THE RIGOROUS DERIVATION OF THE 2D CUBIC FOCUSING NLS ...

THE RIGOROUS DERIVATION OF THE 2D CUBIC FOCUSING NLS FROM QUANTUM MANY-BODY EVOLUTION XUWEN CHEN AND JUSTIN HOLMER Abstract. We consider a 2D time-dependent quantum system of N -bosons with harmonic external con…ning and attractive interparticle interaction in the Gross-Pitaevskii scaling. We derive stability of matter type estimates showing that the k-th power of the energy controls the H 1 Sobolev norm of the solution over k-particles. This estimate is new and more di¢ cult for attractive interactions than repulsive interactions. For the proof, we use a version of the …nite-dimensional quantum di Finetti theorem from [49]. A high particle-number averaging e¤ect is at play in the proof, which is not needed for the corresponding estimate in the repulsive case. This a priori bound allows us to prove that the corresponding BBGKY hierarchy converges to the GP limit as was done in many previous works treating the case of repulsive interactions. As a result, we obtain that the focusing nonlinear Schrödinger equation is the mean-…eld limit of the 2D time-dependent quantum many-body system with attractive interatomic interaction and asymptotically factorized initial data. An assumption on the size of the L1 -norm of the interatomic interaction potential is needed that corresponds to the sharp constant in the 2D Gagliardo-Nirenberg inequality though the inequality is not directly relevant because we are dealing with a trace instead of a power.

Contents 1. Introduction 2 1.1. Organization of the paper 8 1.2. Acknowledgements 8 2. Stability of Matter / Energy Estimates for Focusing Quantum Many-body System 8 2.1. Stability of Matter when k = 1 9 2.2. High Energy Estimates when k > 1 13 2.3. Remark on higher 23 3. Derivation of the 2D Focusing NLS 25 3.1. Proof of Theorem 1.2 25 3.2. Proof of Theorem 1.1 31 References 33

Date: 08/28/2015. 2010 Mathematics Subject Classi…cation. Primary 35Q55, 35A02, 81V70; Secondary 35A23, 35B45, 81Q05. Key words and phrases. BBGKY Hierarchy, Focusing Many-body Schrödinger Equation, Focusing Nonlinear Schrödinger Equation (NLS), Quantum de Finetti Theorem. 1

2

XUWEN CHEN AND JUSTIN HOLMER

1. Introduction Bose-Einstein condensate (BEC) is a state of matter occurring in a dilute gas of bosons (identical particles with integer spin) at very low tempertures, where all particles fall into the lowest quantum state. This form of matter was predicted in 1924 by Einstein, inspired by calculations for photons by Bose. In 1995, BEC was …rst produced experimentally by Cornell and Wieman [4] at the University of Colorado at Boulder NIST–JILA lab, in a gas of rubidium cooled to 20 nK. Shortly thereafter, Ketterle [41] at MIT demonstrated important properties of a BEC of sodium atoms. For this work, Cornell, Weiman, and Ketterle received the 2001 Nobel Prize in Physics1. Since then, this new state of matter has attracted a lot of attention in physics and mathematics as it can be used to explore fundamental questions in quantum mechanics, such as the emergence of interference, decoherence, super‡uidity and quantized vortices. Let us lay out the quantum mechanical description of the N -body problem. Let t 2 R be the time variable and rN = (r1 ; : : : ; rN ) 2 RnN be the position vector of N particles in Rn . The dynamic of N bosons are described by a symmetric N -body wave function N (rN ; t) evolving according to the linear N -body Schrödinger equation i@t

N

= HN

N

with Hamiltonian HN given by (1.1)

HN =

N X j=1

rj

1 + N

X

N

n

V (N (ri

rj )) +

N X

W (rj )

j=1

1 i<j N

where V represents the interparticle attraction/repulsion and W represents the external con…ning potential. Informally, BEC means that, up to a phase factor depending only on t, the N -body wave function nearly factorizes (1.2)

N (rN ; t)

N Y

'(rj ; t)

j=1

In the simplest cases, where is it assumed that interactions between condensate particles are of the contact two-body type and also anomalous contributions to self-energy are neglected, it is widely believed, based upon heuristic and formal calculations, that (1.2) is valid and the one-particle state ' evolves according to the nonlinear Schrödinger equation (NLS) (1.3)

i@t ' = (

+ W (r))' + 8

j'j2 '

This is one of the main motivations for studying the NLS equation, and there is now a wide body of literature on well-posedness [7, 61], the long-time asymptotics of global-in-time solutions [43], the possibility and structure of …nite-time blow-up solutions [60], and the stability and dynamics of coherent solutions called solitary waves [63, 5]. In particular, blow-up and solitary waves only exist in the case of < 0, called the focusing case. 1 http://www.nobelprize.org/nobel_prizes/physics/laureates/2001/press.html.

