Derivation in strong topology and global well-posedness of solutions ...

arXiv:1305.1404v4 [math-ph] 7 Apr 2014

DERIVATION IN STRONG TOPOLOGY AND GLOBAL WELL-POSEDNESS OF SOLUTIONS TO THE GROSS-PITAEVSKII HIERARCHY THOMAS CHEN AND KENNETH TALIAFERRO Abstract. We derive the cubic defocusing GP hierarchy in R3 from a bosonic N -particle Schr¨ odinger equation as N → ∞, in the strong topology corresponding to the space H1ξ introduced in [11]. In particular, we thereby eliminate the requirement of regularity H1+ for the initial data used in [15]. Moreover, the ξ marginal density matrices obtained in this strong limit are allowed to be of infinite rank. This contrasts previous results where weak-* limits were derived, and subsequently enhanced to strong limits based on the condition that the limiting density matrices have finite rank. Furthermore, we prove that positive semidefiniteness of marginal density matrices is preserved over time, which we combine with results in [14], to obtain the global well-posedness of solutions.

1. Introduction The Gross-Pitaevskii (GP) hierarchy emerges, in the limit as N → ∞, from an N -body Schrödinger equation describing an interacting Bose gas under GrossPitaevskii scaling, via the associated BBGKY (Bogoliubov-Born-Green-KirkwoodYvon) hierarchy. Factorized solutions to the GP hierarchy are determined by a nonlinear Schrödinger equation (NLS). Through this procedure, one obtains a rigorous derivation of the NLS as a mean field description of the dynamics of a Bose-Einstein condensate. In this paper, we extend previous results proven in [15, 14] concerning the derivation and global well-posedness of the cubic GP hierarchy in R3 . We derive the GP hierarchy from the BBGKY hierarchy as N → ∞ in the strong topology relative to the space Hξ1 defined in (1.21), below, and we remove the requirement of regularity Hξ1+δ , with an arbitrary δ > 0, for the initial data used in [15]. Moreover, we prove that solutions to the cubic defocusing GP hierarchy remain positive semidefinite over time, which we use, in combination with the higher order energy functionals and related results from [14], to prove global well-posedness. The first derivation of the nonlinear Hartree equation (NLH) as a mean field description of a quantum manybody theory was given by Hepp in [37] using the Fock space formalism and coherent states. Subsequently, Spohn provided a derivation of the NLH using the BBGKY hierarchy in [51]. More recently, this topic was revisited by Fröhlich, Tsai and Yau in [32]. In a series of very important works, Erd¨ os, Schlein and Yau gave the derivation of the NLS and NLH for a wide range of situations [24, 25, 26, 27]; we will outline the main steps of their construction below, and will also mention related works of other authors. 1

2

THOMAS CHEN AND KENNETH TALIAFERRO

The problems studied here are closely related to the study of Bose-Einstein condensation, where fundamental progress was made in recent years, see [2, 43, 42, 44, 45] and the references therein. 1.1. Definition of the Model and Background. We consider a system of N bosons in R3 described by a wave function ΦN ∈ L2sym (R3N ) that satisfies the N -body Schrödinger equation i∂t ΦN = HN ΦN ,

(1.1)

where the Hamiltonian HN is the self-adjoint operator on L2 (R3N ) given by HN =

N X

(−∆xj ) +

j=1

1 N

X

1≤i 0 ,

kγ (k) kH α := kS (k,α) γ (k) kL2 (R3k ×R3k )

(1.21)

(1.22)

(1.23)

is of Hilbert-Schmidt type, as in (1.19). Those spaces are equivalent to those considered by Klainerman and Machedon in [40]. Here, we call Γ = (γ (k) )k∈N symmetric

DERIVATION AND GWP OF THE GP HIERARCHY

5

if each γ (k) (xk , x′k ) satisfies (1.8). Moreover, we call Γ positive semidefinite if γ (k) defines a positive semidefinite integral operator on L2 (R3k ) for all k. We also define the spaces ∞ n o M 2 3k 3k α < ∞ Hα := symmetric Γ ∈ L (R × R ) kΓk (1.24) H ξ ξ k=1

where for α > 0,

kΓkHαξ = 1

∞ X

k=1

ξ k Tr(|S (k,α) γ (k) |)

is of trace-norm type. Clearly, H = ∪ξ>0 H1ξ corresponding to (1.16). Moreover, α Hα ξ ⊂ Hξ holds for any α > 0 and 0 < ξ < 1. In [15], Chen and Pavlović proved that solutions of the N -body Schrödinger equation converge to the solution of the GP hierarchy in Hξ1 , for values β ∈ (0, 1/4), provided that the initial data satisfies Γ0 ∈ H1+δ for an arbitrary δ > 0, and for ξ′ ′ ξ sufficiently small (depending on 0 < ξ < 1). No factorization of solutions was assumed. More recently, a derivation of the GP hierarchy in Klainerman-Machedon type spaces was given by X. Chen and J. Holmer in [21], for values β ∈ (0, 2/3). Assuming a regularity requirement that follows from the condition (2.1) in Theorem 2.1 of our paper, they prove that solutions to the BBGKY hierarchy converge to solutions of the GP hierarchy satisfying the Klainerman-Machedon condition (1.20). This convergence is shown in the weak-* topology on the space of trace class marginal density matrices. See also [20]. Rodnianski and Schlein in [49] investigated the rate of convergence to the NLH, based on the approach of Hepp [37], which led to many further developments, including works of Grillakis-Machedon and G-M-Margetis [33, 36, 34, 35], X.Chen [18], and Lee-Li-Schlein [9]. Many authors have contributed to this very active research field, and introduced a variety of different approaches; see for instance [1, 5, 23, 29, 30, 31, 46]. 1.2. Outline of Main Results. The main results of this paper can be summarized as follows: • Derivation in Hξ1 : We show that solutions to the N -BBGKY hierarchy with initial data Γ0,N converge to those of the GP hierarchy strongly in C([0, T ], Hξ1 ) as N → ∞ when the intial data is in H1ξ , see (3.3). In [15], this convergence is obtained with initial data in Hξ1+δ , for an arbitrary, ′ small δ > 0 extra regularity. In this paper, we eliminate this condition, and provide the derivation of the GP hierarchy in the energy space. The detailed discussion is given in Section 4. • Strong convergence for limits of infinite rank: The convergence proven in the work at hand is established in the strong topology on Hξ1 . We note that in previous works following the BBGKY approach (except for [15]), including [24, 25, 26, 27, 21], convergence along a subsequence is shown in the weak-* topology on the space of trace class marginal density matrices, using a compactness argument. Subsequently, this weak-* convergence is enhanced to strong convergence in trace norm, due to the

6


special case of the limiting density matrices being of finite rank (since factorized initial data are considered). In the finite rank case (i.e., the rank of γ (k) is bounded uniformly in k), the trace norm, corresponding to H1ξ , is equivalent to the Hilbert-Schmidt norm, corresponding to Hξ1 . In contrast, we obtain a strong limit in Hξ1 without any finite rank requirement. • Global well-posedness: Combining the higher order energy functional introduced in [14], combined with the quantum de Finetti theorem as formulated by Lewin-Nam-Rougerie in [41], we prove that solutions to the cubic defocusing GP hierarchy are globally well-posed. To this end, we prove that those solutions remain positive semidefinite over time if the initial data are positive semidefinite; this allows us to invoke a key result in [14] to arrive at global well-posedness. This is carried out in Section 5. • Global in time derivation of GP hierarchy: By combining our derivation of the GP hierarchy locally in time, and global well-posedness of the GP hierarchy, we arrive at a derivation of the GP hierarchy on arbitrarily large time intervals [0, T ]; the details are presented in Section 6.

