particular, (P >K B B P >K+1 ) Γ K =, thanks to which we observe that P >K Γ K by itself satisfies a closed sub-hierarchy of the -BBGKY hierarchy, where clearly, with initial data i t (P >K Γ K ) = ± (P >K Γ K ) + B (P >K+1 Γ K ), (4.17) P >K+1 Γ K = P >K+1 (P >K Γ K ), (4.18) (P >K Γ K )() = P >K (Γ K ()) =. (4.19) Here we recall that the initial data is truncated for k > K. Accordingly, by the same argument as above, there exists a unique solution (P >K Γ K ) L t I H with B (P >K+1 Γ K ) L2 t I H such that B (P >K+1 Γ K ) L 2 t I H C (T,, η) (P >K Γ K )() H 1+δ =, (4.2) for < η 2. Moreover, P >K Γ K L t I H C 1 (T,, η) (P >K Γ K )() H 1+δ =. (4.21) This implies that (P >K Γ K )(t) = for t I, as claimed.
22 T. CHE AD. PAVLOVIĆ 5. From (K, )-BBGKY to K-truncated GP hierarchy In this section, we control the limit of the truncated BBGKY hierarchy, at fixed K. Proposition 5.1. Assume that V (x) = dβ V ( β x) with V C δ L for some arbitrary but fixed, small δ >. Moreover, assume that Γ K W 1+δ (I) (see (1.24)) is the solution of the GP hierarchy with truncated initial data Γ K = P K Γ H 1+δ constructed in [8]. Let Γ K solve the (K, )-BBGKY hierarchy with initial data ΓK, := P KΓ, H 1+δ. Let K() := b log (5.1) for some finite constant b >. Then, as, the strong limits and hold, for < T < T (). lim ΓK() Γ K() L = (5.2) t [,T ] H1 lim B Γ K() BΓ K() L 2 = (5.3) t [,T ] H1 Proof. In [8], we constructed a solution Γ K of the full GP hierarchy with truncated initial data, Γ() = Γ K H 1+δ, satisfying the following: for an arbitrary fixed K, Γ K satisfies the GP-hierarchy in integral representation, Γ K (t) = U(t)Γ K + i t and in particular, (Γ K ) (k) (t) = for all k > K. Accordingly, we have B Γ K BΓ K U(t s) BΓ K (s) ds, (5.4) = B U(t)Γ K, BU(t)Γ K t ( + i B U(t s)b Γ K BU(t s)bγ K ) (s)ds = (B B)U(t)Γ K, + BU(t)(Γ K, Γ K ) + i + i t t ( B B ) U(t s)bγ K (s) ds B U(t s) ( B Γ K BΓ K ) (s) ds. (5.5) Here, we observe that we can apply Lemma B.4 with Θ K := B Γ K BΓ K (5.6)
DERIVATIO OF THE CUBIC GP HIERARCHY 23 and Ξ K := (B B)U(t)Γ K, + BU(t)(Γ K, Γ K ) + i t Given, we introduce parameters,, satisfying ( B B ) U(t s)bγ K (s) ds. (5.7) < η < η 2 < η 3 (5.8) where < η < 1 is as in Lemma B.3. Accordingly, Lemma B.3 implies that B Γ K BΓ L 2 t I H 1 ( C (T,, ) BU(t)(Γ K, Γ K ) L 2 t I H 1 ( C 1 (T,,, ) Γ K, Γ K L 2 t I H 1 + R K () where we used Lemma A.1 to pass to the last line. Here, with and R K 2 () := ) + R K () ), (5.9) R K () = R K 1 () + R K 2 (), (5.1) R K 1 () := (B B)U(t)Γ K, L 2 t I H 1 (5.11) t ( B B ) U(t s)bγ K (s) ds L 2 t I H 1. (5.12) ext, we consider the limit with K() = b log, for some finite constant b <. To begin with, we note that lim Γ, Γ H 1+δ =. (5.13) Including the truncation at K(), it is easy to see that follows. lim ΓK(), ΓK() H 1+δ = lim P K() ( Γ, Γ ) H 1+δ lim Γ, Γ H 1+δ = (5.14) To control R K() (), we invoke Lemma 5.2 below, which implies that for an arbitrary but fixed δ >, lim RK() 1 () lim C V,δ 1 δβ Γ K(), L 2 t I H1+δ =, (5.15) for a constant C V,δ that depends only on V and δ, since and Γ H 1+δ <. lim ΓK(), Γ H 1+δ =, (5.16)