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM

3

Before proceeding, let us remark on the choice of scaling in the interparticle interaction term. In 2D, it is taken as N 1 VN (r) in (1.1), where VN (r) = N 2 V (N r), for > 0.2 This scaling is intended to capture the so-called Gross-Pitaevskii limit, in which the ground-state asymptotics are described by the one-particle Gross-Pitaevskii (GP) energy functional Z E(') = (jr'j2 + W j'j2 + 4 j'j4 )

In the case of repulsive interactions > 0, in the stationary case, the ground state energy asymptotics in the 2D Gross-Pitaevskii limit from the 2D N -body quantum setting, are discussed in [50, Theorem 6.5]. It is found that aN , the 2D scattering length of the microscopic interaction, should scale as aN = N 1=2 e N=2 . The scattering length associated to a potential is the radius of the hard-sphere potential that gives the same low-wave number phase shifts as the given potential. A precise de…nition in 2D isR given in [50, §9.3]. If we take VN (x) = N 2 V (N x), then by [50, Corollary 9.4] with = ( V )N 1 and R = N , we have (1.4)

aN

N

exp

4 N R (1 + (N )) V

where (N ) ! 0 as N ! 1. Thus = 12 gives the correct N -dependence for aN . Other values of could be produced by modifying N 2 V (N r) to (1 + c lnNN )N 2 V (N r) for appropriate c, and thus changing corresponds to aR lower-order correction in the scaling.3 Moreover, the analysis shows that we have = 81 V . The corresponding time-dependent problem, for > 0, was studied by Kirkpatrick-Schlein-Sta¢ lani [44] in the periodic setting and by X.Chen [15] in the trapping setting(W 6= 0). Another way to obtain a 2D limit is to start with a 3D quantum N -body system with strong con…ning in one-dimension (say the z-direction). In the stationary repulsive case, this was explored by Schnee-Yngvason [54]. If external con…ning in the z-direction is imposed to give the system an e¤ective width ! 1=2 , then one should take the 3D interaction potential p p to be (N !)3 1 V ((N !) r), where r = (x; y; z), in place of the 2D interaction potential N 2 1 V (N r), where r = (x; y), and take ! ! 1 as N ! 1. In the repulsive case ( > 0), the corresponding time-dependent problem was studied by X.Chen-Holmer [17]. We will not consider the dimensional reduction problem here. As indicated earlier, one expects that the nonlinear coe¢ cent in (1.3) is given by R 1 1 =8 V , or expressed in terms of the scattering length aN of N VN (r), the relation is = N [ln(N a2N )] 1 .4 The scattering length can be adjusted experimentally by the method of Feshbach resonance, which exploits the hyper…ne R structure ofR the atoms in the condensate. Speci…cally, we see that the sign of depends on V , and that V < 0 leads to focusing NLS 2 We

consider the > 0 case solely in this paper. For = 0 (Hartree dynamic), see [34, 28, 47, 53, 51, 37, 38, 14, 2, 3]. 3 We note, in particular, that, unlike the 3D case, nothing special happens at = 1. Exponential in N scaling would allow one to shift the value of , but still would not yield the 2D scattering length of V itself. 4 Although the de…nition of scattering length in [50, §9.3] is given in the case of repulsive potentials V 0, it can be adjusted to the case of attractive potentials V 0, in which the requirement that (x) = ln jxj for a jxj ! 1 is replaced with (x) = ln jxj for jxj ! 1. Then (1.4) is changed by the reciprocal, so we still a obtain an exponentially small quantity, rather than an unphysical exponentially large quantity in N .