2. Statement of Main Theorems In this section, we present the main theorems proven in this paper. Our first main result provides the derivation of the GP hierarchy from a bosonic N -body Schrödinger system via the associated BBGKY hierarchy as N → ∞, in the strong 1 topology with respect to the energy space L∞ t∈[0,T ] Hξ , for a suitable 0 < ξ < 1. In particular, this strong limit yields solutions Γ(t) = (γ (k) (t))k to the GP hierarchy where γ (k) (t) does not need to be of finite rank. As noted above, a strong limit was obtained in the previous works [24, 25, 26, 27, 21] only for the special case where the density matrices (γ (k) (t))k have finite rank. Moreover, our result removes an extra regularity condition on the initial data which was assumed in [15]. In [15], solutions to the GP hierarchy were derived from the BBGKY hierarchy under the requirement that Γ0 ∈ H1+δ for an arbitrarily ξ′ small, but positive δ > 0. Here, we assume that Γ0 ∈ H1ξ′ . Theorem 2.1. Let (ΦN )N be a sequence of solutions to the N-body Schr¨ odinger (k) equation (1.1) with the corresponding marginal density matrices γΦN (t) given by (1.6). Suppose that k hΦN (0), HN ΦN (0)i < C k N k

(2.1)

and kΦN (0)kL2 = 1 for all N ∈ N and k ≤ N , where C does not depend on k or N . Moreover, assume that for some 0 < ξ ′ < 1, and every N ∈ N, we have (N )

(1)

ΓΦN (0) = (γΦN (0), . . . , γΦN (0), 0, · · · ) ∈ Hξ1′ and that Γ0 := lim ΓΦN (0) N →∞


7

exists in Hξ1′ . Define the truncation operator P≤K by P≤K(N ) Γ = (γ (1) , . . . , γ (K(N )) , 0, . . . ), where 21 b1 log N ≤ K(N ) ≤ b1 log N for some b1 > 0. Then, for sufficently small b1 > 0 (depending only on β (see (1.4))) and sufficiently small ξ > 0 (depending on only on ξ ′ ) and sufficiently small T > 0 (depending only on ξ), the limit Γ := lim P≤K(N ) ΓΦN N →∞

exists in over,

1 L∞ t∈[0,T ] Hξ

and satisfies the GP hierarchy with initial data Γ0 ∈ H1ξ′ . MoreBΓ = lim BN PK≤N ΓΦN N →∞

L2t∈[0,T ] Hξ1 .

holds in The abbreviated notations B and BN for the interaction operators are defined in (3.8) and (3.22), below. Our second main result establishes global well-posedness for solutions to the GP hierarchy, where our proof uses the quantum de Finetti theorem in the formulation presented in a recent paper by Lewin, Nam and Rougerie [41], which we quote here: Theorem 2.2. (Quantum de Finetti Theorem) Let H be a separable Hilbert space Nk and let Hk = sym H denote the corresponding bosonic k-particle space. Let Γ denote a collection of bosonic density matrices on H, i.e., Γ = (γ (1) , γ (2) , . . . )

with γ (k) a non-negative trace class operator on Hk . Then, the following hold: • (Strong Quantum de Finetti theorem, [38, 52, 41]) Assume that Γ is admissible, i.e., γ (k) = Trk+1 γ (k+1) , where Trk+1 denotes the partial trace over the (k + 1)-th factor, ∀k ∈ N. Then, there exists a unique Borel probability measure µ, supported on the unit sphere in H, and invariant under multiplication of φ ∈ H by complex numbers of modulus one, such that Z γ (k) = dµ(φ)(|φihφ|)⊗k , ∀k ∈ N . (2.2) (N )

• (Weak Quantum de Finetti theorem, [41, 3, 4]) Assume that γN is an (N ) arbitrary sequence of mixed states on HN , N ∈ N, satisfying γN ≥ 0 (N ) and TrHN (γN ) = 1, and assume that its k-particle marginals have weak-* limits (k)

(N )

γN := Trk+1,··· ,N (γN ) ⇀∗ γ (k)

(N → ∞) ,

(2.3)

(N ) Trk+1,··· ,N (γN )

in the trace class on Hk for all k ≥ 1 (here, denotes the partial trace in the (k + 1)-st up to N -th component). Then, there exists a unique Borel probability measure µ on the unit ball in H, and invariant under multiplication of φ ∈ H by complex numbers of modulus one, such that (2.2) holds for all k ≥ 0. In the context of Theorem 2.1 proven here, strong convergence in Hilbert Schmidt norm implies weak-* convergence in the trace norm topology, provided that the limit point is trace class, see Proposition A.1 in the Appendix. Therefore, the solutions to the GP hierarchy obtained in Theorem 2.1 from the BBGKY hierarchy (1.11),

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in the limit as N → ∞, satisfy the conditions of the quantum de Finetti theorem, either in its strong or its weak form. Moreover, due to the bound (2.1), it follows that the initial data for the GP hiearchy satisfies Γ0 ∈ H1ξ′ for some 0 < ξ ′ < 1. This implies that the higher order energy functionals introduced in [14], which correspond to Γ0 , are well-defined; see Section 5. We are therefore able to combine an application of the quantum de Finetti theorem with the higher order energy functionals for the cubic GP hierarchy that were introduced in [14], to prove the following result on the global well-posedness of solutions to the defocusing cubic GP hierarchy. Theorem 2.3. Assume that Z (k) γ0 = dµ(φ)(|φihφ|)⊗k , k ∈ N

(2.4)

(k)

1 ′ satisfies Γ0 = (γ0 )∞ k=1 ∈ Hξ ′ for some 0 < ξ < 1, where dµ is a probability measure supported either on the unit sphere, or on the unit ball in L2 (R3 ). For I ⊆ R, we denote by

Wξα (I) := {Γ ∈ C(I, Hξα ) | B + Γ, B − Γ ∈ L2loc (I, Hξα )} ,

(2.5)

the space of local in time solutions of the GP hierarchy, with t ∈ I, following [12]. Then, for a sufficiently small ξ1 > 0 (depending only on ξ ′ ), there is a unique global solution Γ ∈ Wξ11 (R) to the cubic defocusing GP hierarchy (3.4) in R3 with initial data Γ0 . Moreover, Γ(t) is positive semidefinite and satisfies kΓ(t)kH1ξ ≤ kΓ0 kH1′ 1

ξ

for all t ∈ R. The dependence of ξ1 on ξ ′ is detailed in Section 4.2, below. 2.1. Remarks. • We note that, by combining Theorem 2.1 and 2.3, one can show that Theorem 2.1 actually holds for T arbitrarily large, provided that Γ0 ∈ H1ξ′ , and that ξ is sufficiently small. This is addressed in detail in Section 6. • Although we only address the the cubic defocusing GP hierarchy in Rd for d = 3, we note that Theorem 2.3 can be proved in the same way for the more general cases considered in Theorem 7.2 of [14]. Let κ0 be the constant in (1.14), and let p = 2, 4 correspond to the cubic and quintic GP hierarchies, respectively. Then, we have global well-posedness for the following cases: 4 and κ0 = – Energy subcritical, defocusing GP hierarchy with p < d−2 +1. – L2 subcritical, focusing GP hierarchy with p < d4 and κ0 < 0 with |κ0 | sufficiently small (see Theorem 7.2 in [14] for an explicit bound on |κ0 |). • A solution to the GP hierarchy obtained in a weak-* limit does not necessarily satisfy admissibility, even if the system at finite N does. However, we note that solutions to the GP hierarchy preserve admissibility (1.7), provided that it holds at the initial time t = 0; see Proposition B.1 in the appendix.


9

3. Notations for the GP and BBGKY Hierarchy For convenience, we introduce additional notations for the GP and the BBGKY hierarchy in this section, mostly adopted from [12], which allow us to discuss them both on the same setting. Let 0 < ξ < 1. We recall that ∞ o n M (3.1) Hξα = symmetric Γ ∈ L2 (R3k × R3k ) kΓkHαξ < ∞ k=1

where

kΓkHαξ = with

∞ X

k=1

ξ k k γ (k) kH α (R3k ×R3k ) ,

kγ (k) kH α := kS (k,α) γ (k) kL2 (R3k ×R3k )

Qk

where S (k,α) = j=1 (1 − ∆xj )α/2 (1 − ∆x′j )α/2 . We also recall the spaces ∞ o n M 2 3k 3k α < ∞ Hα = symmetric Γ ∈ L (R × R ) kΓk H ξ ξ

(3.2)

(3.3)

k=1

where

kΓkHαξ = is of trace-norm type.