24 T. CHE AD. PAVLOVIĆ Moreover, invoking Lemma 5.3 below, we find because lim RK() 2 () lim C V,δ 1 δβ BΓ K() (s) L 2 t I H 1+δ =, (5.17) BΓ K() L 2 t I H 1+δ < C(T,, ) Γ H 1+δ (5.18) is uniformly bounded in, as shown in [8]. Lemma 5.2. Let δ > be an arbitrary, but fixed, small number. Assume that V (x) = dβ V ( β x) with V C δ L. Then, with < η as in (5.8), (B B)U(t)Γ K, L 2 t R H 1 < C V,δ 1 δβ Γ K, H 1+δ (5.19) for a constant C V,δ depending only on V and δ, but not on K or. Proof. In a first step, we prove that (B + ;k+1 B+ k+1 ) U (k+1) (t)γ (k+1) L 2 t R H C 1 k 2 δβ γ (k+1) L 2 t R H1+δ (5.2) holds, for V C δ L with δ >. To this end, we note that V () = V ( β ), V () = V (x)dx = V (x)dx = V () = 1, (5.21) and we define We have where χ () := k V () V (), (5.22) χ (q q ) = χ 1 (q q ) + χ 2 (q q ) (5.23) χ 1 (q q ) := V (q q ) V (), χ 2 (q q ) := k V (q q ). (5.24) Clearly, we have that for δ > small, δ-holder continuity of V implies χ 1 (q q ) V C δ δβ q q δ V C δ δβ ( q δ + q δ ), (5.25) and χ 2 (q q ) V L k 1 (5.26) is clear. ext, we let (τ, u k, u k ), q and q denote the Fourier conjugate variables corresponding to (t, x k, x k ), x k+1, and x k+1, respectively. Without any loss of generality, we may assume that j = 1 in B ;j;k+1 and B j;k+1. Then, abbreviating k δ( ) := δ( τ + (u 1 + q q ) 2 + u 2 j + q 2 u k 2 (q ) 2 ) (5.27) j=2
DERIVATIO OF THE CUBIC GP HIERARCHY 25 we find S (k,1) (B ;1;k+1 B 1;k+1 )U (k+1) (t)γ (k+1) = R dτ du k du k, 2 L 2 (R R dk R dk ) k 2 uj u 2 j (5.28) ( ) 2 dqdq δ( ) χ (q q ) γ (k+1) (τ, u 1 + q q, u 2,..., u k, q; u k, q ), similarly as in [23, 24]. Using the Schwarz inequality, this is bounded by dτ du k du k J(τ, u k, u k) dqdq δ( ) R u1 + q q 2 2 q q 2 k 2 k uj u 2 j χ (q q ) 2 j=2 j =1 γ (k+1) (τ, u 1 + q q, u 2,..., u k, q; u k, q ) 2 (5.29) where J(τ, u k, u k) := 2 dq dq δ( ) u1 R 3 R 3 u1 + q q 2 2 q q 2. (5.3) The boundedness of C J := ( ) 1 sup J(τ, u k, u k ) 2 τ,u k,u k < (5.31) is proven in [23] for dimension 3, and in [6, 24] for dimension 2. Using (5.25) and (5.26), we obtain, from the Schwarz inequality, that (5.29) C V,J dτ du k du k dqdq ( 2δβ ( q 2δ + q 2δ ) + k 2 2 ) R u1 + q q 2 2 q q 2 k 2 k uj u 2 j where C V,J := C V C J, and C V j=2 j =1 γ (k+1) (τ, u 1 + q q, u 2,..., u k, q; u k, q ) 2 (5.32) S (k,1) (B ;1;k+1 B 1;k+1 )U (k+1) (t)γ (k+1) is a finite constant depending on V. Hence,, 2 L 2 (R R dk R dk ) C V,J 2δβ γ (k+1), 2 H 1+δ + C V,J k 2 2 γ (k+1), 2 H 1 C V,J k 2 2δβ γ (k+1), 2 H1+δ (5.33) follows, given that δβ < 1 for δ > sufficiently small.