4

XUWEN CHEN AND JUSTIN HOLMER

with < 0. BEC with < 0 has been produced in laboratory experiments [24, 26, 59, 42] in di¤erent contexts, and solitary waves and blow-up have been observed. Thus there is strong motivation for determining whether the mean-…eld approximation (1.3) is theoretically valid in di¤erent contexts. We are concerned here with precise conditions under which (1.2) and (1.3) hold, and the rigorous demonstration of this result. For our quantitive formulation of the N ! 1 limit, we use the BBGKY framework. Speci…cally, let N be the projection operator in L2 (R2N ) onto the one-dimensional space spanned by N . The kernel is 0 N (t; rN ; rN )

(1.5)

=

N (t; rN )

0 N (t; rN )

(k)

Let N denote the trace of N over the last (N k) particles, called the k-th marginal (k) density. Then N is a trace-class operator on L2 (R2k ) with kernel given by Z (k) 0 0 (1.6) N (t; rk ; rN k ; rk ; rN k ) drN k N (t; rk ; rk ) = rN

k

In this language, (1.2) becomes the informal statement (k) 0 N (t; rk ; rk )

k Y

'(rj ; t)'(rj0 ; t)

j=1

Our main result demonstrates that this holds, in the sense of convergence as N ! 1 in the trace norm. Our result covers the focusing case in 2D, also known as the mass-critical focusing case in the NLS literature. Previous results either dealt with the defocusing case in dimensions 1,2, or 3, or the focusing case in dimension 1 (obtained either as limit from 1D or 3D quantum many-body dynamics). De…nition 1. We denote Cgn the sharp constant of the 2D Gagliardo–Nirenberg estimate: 1

1

k kL4 6 Cgn k kL2 2 kr kL2 2 :

(1.7)

Theorem 1.1 (Main Theorem). Assume that the focusing pair interaction V is an even nonpositive Schwartz class function such that kV kL1 < C24 for some 2 (0; 1). Let N (t; xN ) gn

be the N

body Hamiltonian evolution eitHN

where

1 X 2 N V (N (xi xj )) N i<j j=1 n o (k) 1 for some nonzero ! 2 R=f0g and for some 2 0; 6 ; and let be the family of N marginal densities associated with N . Suppose that the initial datum N (0) veri…es the following conditions: (a) the initial datum is normalized, that is (1.8)

HN =

k X

N (0),

4xj + ! 2 jxj j2 +

k

N (0)kL2

= 1;

(b) the initial datum is asymptotically factorized, in the sense that, (1.9)

lim Tr

N !1

(1) 0 N (0; x1 ; x1 )

0 0 (x1 ) 0 (x1 )

= 0;

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM

for some one particle wave function

0

4x + ! 2 jxj2

s.t.

1 2

0

L2 (R)

5

< 1.

(c) initially, each particle’s energy, though may not be positive, is bounded above (1.10)

sup N

1 h N

N (0); HN

N (0)i

< 1:

Then 8t > 0, 8k > 1, we have the convergence in the trace norm or the propagation of chaos that k Y (k) 0 lim Tr N (t; xk ; xk ) (t; xj ) (t; x0j ) = 0; N !1

j=1

where (t; x) is the solution to the 2D focusing cubic NLS (1.11)

i@t

4x + ! 2 jxj2

=

(0; x) = and the coupling constant b0 =

R

R2

b0 j j2

0 (x)

in R2+1

V (x)dx :

Theorem 1.1 is equivalent to the following theorem. Theorem 1.2 (Main Theorem). Assume that the focusing pair interaction V is an even nonpositive Schwartz class function such that kV kL1 < C24 for some 2 (0; 1). Let N (t; xN ) gn

be the N body Hamiltonian evolution eitHN N (0) n with o HN given by (1.8) for some nonzero (k) ! 2 R=f0g and for some 2 (0; 1=6) ; and let be the family of marginal densities N associated with N . Suppose that the initial datum N (0) is normalized and asymptotically factorized in the sense of (a) and (b) in Theorem 1.1 and veri…es the following energy condition: (c’) there is a C > 0 independent of N or k such that k N (0); HN

(1.12)

N (0)

< C k N k ; 8k > 1;

though the quantity N (0); HNk N (0) may not be positive. Then 8t > 0, 8k > 1, we have the convergence in the trace norm or the propagation of chaos that k Y (k) lim Tr N (t; xk ; x0k ) (t; xj ) (t; x0j ) = 0; N !1

j=1

where (t; x) is the solution to the 2D focusing cubic NLS (1.11).