∞ X

k=1

ξ k Tr(|S (k,α) γ (k) |) .

3.1. The GP hierarchy. The cubic defocusing GP hierarchy is given by i∂t γ (k) =

k X

[−∆xj , γ (k) ] + Bk+1 γ (k+1)

(3.4)

j=1

for k ∈ N, where where

+ − Bk+1 γ (k+1) = Bk+1 γ (k+1) − Bk+1 γ (k+1) ,

+ Bk+1 γ (k+1) =

k X

(3.5)

+ Bj;k+1 γ (k+1) ,

(3.6)

− Bj;k+1 γ (k+1) ,

(3.7)

j=1

and − Bk+1 γ (k+1) =

k X j=1

with

+ Bj;k+1 γ (k+1) (t, x1 , . . . , xk ; x′1 , . . . , x′k ) Z = dxk+1 dx′k+1

δ(xj − xk+1 )δ(xj − x′k+1 )γ (k+1) (t, x1 , . . . , xk+1 ; x′1 , . . . , x′k+1 ),

10


and

− Bj;k+1 γ (k+1) (t, x1 , . . . , xk ; x′1 , . . . , x′k ) Z = dxk+1 dx′k+1

δ(x′j − xk+1 )δ(x′j − x′k+1 )γ (k+1) (t, x1 , . . . , xk+1 ; x′1 , . . . , x′k+1 ).

The GP hierarchy can be rewritten in the following compact manner: b ± Γ = BΓ i∂t Γ + ∆

Γ(0) = Γ0 ,

where

and

b ± Γ := ( ∆(k) γ (k) )k∈N , ∆ ±

(k)

with ∆± =

(3.8) k X ∆xj − ∆x′j , j=1

BΓ := ( Bk+1 γ (k+1) )k∈N .

(3.9)

We will also use the notation + − B + Γ := ( Bk+1 γ (k+1) )k∈N , B − Γ := ( Bk+1 γ (k+1) )k∈N .

Moreover, we define the free evolution operator U (t) by (U (t)Γ)(k) = U (k) (t)γ (k) , where (U (k) (t)γ (k) )(xk , x′k ) = eit∆xk e

−it∆x′

k

γ (k) (xk , x′k )

corresponds to the k-th component. 3.2. The BBGKY hierarchy. The cubic defocusing BBGKY hierarchy in R3 is given by (k)

i∂t γN (t) =

k X

(k)

[−∆xj , γN (t)] +

j=1

1 X (k) [VN (xj − xk ), γN (t)] N 1≤j N , and write (k)

i∂t γN =

k X

(k)

[−∆xj , γN ] + (BN ΓN )(k)

(3.11)

j=1

(k)

for k ∈ N. Here, we have γN = 0 for k > N , and we define  (k+1) main error (k)  + BN if k ≤ N BN ;k+1 γN ;k γN (BN ΓN )(k) :=   0 if k > N.

(3.12)


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The interaction terms on the right hand side are defined by (k+1)

main BN ;k+1 γN

(k+1)

+,main = BN ;k+1 γN

(k+1)

−,main − BN ;k+1 γN

,

(3.13)

and (k)

(k)

(k)

−,error +,error error γN , γN − BN BN ;k γN = BN ;k ;k

(3.14)

where (k+1)

±,main BN ;k+1 γN

:=

k N − k X ±,main (k+1) B γ , N j=1 N ;j;k+1 N

(3.15)

and (k)

±,error BN γN := ;k

k 1 X ±,error (k) B γ , N i N . Then, we can write the BBGKY hierarchy compactly in the form

b ± Γ N = BN Γ N i∂t ΓN + ∆ ΓN (0) ∈ Hξα ,

where

and

(k) b ± ΓN := ( ∆(k) ∆ ± γN )k∈N ,

(k)

with ∆± =

k X j=1

(k+1)

BN ΓN := ( BN ;k+1 γN

)k∈N .

(3.21)

∆xj − ∆x′j

,

(3.22)

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In addition, we introduce the notation (k+1)

+ + BN ΓN := ( BN ;k+1 γN − BN ΓN

:=

)k∈N

(k+1) − )k∈N ( BN ;k+1 γN

adapted to (3.18) and (3.20).

4. Derivation of GP from BBGKY hierarchy In this section, we prove Theorem 2.1. 4.1. Local well-posedness of the BBGKY hierarchy in Hξ1 . Taking δ = 0 in Lemma 4.1 of [15] gives us local well-posedness of the K-truncated N -BBGKY hierarchy: Lemma 4.1. Let K < b1 log N , for some constant b1 > 0. Let (k)

1 K ΓK 0,N := P≤K Γ0,N = (γ0,N )k=1 ∈ Hξ ′ ,

(4.1)

and T0 (ξ) := ξ 2 /c0 , ξ ∈ R+ ,

(4.2)

where the constant c0 > 0 is defined in Lemma D.1. Then, for 0 < ξ ′ < 1 and ξ satisfying (4.5), the following holds: For 0 < T < T0 (ξ), and for b1 > 0 sufficiently small (see (4.7) below), there exists ∞ 1 a unique solution ΓK N ∈ Lt∈I Hξ of the BBGKY hierarchy (3.10) for I = [0, T ] such 1 2 K that BN ΓN ∈ Lt∈I Hξ . Moreover, ′ K 1 ≤ C0 (T, ξ, ξ ) kΓ kΓK N kL∞ 0,N kH1′ t∈I Hξ

(4.3)

ξ

and ′ K kBN ΓK N kL2t∈I H1ξ ≤ C0 (T, ξ, ξ ) kΓ0,N kH1′ ξ

(4.4)

hold. The constant C0 = C0 (T, ξ, ξ ′ ) is independent of N . (k) Furthermore, (ΓK = 0 for all K < k ≤ N , and t ∈ I. N) 4.2. Constants. For convenience, we collect here the interdependence of various constants that appear in the formulation of the main theorems above. Throughout this paper, we will require that, given ξ ′ > 0, the real, positive constants ξ and ξ1 satisfy n o ( ξ < η min ξ1′ e−2β/b1 , e−24β/b1 and (4.5) 3 6 ′ 0 < ξ1 < θ ξ < θ ξ , where θ := min{η, (1 + 52 CSob )−2/5 }; the constant η > 0 is defined in Lemma D.2, and CSob > 0 is the constant in the trace Sobolev inequality Z Z 12

2 12

≤ CSob (4.6) dx1 dx2 ∇x1 ∇x2 f (x1 , x2 ) dx|f (x, x)|2 for x1,2 ∈ R3 , see [14]. This will ensure that ξ1 and ξ are small enough so that the results of both [15] and [14] hold.


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In (4.5), b1 > 0 is a constant chosen sufficiently small that Lemma D.2 holds for all K, N satisfying K ≤ b1 log N .

(4.7)

To satisfy this requirement, b1 only depends on β (see (1.4)).

4.3. From (K, N )-BBGKY to K-truncated GP hierarchy. In this section, we show that solutions to the (K, N )-BBGKY hierarchy approach those of the K-Truncated GP hierarchy as N → ∞. (k)

1 K Proposition 4.2. Suppose that Γ0 = (γ0 )∞ ∈ {Γ ∈ k=1 ∈ Hξ ′ . Moreover, let Γ ∞ 1 2 1 Lt∈[0,T ] Hξ | BΓ ∈ Lt∈[0,T ] Hξ } be the solution of the GP hierarchy (3.4) with trun′ cated initial data ΓK 0 = P≤K Γ0 constructed in [13], where 0 < ξ < 1 and ξ satisfy K (4.5), and 0 < T < T0 (ξ) (see (4.2)). Let ΓN solve the (K, N )-BBGKY hierarchy (3.10) with the same initial data ΓK 0,N := P≤K Γ0 . Let

K(N ) ≤ b1 log N

(4.8)

as in Lemma 4.1. Then, K(N )

lim k ΓN

N →∞

− ΓK(N ) kL∞ H1 = 0 t∈[0,T ] ξ

(4.9)

and K(N )

lim k BN ΓN

N →∞

− BΓK(N ) kL2t∈[0,T ] H1ξ = 0.