26 T. CHE AD. PAVLOVIĆ Therefore, we conclude that (B B) U(t) Γ K = K k=1 L 2 t R H 1 k (B + ;k+1 B+ k+1 ) U (k+1) (t)γ (k+1) L 2 t R H 1 C δβ 1 K k=1 k 2 k+1 γ (k+1) H 1+δ k+1 ( C δβ sup k 2( ) k ) k 1 Γ K H 1+δ C δβ 1 Γ K H 1+δ, (5.34) for <. This proves the Lemma. Lemma 5.3. Assume that V (x) = dβ V ( β x) with V C δ L. Then, t ( B B ) U(t s)bγ K (s) ds L 2 t I H 1 < C V,δ 1 T 1 2 δβ BΓ K L 2 t I H 1+δ (5.35) where the constant C V,δ > depends only on V and δ, and < η as in (5.8). Proof. Using Lemma 5.2, t ( B B ) U(t s)bγ K (s) ds L 2 t I H 1 T ( B B ) U(t s)bγ K (s) ds C V,δ 1 δβ T BΓ K (s) ds H 1+δ L 2 t R H 1+δ < C V,δ 1 T 1 2 δβ BΓ K L 2 t I H, 1+δ (5.36) for C V,δ as in the previous lemma. This proves the claim.
DERIVATIO OF THE CUBIC GP HIERARCHY 27 6. Comparing the (K, )-BBGKY with the full -BBGKY hierarchy In this section, we compare solutions Γ K of the (K, )-BBGKY hierarchy to solutions Γ Φ to the full -BBGKY hierarchy obtained from Φ which solves the -body Schrödinger equation (2.1). Lemma 6.1. There is a finite constant C(T, ) independent of K, such that the estimate B Γ K B P K Γ Φ L 2 t I H 1 C(T, ) K K (B Γ Φ ) (K) L 2 t I H 1 (6.1) (and the only non- holds, where (B Γ Φ ) (K) is the K-th component of B Γ Φ vanishing component of P K B Γ Φ ). Proof. We have already shown that B Γ K L2 t I H1. Moreover, it is easy to see that B Γ K L 2 t I H 1 < C(, K, T ). (6.2) The easiest way to see this is to use the trivial bound V L fact that I = [, T ] is finite. < c(), and the Thus, B Γ K B P K Γ Φ L 2 t IH 1 (6.3) follows. ext, we observe that (B Γ K B P K Γ Φ )(t) = (B Γ K P K 1 B Γ Φ )(t) (6.4) = B U(t)Γ K () P K 1 B U(t)Γ () + i t B U(t s)b Γ K (s) ds i = (B P K P K 1 B ) U(t)Γ () + i (B P K P K 1 B ) + i t t B U(t s)b Γ K (s) ds t t i B P K U(t s)b Γ Φ (s) ds = (B P K P K 1 B ) Γ Φ (t) + i t B U(t s) ( B Γ K + i (B P K 1 B P K ) P K 1 B U(t s)b Γ Φ (s) ds U(t s)b Γ Φ (s) ds P K 1 B Γ Φ ) (s) ds t U(t s) B Γ Φ (s) ds, (6.5)
28 T. CHE AD. PAVLOVIĆ where to obtain (6.4) we used the fact that (B P K P K 1 B ) Γ Φ =, (6.6) which follows, as (4.16), based on the componentwise definition of B in (2.22) and (2.23). ow we notice that B P K B P K 1 = B P K. Hence (6.5) implies that (B Γ K P K 1 B Γ Φ )(t) = (B P K P K 1 B ) Γ Φ (t) i t B U(t s) P K B Γ Φ (s) ds + i t which thanks to (6.6) simplifies to (B Γ K P K 1 B Γ Φ )(t) = i