(4.10)

Proof. In [13], the authors constructed a solution ΓK of the full GP hierarchy with 1 K truncated initial data, Γ(0) = ΓK 0 ∈ Hξ ′ , such that for an arbitrary fixed K, Γ satisfies the GP-hierarchy in integral representation, Z t U (t − s) BΓK (s) ds . (4.11) ΓK (t) = U (t)ΓK + i 0 0

and, in particular, (ΓK )(k) (t) = 0 for all k > K. Accordingly, we have K BN Γ K N − BΓ

K = BN U (t)ΓK 0,N − BU (t)Γ0 Z t K (s)ds BN U (t − s)BN ΓK +i N − BU (t − s)BΓ 0

K K = (BN − B)U (t)ΓK 0,N + BU (t)(Γ0,N − Γ0 ) Z t BN − B U (t − s)BΓK (s) ds +i 0 Z t K (s) ds . +i BN U (t − s) BN ΓK N − BΓ

(4.12)

0

Here, we observe that we can apply Lemma D.2 with K K eK Θ N := BN ΓN − BΓ

(4.13)

14


and K K K ΞK N := (BN − B)U (t)Γ0,N + BU (t)(Γ0,N − Γ0 ) Z t +i BN − B U (t − s)BΓK (s) ds .

(4.14)

ξ < θ ξ ′′ < θ2 ξ ′′′ < θ3 ξ ′

(4.15)

0

Given ξ ′ , we introduce parameters ξ, ξ ′′ , ξ ′′′ satisfying

where the constant θ is defined as in (4.5), so that 0 < θ ≤ η, where η is defined as in Lemma D.2. Accordingly, Lemma D.2 implies that K kBN ΓK N − BΓ kL2t∈[0,T ] H1ξ K 2 1 ≤ C0 (T, ξ, ξ ′′ ) kBU (t)(ΓK − Γ )k + R (N ) + R (N ) 1 2 L H 0,N 0 t∈[0,T ] ξ ′′ K ′ ′′ K ≤ C1 (T, ξ, ξ , ξ ) kΓ0,N − Γ0 kL2t∈[0,T ] H1′ + R1 (N ) + R2 (N ) , (4.16) ξ

where we used Lemma A.1 in [15] to pass to the last line. Here, R1 (N ) := k(BN − B)U (t)ΓK 0,N kL2t∈[0,T ] H1′′

(4.17)

ξ

and

Z

R2 (N ) :=

BN − B U (t − s)BΓK (s) ds

t

L2t∈[0,T ] H1ξ′′

0

.

(4.18)

Next, we consider the limit N → ∞ with K(N ) as given in (4.8). We have K(N )

lim kΓ0,N

N →∞

K(N )

− Γ0

kH1′ = lim k P≤K(N ) ( Γ0,N − Γ0 ) kH1′ N →∞

ξ

ξ

≤ lim k Γ0,N − Γ0 kH1′ N →∞

= 0.

ξ

(4.19)

By Lemmas 4.3 and 4.4 below, we have that lim R1 (N ) = 0

N →∞

and lim R2 (N ) = 0.

N →∞

Thus (4.16) → 0 as N → ∞, and hence the limit (4.10) holds. To prove (4.9), we observe that K(N )

(t) − ΓK(N ) (t) Z t K(N ) K(N ) K(N ) U (t − s) BN ΓN (s) − BΓK(N ) (s) ds, (0) + i = U (t) ΓN (0) − Γ

ΓN

0


15

and hence, for 0 < t < T , K(N )

(t) − ΓK(N ) (t)kH1ξ K(N ) ≤ kU (t) ΓN (0) + ΓK(N ) (0) kH1ξ K(N ) + t1/2 kU (t − s) BN ΓN (s) − BΓK(N ) (s) kL2s∈[0,t] H1ξ

kΓN

K(N )

K(N )

(0) + ΓK(N ) (0)kH1ξ + t1/2 kBN ΓN

K(N )

(0) + ΓK(N ) (0)kH1ξ + T 1/2 kBN ΓN

= kΓN

≤ kΓN

− BΓK(N ) kL2[0,t] H1ξ

K(N )

− BΓK(N ) kL2[0,T ] H1ξ

(4.20)

→ 0 as N → ∞ by (4.10).

Since the last line (4.20) is independent of t, the result (4.9) follows. Lemma 4.3. Under the same assumptions as in Proposition 4.2,

K(N ) = 0. lim (BN − B)U (t) Γ0,N 2 1 N →∞

Lt∈R Hξ′′

(4.21)

Proof. We recall that for g : Rn → C of the form g(x) = f (x, x) for some Schwartz class fuction f : Rn × Rn → C, one has Z (4.22) g(ξ) = fb(ξ − η, η) dη. b (k+1) (xk+1 , x′k+1 ) with respect We note that the Fourier transform of U (k+1) (t)γ0 to the variables (t, xk+1 , x′k+1 ) is given by

(k+1)

δ(τ + |ξ k+1 |2 − |ξ ′k+1 |2 )b γ0

(ξ k+1 , ξ ′k+1 ).

(4.23)

(k+1)

+,main Recall that BN,1,k+1 U (t)(γ0,N ) is given by Z (k+1) VN (x1 − xk+1 )γN (t, x1 , ..., xk , xk+1 ; x′1 , ..., x′k , xk+1 )dxk+1

and hence its Fourier transform with respect to the variables (t, xk , x′k ) is given by Z (k+1) )(τ, u1 , ..., uk , xk+1 ; u′1 , ..., u′k , xk+1 ) dxk+1 e−ixk+1 u1 VbN (u1 ) ∗u1 (F γN Z Z (k+1) )(τ, u1 − η, u2 , ..., uk , xk+1 ; u′1 , ..., u′k , xk+1 ) dη dxk+1 = e−ixk+1 η VbN (η)(F γN Z Z (k+1) VbN (η)b γN (τ, u1 − η, u2 , ..., uk , η − ν; u′1 , ..., u′k , ν) dν dη (by (4.22)) = Z Z (k+1) VbN (η + ν)b γN (τ, u1 − η − ν, u2 , ..., uk , η; u′1 , ..., u′k , ν) dν dη =

where we substituted η → η + ν. Thus, the above equals Z Z VbN (uk+1 + u′k+1 ) = (k+1)

bN γ (τ, u1 − uk+1 − u′k+1 , u2 , ..., uk , uk+1 ; u′1 , ..., u′k , u′k+1 ) du′k+1 duk+1 Z Z VbN (uk+1 + u′k+1 )δ(· · · ) = (k+1)

γ0,N (u1 − uk+1 − u′k+1 , u2 , ..., uk , uk+1 ; u′k+1 ) du′k+1 duk+1 b

(4.24)

16


where the operator F is the Fourier transform with respect to the variables (t, xk , x′k ) and δ(...) := δ(τ + |u1 − uk+1 − u′k+1 |2 + |uk+1 |2 − |u1 |2 − |u′k+1 |2 ). Equation (4.23) was used to pass to the last line (4.24). Similarly, the Fourier (k+1) + transform of Bk+1 U (t)(γ0,N ) with respect to the variables (t, xk , x′k ) is given by Z Z (k+1) δ(...)b γ0,N (u1 − uk+1 − u′k+1 , u2 , ..., uk+1 , u′k+1 )duk+1 du′k+1 . Thus, (k+1)

+,main + 2 k(BN ;1;k+1 − B1;k+1 )U (t)γ0,N kL2t H 1 Z Z Z Z Z Y k k Y ′ 2 2 huj i (1 − VbN (uk+1 + u′k+1 ))δ(...) huj i = j=1

j=1

2 (k+1) γ b0,N (u1 − uk+1 − u′k+1 , u2 , ..., uk , uk+1 ; u′k+1 ) du′k+1 duk+1 duk du′k dτ Z Z Z Z Z ≤ J(τ, uk , u′k ) δ(...)hu1 − uk+1 − u′k+1 i2 huk+1 i2 hu′k+1 i2 k Y

j=2

huj i2

k Y

hu′j i2 |1 − VbN (uk+1 + u′k+1 )|2

j ′ =1

(k+1)

|b γ0,N (u1 − uk+1 − u′k+1 , u2 , ..., uk , uk+1 ; u′k+1 )|2

du′k+1 duk+1 duk du′k dτ

where J(τ, uk , u′k )

:=

Z Z

(4.25)

δ(...)hu1 i2 duk+1 du′k+1 hu1 − uk+1 − u′k+1 i2 huk+1 i2 hu′k+1 i2

and J(τ, uk , u′k ) is bounded uniformly in τ, uk , u′k , see Proposition 2.1 of [40]. Let δ satisfy 0 < δ < β. Recall that VbN (u) = Vb (N −β u). The integral (4.25) can now be separated into the regions {|uk+1 + u′k+1 | < N δ } and {|uk+1 + u′k+1 | ≥ N δ }. The portion of the integral (4.25) over {|uk+1 + u′k+1 | < N δ } is bounded by (k+1)

CV N 4(δ−β) kγ0,N k2H 1

(4.26)

because ∇Vb (0) = 0 and Vb ∈ C 2 , so by bounding the Taylor remainder term, sup

|uk+1 +u′k+1 | 0 satisfies 1 −8β/b1 −24β/b1 ξ < η min , (4.32) ,e e C where

(K)

Tr S (k,1) γN (0) < C K .

(4.33)

Then K(N )

lim kBN ΓN

N →∞

− P≤K(N )−1 BN ΓΦN kL2t∈I H1ξ = 0.

Proof. From Lemma 6.1 in [15], we have that ΦN kL2t∈I H1ξ ≤ C(T, ξ)(η −1 ξ)K Kk(BN ΓΦN )(K) kL2t∈I H 1 kBN ΓK N − P≤K−1 BN Γ (4.34)

holds for a finite constant C(T, ξ) independent of K, N . It follows immediately from the definition of VN that 4β [ k∇V . N kL1 ≤ CN

Thus we have that + ΦN (K) 2 k(BN Γ ) kL2

t∈I H

=

Z

I

dt

Z

1

  K Z k X Y  h∇xj ih∇x′ i VN (xℓ − xK+1 ) dxK dx′K j ℓ=1

j=1

2 ΦN (t, xN )ΦN (t, x′K , xK+1 , . . . , xN ) dxK+1 . . . dxN

2 2 (k,1) 2 (k,1) [ 2 2 ≤ CT (kVN k2L∞ + k∇V )K sup kR Φ k k kR Φ k 1 N L N L N L t∈I

2 (K) = CT N 8β K 2 sup Tr(S (K,1) γN (t)) . t∈I

K Since hΦN (0), HN , ΦN (0)i < C k N K , it follows from Proposition 4.5, that (K)

(4.35)

Tr( S (K,1) γN (t) ) = hΦN (t), R(K,2) ΦN (t)i 1 ≤ k k hΦN (t), (HN + N )k ΦN (t)i N C 1 = k k hΦN (0), (HN + N )k ΦN (0)i N C 1 k ≤ k k (2k hΦN (0), HN ΦN (0)i + 2k N k hΦN (0), ΦN (0)i) N C ≤ Ck. (4.36)

20


Combining (4.34), (4.35), and (4.36) yields ΦN kL2t∈I Hξ1 kBN ΓK N − P≤K−1 BN Γ

≤ C(T, ξ)(η −1 ξ)K Kk(BN ΓΦN )(K) kL2t∈I H 1

by (4.34) (K)

≤ C(T, ξ)(η −1 ξ)K KCT 1/2 N 4β K sup Tr(S (K,1) γN (t))

by (4.35)

≤ C(T, ξ)(η −1 ξ)K KCT 1/2 N 4β C K

by (4.36)

t∈I

e ≤ C(T, ξ)(η

−1

K

ξ) KN

4β

C

K

→ 0 as N → ∞

because K(N ) ≥ 21 b1 log(N ) and ξ satisfies (4.32).

4.5. Proof of Theorem 2.1. We are now ready to conclude the proof of Theorem 2.1. To this end, we recall again the solution ΓK of the GP hierarchy with truncated initial data, ΓK (t = 0) = P≤K Γ0 ∈ Hξ1 . In [13], the authors proved the existence of a solution ΓK that satisfies the K-truncated GP-hierarchy in integral form, Z t K K Γ (t) = U (t)Γ (0) + i U (t − s) BΓK (s) ds (4.37) 0

K (k)

where (Γ ) (t) = 0 for all k > K. Moreover, it is shown in [13] that this solution satisfies BΓK ∈ L2t∈I Hξ1 , where I := [0, T ]. Additionally, the following convergence was proved in [13]: (a) The limit Γ := lim ΓK

(4.38)

Θ := lim BΓK

(4.39)

K→∞

1 exists in L∞ t Hξ .

(b) The limit K→∞

exists in L2t Hξ1 , and in particular, Θ = BΓ .

(4.40)

(c) The limit Γ in equation (4.38) satisfies the full GP hierarchy with initial data Γ0 . Clearly, we have that kBΓ − BN P≤K(N ) ΓΦN kL2t∈I H1ξ ≤ kBΓ − BΓK(N ) kL2t∈I H1ξ

K(N )

+ kBΓK(N ) − BN ΓN K(N )

+ kBΓN

(4.41) kL2t∈I H1ξ

− BN P≤K(N ) ΓΦN kL2t∈I H1ξ .

(4.42) (4.43)

In the limit N → ∞, we have that (4.41) → 0 from (4.39) and (4.40). By Proposition 4.2, (4.42) → 0. (4.43) → 0 follows from Proposition 4.6. This is


21

because Γ0 ∈ H1ξ′ and hence (4.33) holds. Therefore, lim kBΓ − BN ΓΦN kL2t∈I H1ξ = 0.

N →∞

Moreover, we have that 1 kP≤K(N ) ΓΦN − ΓkL∞ t∈I Hξ

K(N )

≤ kP≤K(N ) ΓΦN − ΓN

1 kL∞ t∈I Hξ

1 + kΓK(N ) − ΓkL∞ t∈I Hξ

K(N )

+ kΓN

1 − ΓK(N ) |L∞ t∈I Hξ

(4.44) (4.45) (4.46)

By the Duhamel formula, and applying the Cauchy-Schwarz inequality in time, we have Z t K(N ) 1 (4.44) = k U (t − s)BN (P≤K(N ) ΓΦN − ΓN )(s) dskL∞ t∈I Hξ ≤T

0 1/2

K(N )

kBN ΓN

− BN P≤K(N ) ΓΦN kL2t∈I H1ξ

→ 0 as N → ∞ by Proposition 4.6.

(4.45) → 0 as N → ∞ by (4.38). Finally, (4.46) → 0 as N → ∞ follows from proposition 4.2. Thus 1 = 0. lim kP≤K(N ) ΓΦN − ΓkL∞ t∈I Hξ

N →∞

This completes the proof of Theorem 2.1.

5. Global Well-Posedness In this section, we prove Theorem 2.3. To this end, we first prove positive semidefiniteness of solutions to the GP hieararchy in Theorem 5.2, below, and subsequently global well posedness of the GP hierarchy in Theorem 5.4. To prove positive semi-definiteness, we recall the quantum de Finetti theorem, Theorem 2.2, and we invoke the following lemma from [10]. Lemma 5.1. Let µ be a Borel probability measure in L2 (R3 ), and assume that Z 2k (5.1) dµ(φ)kφk2k H1 ≤ M holds for some finite constant M > 0, and all k ∈ N. Then, µ φ ∈ L2 (R3 ) kφkH 1 > M = 0.