t B U(t s) ( B Γ K P K 1 B Γ Φ ) (s) ds, (6.7) B U(t s) P K B Γ Φ (s) ds + i t B U(t s) ( B Γ K P K 1 B Γ Φ ) (s) ds. (6.8) We observe that the term in parenthesis on the last line corresponds to (6.4), which is the same as (6.3). Therefore, we can apply Lemma B.3 with and Θ K (t) := B Γ K (t) P K 1 B Γ Φ (t) (6.9) Ξ K (t) := i t We note that for the integral on the rhs of (6.1), t B U(t s)p K B Γ Φ (s) ds. (6.1) B U(t s)p K B Γ Φ (s) ds L 2 t I H 1 C T 1 2 K PK B Γ Φ L 2 t I H 1, (6.11) for a constant C uniformly in and K, based on similar arguments as in the proof of Lemma 5.3, and using the Strichartz estimates (A.19) and (A.31). Accordingly, Lemma B.3 implies that B Γ K B P K Γ Φ L 2 t I H 1 = B Γ K P K 1 B Γ Φ L 2 t I H 1 C (T, ) Ξ K L 2 t I H 1 C(T, ) K P K B Γ Φ L 2 t I H 1, (6.12) where P K = P K P K 1. This immediately implies the asserted estimate, for T sufficiently small (depending on K). Clearly, P K B Γ Φ L 2 t I H 1 = K (B Γ Φ ) (K) L 2 t I H 1. (6.13)
DERIVATIO OF THE CUBIC GP HIERARCHY 29 Therefore, B Γ K B P K Γ Φ L 2 t I H 1 C(T, ) K K (B Γ Φ ) (K) L 2 t I H 1, (6.14) as claimed. Here, we have modified the result of Lemma B.3 by setting =, due to the fact that P K B Γ Φ has a single nonzero component.
3 T. CHE AD. PAVLOVIĆ 7. Control of Γ Φ and Γ K as In this section, we control the comparison between Γ Φ and Γ K in a limit where simultaneously, K = K() at a suitable rate. Proposition 7.1. Assume that K() := b log, (7.1) for a sufficiently large constant b >, and that < 1 given in Lemma 7.2 below. Then, 2b 1 where b 1 is the constant lim B Γ K() P K() 1 B Γ Φ L 2 t I H 1 = (7.2) holds. Proof. From Lemma 7.2 below, we have the estimate B Γ K P K 1 B Γ Φ L 2 t I H 1 C(T, ) (d+1)β K 2 (b 1 ) K (7.3) where b 1, C are independent of K,. By assumption, b 1 < 1 2. Therefore, we have that for sufficiently large b, (d+1)β K 2 (b 1 ) K() < C (d+1)β (b log ) 2 b log 2 < ɛ, (7.4) for some ɛ >. This immediately implies the claim. Lemma 7.2. The estimate B Γ K P K 1 B Γ Φ L 2 t I H 1 C(T, ) (d+1)β K 2 (b 1 ) K (7.5) holds for finite constants b 1, C independent of K,, and T. The constant b 1 only depends on the initial state Φ () of the -body Schrödinger problem. Proof. From Lemma 6.1, we have that B Γ K P K 1 B Γ Φ L 2 t I H 1 C(T, ) K K (B Γ Φ ) (K) L 2 t I H 1 (7.6) holds for a finite constant C(T, ) independent of K,.