Proof. From Chebyshev’s inequality, we have that Z M 2k 1 dµ(φ)kφk2k µ φ ∈ L2 (R3 ) kφkH 1 > λ ≤ 2k H1 ≤ λ λ2k

for any k > 0. For λ > M , the right hand side tends to zero when k → ∞.

22


We recall that, for I ⊆ R,

Wξα (I) = {Γ ∈ C(I, Hξα ) | B + Γ, B − Γ ∈ L2loc (I, Hξα )}.

We are now ready to prove positive semidefiniteness of solutions to the GP hierarchy. Theorem 5.2. Assume that Z (k) γ0 = dµ(φ)(|φihφ|)⊗k , k ∈ N

(5.2)

(k)

1 ′ satisfies Γ0 = (γ0 )∞ k=1 ∈ Hξ ′ for some 0 < ξ < 1, where dµ is a probability measure supported either on the unit sphere, or on the unit ball in L2 (R3 ). Then, for 0 < ξ ′ < 1 and ξ > 0 satisfying (4.5), and for 0 < T < min{T0 (ξ), T1 (ξ)} (see (4.2) and (5.13)), there is a unique solution Γ ∈ Wξ1 ([0, T ]) to the cubic defocusing GP hierarchy (3.4) in R3 with initial data Γ0 . Moreover, Γ(t) is positive semidefinite for t ∈ [0, T ].

Proof. By [12] and Proposition C.3, there exists a unique solution Γ to the GP hierarchy in Wξ1 ([0, T ]) with initial data Γ0 . By the quantum de Finetti theorem (Theorem 2.2) and Lemma 5.1, there exists a positive semidefinite Borel probability measure µ on the unit sphere in L2 (R3 ) such that Z (k) γ0 = dµ(φ)(|φihφ|)⊗k (5.3)

and kφk2H 1 ≤ (ξ ′ )−1 kΓ0 kH1′ µ-almost everywhere. Let St be the flow map of the ξ

cubic defocusing NLS. Since the NLS is well-posed in H 1 , Z γ˜ (k) (t) := dµ(φ)(|St φihSt φ|)⊗k

(5.4)

˜ := {˜ is well-defined, positive semidefinite, and Γ γ (k) }∞ k=1 satisfies the cubic defocusing GP hierarchy. ˜ ∈ W 1 ([0, T ]). To prove this fact, let hK(m) iΓ(t) , Moreover, we claim that Γ ξ m ∈ N, denote the higher order energy functionals for the cubic GP hierarchy introduced in [14]. They are given by for m ∈ N, where

hK (m) iΓ(t) := Tr1,3,5,...,2m+1 (K (m) γ (2m) (t))

(5.5)

1 + 1 (1 − ∆xℓ )Trℓ+1 + Bℓ;ℓ+1 , ℓ ∈ N, 2 4 := K1 K3 · · · K2m−1 .

Kℓ := K (m)

In [14], it is shown that these higher order energy functionals are conserved: Proposition 5.3. (C-Pavlović [14]) Suppose that Γ ∈ H1ξ is symmetric, admissible, and solves the GP hierarchy. Then, for all m ∈ N, the higher order energy functionals (5.5) are bounded and conserved, hK (m) iΓ(t) = hK (m) iΓ(0) . Using the de Finetti theorem, we can eliminate the requirement of admissibility. We write E[φ] := 12 kφk2H 1 kφk2L2 + 41 kφk4L4 = E[St φ]

(5.6)


23

for the conserved energy of the solution of the NLS. Then, it can be easily checked that Z m 1 . (5.7) + E[St φ] hK(m) iΓ(t) = dµ(φ) ˜ 2 We have that the sequence of higher energy functionals hK(m) iΓ(t) , for m ∈ N, satisfies X kΓ(t)kH1ξ ≤ (2ξ)m hK(m) iΓ(t) m∈N

=

X

m∈N

(2ξ)m hK(m) iΓ(0)

≤ kΓ(0)kH1′ , ξ

by Theorem 6.2 in [14]. As a consequence, we find that ˜ ˜ kΓ(t)k H1ξ ≤ kΓ(t)kH1ξ X ≤ (2ξ)m hK(m) iΓ(t) ˜ m∈N

=

X

m∈N

(2ξ)m hK(m) iΓ(0) ˜

≤ kΓ0 kH1′ ξ

< ∞. Moreover, ˜ L2 kB Γk H1 t∈[0,T ] ξ Z ∞ X ≤ (ξ)k dµ(φ)kh∇i(|St φ|2 St φ)kL2t∈[0,T ] L2 (R3 ) kh∇iSt φk2k−1 L∞ ≤ ≤

k=1 ∞ X

k=1 ∞ X k=1

t∈[0,T ]

(ξ)k (ξ)k

Z

Z

L2 (R3 )

2k−1 3 3 kh∇iSt φkL2 dµ(φ)k|St φ|2 kL∞ L6 (R3 ) kh∇iSt φkL∞ t L (R ) t∈[0,T ]

t∈[0,T ]

dµ(φ)kSt φk2L∞

t∈[0,T ]

(5.8)

L2 (R3 )

2k−1 L6 (R3 ) kh∇iSt φkL2t∈[0,T ] L6 (R3 ) kh∇iSt φkL∞ L2 (R3 ) t∈[0,T ]

.

(5.9)

Here, we use the bound p kh∇iSt φkL2t∈[0,T ] L6 (R3 ) ≤ C(T )kh∇iφkL2 ≤ C(T ) 1 + 2E[φ] ,

(5.10)

p p 1 + 2E[St φ] = 1 + 2E[φ]

(5.11)

with T > 0 as in (5.13), below; see for instance [40] or [8] for details. Moreover, kh∇iSt φkL∞ L2 (R3 ) ≤ sup t∈[0,T ]

t∈[0,T ]

24


We then obtain that (5.9) ≤ C

∞ X

k

(2ξ)

k=1

= Cξ −1

∞ X

k=2

Z

k+1 1 + E[φ] dµ(φ) 2

(2ξ)k hK(k) iΓ(0) ˜

≤ Cξ −1 kΓ0 kH1′ ξ

< ∞.

(5.12)

Finally, we pick T1 (ξ) > 0 sufficiently small that (5.10) above holds for 0 < T < T1 (ξ) ,

(5.13)

noting that the constant C(T ) in (5.10) depends on kφkH 1 < (ξ ′ )−1/2 kΓ0 kH1′ and ξ thus on ξ, where ξ and ξ ′ are related as in (4.5). ˜ ∈ W 1 ([0, T ]). By uniqueness of solutions to the GP Thus, we have shown that Γ ξ ˜ hierarchy in Wξ1 ([0, T ]), we conclude that Γ = Γ. In particular, we note that Γ(t) is positive semidefinite for t ∈ [0, T ]. Now that we have positive semidefinitenss of solutions to the GP hierarchy, we are able to to global well posedness of solutions to the GP hierarchy, using an induction argument as in [14] below. (k)

1 Theorem 5.4. Suppose that Γ0 = (γ0 )∞ k=1 ∈ Hξ ′ is as in Theorem 5.2. Then, for ′ 0 < ξ < 1 and ξ1 satisfying (4.5), there is a unique global solution Γ ∈ Wξ11 (R) to the cubic defocusing GP hierarchy (3.4) in R3 with initial data Γ0 . Moreover, Γ(t) is positive semidefinite and satisfies

kΓ(t)kH1ξ ≤ kΓ0 kH1′ ξ

1

(5.14)

for all t ∈ R. Proof. Let Ij be the time interval [jT, (j +1)T ], where 0 < T < min{T0 (ξ1 ), T1 (ξ1 )} (see (4.2) and (5.13)) and ξ, ξ1 satisfy (4.5). By [12] and Proposition C.3, we have that there is a unique solution Γ to the GP hierarchy in Wξ1 (I0 ). Moreover, by Theorem 5.2, Γ is positive semidefinite on I0 . It follows as in the proof of Theorem 7.2 in [14] that the higher order energy functionals hK (m) iΓ(t) , which are defined in equation (5.5), are conserved on I0 . Thus, as in inequality (7.18) in [14], we have that on I0 , kΓ(t)kH1ξ ≤ kΓ(t)kH1ξ X ≤ (2ξ)m hK (m) iΓ(t)

(5.15) (5.16)

m∈N

≤

X

m∈N

(2ξ)m hK (m) iΓ0

≤ kΓ0 kH1′ . ξ

(5.17) (5.18)

Note that positive semidefiniteness of Γ is needed to pass from (5.15) to (5.16) because the definition of kΓ(t)kH1ξ involves taking absolute values, but the definition of hK (m) iΓ(t) does not.


25

Therefore Γ(T ) ∈ H1ξ , and so by [12] and Proposition C.3, there is a unique solution Γ ∈ Wξ11 (I1 ) of the GP hierarchy with initial data Γ(T ). By another application of Theorem 5.2 and energy conservation (5.15) ∼ (5.18), Γ is positive semidefinite on I1 and Γ(2T ) ∈ H1ξ . Thus, we can repeat the argument and find that we have a unique solution Γ ∈ Wξ11 (R). Moreover, kΓ(t)kH1ξ ≤ kΓ(t)kH1ξ ≤ kΓ0 kH1′ ξ

1

for all t ∈ R.

6. Global derivation of the GP hierarchy In this section, we show that the validity of Theorem 2.1 can be extended to arbitrarily large values of T , provided that Γ0 ∈ H1ξ′ has the form (2.4), and that ξ is sufficiently small. This is obtained from combining Theorem 2.1 and Theorem 2.3 in a recursive manner. We begin by observing that, in the statement of Theorem 2.1, instead of assuming that Γ0 := lim ΓΦN (0) N →∞

holds in

Hξ1′ ,

we may assume that Γ0 := lim P≤K(N ) ΓΦN (0) N →∞

holds. Indeed, the proof of Theorem 2.1 is unaffected by this replacement. k We also note that initial condition hΦN (0), HN ΦN (0)i implies that ΓΦN (t) ∈ Hξ1′ for any t ∈ R, provided that ξ ′ < (4(C + 1))−1 . This follows from (4.36). In fact, e uniform in N and t, such that given ξ ′ , we have a bound C, ek . kΓΦN (t)(k) kH 1 < C

(6.1)

We also note that, by Theorem 2.3, the solution to the GP hierarchy Γ(t) ∈ Hξ11 for all t ∈ R, provided that ξ1 sufficiently small. Thus, under the assumptions of Theorem 2.1, at time T , we have ( ΓΦN (T ) ∈ Hξ11 and (6.2) Γ(T ) = limN →∞ P≤K(N ) ΓΦN (T ) in Hξ11 , provided that ξ1 is sufficiently small (note that we also require ξ1 < (4(C + 1))−1 ). By another application of Theorem 2.1, we have that at time 2T , ΓΦN (2T ) ∈ Hξ11 and

Γ(2T ) = lim P≤K(N ) ΓΦN (2T ) in Hξ12 , N →∞

provided that ξ2 < ξ1 is sufficiently small. (6.3) says that ∞ X

k=1

ξ2k kΓ(2T )(k) − P≤K(N ) ΓΦN (2T )(k) kH 1 → 0 as N → ∞.

(6.3)

26


However, by (6.1) and the dominated convergence theorem for sequences, we actually have the stronger statement ∞ X ξ1k kΓ(2T )(k) − P≤K(N ) ΓΦN (2T )(k) kH 1 → 0 as N → ∞, k=1

where we have ξ1 instead of ξ2 . Thus, at time 2T we actually have ( ΓΦN (2T ) ∈ Hξ11 and Γ(2T ) = limN →∞ P≤K(N ) ΓΦN (2T ) in Hξ11 .

(6.4)

Note that (6.4) is the same as (6.2), but with T replaced by 2T . Thus, we may iterate the argument again, and conclude that Theorem 2.1 holds for T arbitrarily large, provided that Γ0 ∈ H1ξ′ has the form (2.4), and that ξ is sufficiently small.

Appendix A. Strong vs weak-* convergence (k)

2 k Proposition A.1. Suppose that (γN )∞ N =1 is a sequence of operators on L (R ) (k) (k) (k) such that γN → γ∞ strongly in Hilbert Schmidt norm. Suppose also that γN and (k) (k) (k) (k) γ∞ are trace class operators such Tr|γN | ≤ 1 for all N . Then γN → γ∞ in the weak-* topology induced by the trace norm.

Proof. We follow the usual construction of a metric for the weak-* topology induced by the trace norm, as presented in [25], for example. Let Kk be the space of compact operators on L2 (Rk ) equipped with the operator norm topology. Let L1k be the space of trace class operators on L2 (R2k ). By [47], we have that L1k = Kk∗ . (k) Since Kk is separable, there exists a sequence {Ji }∞ i=1 ∈ Kk of Hilbert Schmidt operators, dense in the unit ball of Kk . Note that Hilbert Schmidt operators are dense in the space of compact operators, because, by [47], every compact operator PN on a Hilbert space is of the form limN →∞ n=1 λn hψn , · iφn , with {ψn }∞ n=1 and ∞ {φn }∞ n=1 orthonormal sets, and {λn }n=1 positive real numbers such that λn → 0. On L1k , we define the metric ηk by ∞ X (k) −i (k) (k) (k) (k) 2 TrJi γ − e γ ηk (γ , e γ ) := . i=1

By [48], the topology induced by the metric ηk is equivalent to the weak-* toplology on L1k . (k) Now, since {Ji }∞ i=1 ∈ Kk are Hilbert Schmidt, we have (k)

(k)

(k)

(k)

(k) 2 1/2 (k) | )) )| ≤ (Tr(|Ji |2 ))1/2 (Tr(|γN − γ∞ Tr|Ji (γN − γ∞

→ 0 as N → ∞.

Moreover, (k)

(k)

(k)

(k)

(k) (k) | )| ≤ kJi kL2 →L2 Tr|γN − γ∞ Tr|Ji (γN − γ∞

≤1+

(k) Tr|γ∞ |.

Thus, by the dominated convergence theorem for sequences, (k) (k) N → ∞, and so γN → γ∞ in the weak-* topology on L1k .

(A.1) (A.2) (A.3) (A.4)

(k) (k) ηk (γN , γ∞ )

→ 0 as


27

Appendix B. Conservation of admissibility for the GP hierarchy In this part of the appendix, we prove that the GP hierarchy conserves admisibility. This result has been used in many papers, but we have not found an explicit proof. For the convenience of the reader, we present it here. (k)

1 Proposition B.1. Suppose that Γ0 = (γ0 )∞ k=1 ∈ Hξ ′ is admissible and satisfies (k)

Tr γ0 = 1 for all k ∈ N. Then, for 0 < ξ ′ < 1 and ξ satisfying (4.5), the unique solution Γ ∈ Wξ1 (I) to the GP hierarchy obtained in [12] is admissible for all t ∈ I, 1 provided that A := {A(k) }∞ k=1 ∈ Wξ (I), where

(k)

A

(t, xk ; x′k )

:= −γ

(k)

(t, xk ; x′k )

+

Z

γ (k+1) (t, xk , xk+1 ; x′k , xk+1 ) dxk+1 .

(B.1)

Proof. We first note that for f ∈ S(Rn × Rn ), we have Z

((∆x1 − ∆x2 )f )(x, x) dx = 0.

(B.2)

Indeed, this follows from Z

((∆x1 − ∆x2 )f )(x, x) dx Z Z = δ(x1 − x2 )(∆x1 − ∆x2 )f (x1 , x2 ) dx1 dx2 Z Z Z Z = δ(x1 − x2 )eiu1 x1 +iu2 x2 ((u2 )2 fˆ(u1 , u2 ) − u21 fˆ(u1 , u2 )) du1 du2 dx1 dx2 Z Z Z = eix1 (u1 +u2 ) ((u2 )2 fˆ(u1 , u2 ) − u21 fˆ(u1 , u2 )) du1 du2 dx1 Z Z = δ(u1 + u2 )((u2 )2 fˆ(u1 , u2 ) − u21 fˆ(u1 , u2 )) du1 du2 Z = (u21 − u21 )fˆ(u1 , −u1 ) du1 = 0,

which implies (B.2). Next, we note that the definition of admissibility implies that γ (k) is admissible at time t if and only if

A(k) (t, xk , x′k ) = 0.