DERIVATIO OF THE CUBIC GP HIERARCHY 31 We have (B + ΓΦ ) (K) 2 L 2 t I H1 C dt dx K dx K I C V 2 C 1 I dt C T 2(d+1)β K 2 sup t I K [ K l=1 ] xj x V j (x l x K+1 )Φ (t, x ) Φ (t, x K, x K+1,..., x ) dx K+1 dx 2 K [ K ] dx xj Φ (t, x ) sup t I ( l=1 dx [ K Tr ( S (K,1) γ (K) 2 ] x Φ j (t, x ) 2 ) ) 2 using Cauchy-Schwarz to pass to (7.7), and admissibility to obtain (7.8). (7.7) (7.8) It remains to bound the term Tr ( S (K,1) γ (K) ) (7.9) in (7.8). To this end, we recall energy conservation in the -body Schrödinger equation satisfied by Φ. Indeed, it is proved in [14, 15, 24] that Φ, ( + H ) K Φ C K K Tr(S (1,K) γ (K) Φ ) (7.1) for some positive constant C > where γ (k) Φ = Tr k+1,..., ( Φ Φ ). (7.11) This implies that Tr ( S (K,1) γ (K) ) < b K 1 (7.12) for some finite constant b 1 >. The fact that follows immediately from the definition of V. V C 1 C (d+1)β (7.13)
32 T. CHE AD. PAVLOVIĆ 8. Proof of the main Theorem 3.1 We may now collect all estimates proven so far, and prove the main result of this paper, Theorem 3.1. To this end, we recall again the solution Γ K of the GP hierarchy with truncated initial data, Γ K (t = ) = P K Γ H 1. In [8], we proved the existence of a solution Γ K that satisfies the K-truncated GP-hierarchy in integral form, Γ K (t) = U(t)Γ K () + i t U(t s) BΓ K (s) ds (8.1) where (Γ K ) (k) (t) = for all k > K. Moreover, it is shown in [8] that this solution satisfies BΓ K L 2 t I H1. Moreover, we proved in [8] the following convergence: (a) The strong limit exists. (b) The strong limit exists, and in particular, Clearly, we have that Γ := s lim K ΓK L t H 1 (8.2) Θ := s lim K BΓK L 2 t H 1. (8.3) Θ = BΓ. (8.4) BΓ B P K() Γ Φ L 2 t I H 1 BΓ BΓ K() L 2 t I H 1 + BΓ K() B Γ K() L 2 t I H 1 (8.5) (8.6) + B Γ K() B P K() Γ Φ L 2 t I H 1. (8.7) In the limit, we have that (8.5) from (8.3) and (8.4). Moreover, (8.6) follows from Proposition 5.1. Finally, (8.7) follows from Proposition 7.1. Therefore, follows. lim BΓ B P K() Γ Φ L 2 t I H 1 = (8.8)
DERIVATIO OF THE CUBIC GP HIERARCHY 33 Moreover, we have that P K() Γ Φ Γ L t I H 1 P K() Γ Φ Γ K() L t I H 1 + Γ K() Γ L t I H 1 (8.9) (8.1) + Γ K() Γ K() L t I H 1. (8.11) In the limit, we have (8.9), as a consequence of Proposition 7.1. Indeed, P K() Γ Φ T 1 2 B Γ K() Γ K() L t I H 1 B P K() Γ Φ L 2 t I H 1 where the rhs tends to zero as, as discussed for (8.7). Moreover, (8.1), as a consequence of (8.2). Finally, (8.11) follows from Proposition 5.1. (8.12) This completes the proof of Theorem 3.1.