28


Since Γ satisfies the GP hierarchy, we have that i∂t A(1) (x1 ; x′1 ) = (∆x1 − ∆x′1 )γ (1) (x1 ; x′1 ) (2) ′ (2) ′ ′ ′ − κ0 γ (x1 , x1 ; x1 , x1 ) − γ (x1 , x1 ; x1 , x1 ) Z (−∆x2 + ∆x′2 )γ (2) (x1 , x2 ; x′1 , x2 ) +

(B.3) (B.4) (B.5)

+ κ0 γ (3) (x1 , x2 , x1 ; x′1 , x2 , x1 )

− κ0 γ (3) (x1 , x2 , x′1 ; x′1 , x2 , x′1 )

=

Z

+ κ0 γ (3) (x1 , x2 , x2 ; x′1 , x2 , x2 ) (3) ′ − κ0 γ (x1 , x2 , x2 ; x1 , x2 , x2 ) dx2

(∆x1 − ∆x′1 )γ (2) (x1 , x2 ; x′1 , x2 ) dx2

(B.6)

− (∆x1 − ∆x′1 )A(1) (x1 ; x′1 ) Z (3) ′ (3) ′ ′ ′ γ (x1 , x1 , x2 ; x1 , x1 , x2 ) − γ (x1 , x1 , x2 ; x1 , x1 , x2 ) dx2 − κ0

(B.7)

+ κ0 A(2) (x1 , x1 ; x′1 , x1 ) − κ0 A(2) (x1 , x′1 ; x′1 , x′1 ) Z + (−∆x1 + ∆x′1 )γ (2) (x1 , x2 ; x′1 , x2 ) + κ0 γ (3) (x1 , x2 , x1 ; x′1 , x2 , x1 ) (3) ′ ′ ′ − κ0 γ (x1 , x2 , x1 ; x1 , x2 , x1 ) dx2

= −(∆x1 − ∆x′1 )A(1) (x1 ; x′1 ),

(B.8)

(B.9)

+ κ0 A(2) (x1 , x1 ; x′1 , x1 ) − κ0 A(2) (x1 , x′1 ; x′1 , x′1 )

where (B.1) was used to pass from (B.3) to (B.6) and from (B.4) to (B.7). Moreover, (B.2) and density of S in H 1 was used to pass from (B.5) to (B.8). Symmetry of γ (k) was used to pass to (B.9). Observe that (B.9) is precisely the right hand side of the first equation in the GP hierarchy. Thus A(1) , and similarly A(k) for k > 1, satisfies the GP hierarchy. A(0) = 0, so by uniquenss of solutions to the GP hierarchy [12], A = 0.

Appendix C. Continuity of solutions to the GP hierarchy In [12], it is shown that there is a unique soution Γ to the GP hierarchy (3.4) in α + − 2 α {Γ ∈ L∞ t∈[0,T ] Hξ | B Γ, B Γ ∈ Lt∈[0,T ] Hξ }. In this part of the appendix, we show that this solution Γ is an element of C([0, T ], Hξ1 ).


29

Lemma C.1. If γ (k) ∈ L2 (Rdk × Rdk ), then lim k U (k) (t) − U (k) (0) γ (k) kL2 (Rdk ×Rdk ) = 0. t→0

−it(−∆

+∆

′

)

xk x k . Since −∆ Proof. We recall that U (k) (t) = e xk + ∆x′k is a self-adjoint (k) operator, we have from theorem VIII.7 in [47] that U (t) is a strongly continuous one-parameter unitary group, and the lemma follows.

Lemma C.2. If Γ ∈ Hξα (Rdk × Rdk ), then lim k (U (t) − U (0)) ΓkHαξ = 0.

t→0

Proof. k (U (t) − U (0)) ΓkHαξ ∞ X = ξ k k U (k) (t) − U (k) (0) S (k,α) γ (k) kL2 k=1

→ 0 as t → 0

by Lemma C.1, the fact that kU (k) (t)kL2 →L2 ≤ 1, and the dominated convergence theorem for series. Proposition C.3. The solution Γ to the GP hierarchy constructed in [12] lies in C([0, T ], Hξ1 ). Proof. As proven in [12], the solution Γ satisfies 1 Γ ∈ L∞ t∈[0,T ] Hξ ,

(C.1)

BΓ ∈ L2t∈[0,T ] Hξ1 , and Z t U (t − s)BΓ(s) ds. Γ(t) = U (t)Γ0 + iκ0

(C.2) (C.3)

0

Thus, in Hξ1 , we have that lim Γ(t + h) − Γ(t) h→0

Z = lim U (t + h)Γ0 + iκ0 h→0

t+h

U (t + h − s)BΓ(s) ds Z t − U (t)Γ0 − iκ0 U (t − s)BΓ(s) ds 0

0

= lim U (h)Γ0 h→0

+ lim (U (h) − U (0)) iκ0 h→0 | + lim iκ0 h→0

Z

t

t+h

Z

0

(C.4)

t

U (t − s)BΓ(s) ds {z }

(C.5)

[∗]

U (t + h − s)BΓ(s) ds.

(C.6)

30


By Lemma C.2, (C.4) = 0. By (C.1) and (C.3), [∗] ∈ Hξ1 , so it follows from Lemma C.2 that (C.5) = 0. Now

Z

t+h

U (t + h − s)BΓ(s) ds

t

1 ≤

Hξ

Z

t+h

kU (t + h − s)BΓ(s)kH1 ds ξ √t ≤ hkU (t + h − s)BΓ(s)kL2s∈[t,t+h] H1ξ √ = hkBΓ(s)kL2s∈[t,t+h] H1ξ → 0 as h → 0

by (C.2), so (C.6) = 0.

Appendix D. Iterated Duhamel formula and boardgame argument In this part of the appendix, we recall a technical result from [15] that is used in parts of this paper. It corresponds to Lemma B.3 in [15]. Let Ξ = (Ξ(k) )n∈N denote a sequence of functions Ξ(k) ∈ L2t∈[0,T ] H 1 (R3k × R3k ), for T > 0. Then, we define the associated sequence Duhj (Ξ) of j-th level iterated main Duhamel terms based on BN (see Section 3.2 for notations), with components given by Duhj (Ξ)(k) (t) Z t Z := ij dt1 · · · 0

(D.1) tj−1

0

main dtj BN ;k+1 e

(k+1) i(t−t1 )∆±

main BN ;k+2 e

(k+j)

i(tj−1 −tj )∆± main BN ;k+2 · · · · · · e

(k+2) i(t1 −t2 )∆±

( Ξ )(k+j) (tj ) ,

with the conventions t0 := t, and

Duh0 (Ξ)(k) (t) := ( Ξ )(k) (t)

(D.2)

for j = 0. Using the boardgame estimates of [24, 25, 40], one obtains: Lemma D.1. For Ξ = (Ξ(k) )k∈N as above, k Duhj (Ξ)(k) (t) kL2t∈I H 1 (R3k ×R3k )

(D.3)

j 2

≤ k C0k (c0 T ) kΞ(k+j) kL2t∈I H 1 (R3(k+j) ×R3(k+j) ) , where the constants c0 , C0 depend only on d, p. For this work, the dimension is given by d = 3 and the nonlinearity is given by p = 2 (cubic GP hierarchy). Lemma D.1 is used for the proof of the next result (by suitably exploiting the main error splitting BN = BN + BN ), which corresponds to Lemma B.3 in [15]. Lemma D.2. Let δ ′ > 0 be defined by 1 − δ′ . 4 Assume that N is sufficiently large that the condition β

K

0 independent of K, N .

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