34 T. CHE AD. PAVLOVIĆ Appendix A. Strichartz estimates for GP and BBGKY hierarchies In this section, motivated by the Strichartz estimate for the GP hierarchy, we establish a Strichartz estimate for the BBGKY hierarchy. A.1. Strichartz estimates for the GP hierarchy. Following [8], we first recall a version of the GP Strichartz estimate for the free evolution U(t) = e it ± = (U (n) (t)) n. The estimate is obtained via reformulating the Strichartz estimate proven by Klainerman and Machedon in [23]. Lemma A.1. Let α A(d) =, ) if d = 1 [, ) if d 2 and d 3 (A.1) 1, ) if d = 3. ( 1 2 ( d 1 2 Then, the following hold: (1) Bound for K-truncated case: Assume that Γ H α for some < < 1. Then, for any K, there exists a constant C(K) such that the Strichartz estimate for the free evolution BU(t)Γ K L 2 t R H α 1 C(K) Γ K H α (A.2) holds. otably, the value of is the same on both the lhs and rhs. (2) Bound for K : Assume that Γ H α for some < < 1. Then, for any < <, there exists a constant C(, ) such that the Strichartz estimate for the free evolution BU(t)Γ L 2 t R H α C(, ) Γ H α (A.3) holds. Proof. From Theorem 1.3 in [23] we have, for α A(d, p), that B k+1 U (k+1) (t)γ (k+1) L 2 t R Hk α k 2 B + j;k+1 U (k+1 (t)γ (k+1) L 2 t R Hk α C k γ (k+1) H α k+1. (A.4)
DERIVATIO OF THE CUBIC GP HIERARCHY 35 Then for any < <, we have: BU(t)Γ L 2 t R H α k 1 k B k+1 U (k+1) (t)γ (k+1) L 2 t R H α k C k 1 k k γ (k+1) H α k+1 (A.5) = C ( ) 1 k 1 k ( ) k ( ) (k+1) γ (k+1) H α k+1 ( ) k C ( ) 1 sup k k 1 ( ) (k+1) γ (k+1) H α k+1 k 1 C(, ) Γ H α, where to obtain (A.5) we used (A.4). On the other hand, we have BU(t)Γ K L 2 t R H α This proves the Lemma. K 1 C k=1 k B k+1 U (k+1) (t)γ (k+1) L 2 t R H α k K 1 k=1 K 1 = C K ( ) 1 k k γ (k+1) H α k+1 (A.6) k=1 C (K) 1 Γ K H α. (k+1) γ (k+1) H α k+1
36 T. CHE AD. PAVLOVIĆ A.2. Strichartz estimates for the BBGKY hierarchy. In this subsection, we prove a new Strichartz estimate for the free evolution U(t) = e it ± in L 2 t I Hα, for the BBGKY hierarchy, at the level of finite. This result parallels the one for the GP hierarchy, which was stated in Lemma A.1. Proposition A.2. Let α A(d) for d 2, and β < 1 d + 2α 1. (A.7) Assume that V L 1 (R d ), and that V decays rapidly outside the unit ball. Letting c := 1 β(d + 2α 1), the following hold: (1) Bound for K-truncated case: Assume that Γ H α for some < < 1. Then, for any K, there exists a constant C(K) such that the Strichartz estimate for the free evolution and B main U(t)P K Γ L 2 t R H α B error U(t)P K Γ L 2 t R H α 1 C K Γ H α (A.8) 1 C K c Γ H α. (A.9) otably, the value of is the same on both the lhs and rhs. (2) Bound for K : Assume that Γ () H α for some < < 1. Then, for any < <, there exists a constant C(, ) such that we have the Strichartz estimates for the free evolution and B main Û(t)Γ () L 2 t R H α B error Û(t)Γ () L 2 t R H α C(, ) Γ () H α C(, ) c Γ () H α. (A.1) (A.11) Proof. We recall that B contains a main, and an error term. We will see that the error term is small only if the condition (A.7) on the values of β holds. This is an artifact of the L 2 -type norms used in this paper; squaring the potential V in the error term makes it more singular to a degree that it can only be controlled for sufficiently small β. (1) The main term. We first consider the main term in B ± ;k;k+1 γ(k+1). We have B +,main ;j;k+1 U (k+1) (t)γ (k+1) () 2 L 2 t R Hα = B +,main ;k;k+1 U (k+1) (t)γ (k+1) () 2 L 2 t R Hα = dt dx k dx k S (k,α) du k+1 du k+1 dx k+1 dq V (q) e iq(x k x k+1 ) R e i k (xjuj x j u j ) e ix k+1(u k+1 u k+1 ) e it k+1 (u2 j (u j )2) γ (k+1) (; u k+1 ; u k+1 ) 2 (A.12)