It follows from the fact that N evolves according to i@t (k) (1.5), (1.6) of the marginal densities N that (k) i@t N

=

k h X j=1

+

N N

2

2

4xj +! jxj j ; k kX j=1

(k) N

h Trk+1 VN (xj

i

1 + N

= HN

N

X h

VN (xi

16i<j6k

xk+1 );

(k+1) N

i

;

N

and the de…nition

xj );

(k) N

i

6

XUWEN CHEN AND JUSTIN HOLMER

This coupled sequence of equations is called the BBGKY hierarchy. The use of the BBGKY hierarchy in the quantum setting was suggested by Spohn [58] and has been employed in rigorous work by Adami, Golse, & Teta [1] and Elgart, Erdös, Schlein, & Yau [27, 29, 30, 31, 32]. The latter series of works rigorously derives the 3D cubic defocusing NLS from a 3D timedependent quantum many-body system with repulsive pair interactions and no trapping (! = 0). Their program consists of two main steps.5 First, they derive H 1 -energy type a priori estimates for the N -body Hamiltonian from which a compactness property, for each k, (k) (k) of the sequence f N g+1 solving the 3D Gross-Pitaevskii N =1 follows, yielding limit points hierarchy (1.13)

i@t

(k)

+

k X j=1

4rk ;

(k)

= b0

k X

Trrk+1 [ (rj

rk+1 );

(k+1)

]; for all k

1:

j=1

Second, they show that (1.13) has a unique solution which satis…es the H 1 -energy type a priori estimates obtained in the …rst step. Since a compact sequence with a unique limit point is, in fact, a convergent sequence, it follows that (in an appropriate weak sense) solutions to (k) the BBGKY hierarchy N converge to solutions to the GP hierarchy (k) . Moreover, it is easily veri…ed that a tensor product of solutions of NLS (1.3) solves the GP hierarchy, and hence this is the unique solution. In the defocusing literature, a major di¢ culty is that the uniqueness theory for the hierarchy (1.13) is surprisingly delicate due to the fact that it is a system of in…nitely many coupled equations over an unbounded number of variables and there has been much work on it. Klainerman & Machedon [45] gave a Strichartz type uniqueness theorem using a collapsing estimate originating from the multilinear Strichartz estimates and a board game argument inspired by the Feynman graph argument in [30]. The method by Klainerman & Machedon [45] was taken up by Kirkpatrick, Schlein, & Sta¢ lani [44], who derived the 2D cubic defocusing NLS from the 2D time-dependent quantum many-body system; by T. Chen & Pavlovi´c [10], who considered the 1D and 2D 3-body repelling interaction problem; by X. Chen [15, 16], who investigated the defocusing problem with trapping in 2D and 3D; by X. Chen & Holmer [17], who proved the e¤ectiveness of the defocusing 3D to 2D reduction problem, and by T.Chen & Pavlovi´c [11] and X.Chen & Holmer [16, 18, 21], who proved the Strichartz type bound conjectured by Klainerman & Machedon. Such a method has also inspired the study of the general existence theory of hierarchy (1.13), see [12, 9, 35, 57]. Recently, using a version of the quantum de Finetti theorem from [48]6, T.Chen, Hainzl, Pavlovi´c, & Seiringer [8] provided an alternative proof to the uniqueness theorem in [30] and showed that it is an unconditional uniqueness result in the sense of NLS theory. With this method, Sohinger derived the 3D defocusing cubic NLS in the periodic case [56]. See also [22, 40]. However, for the focusing case, things are di¤erent. How to obtain the needed H 1 energy type a priori estimates is the central question. To be precise, without such a priori estimates, one cannot check the requirements of the various uniqueness theorems 5 See 6 See

[6, 36, 52, 46] for di¤erent approaches. also [3, 2].

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM

7

[30, 45, 44, 15, 16, 56, 8, 22, 40] at all.7 It is already highly nontrivial and may not be possible to even prove the weaker type II stability of matter estimate (1.14)

h

N ; HN

Ni

>

CN for all

N

2 L2s (RnN )

when HN is given by (1.1) with V < 0 while it is obviously true when V > 0. The …rst complete work on the focusing problem was done by X.Chen and Holmer [19, 20] for the time-dependent 1D problem. The key is to explore the structure of the 2-body operator (1.15)

H+ij =

ri

+ W (ri )

rj

+ W (rj ) +

N

1 N

N n V (N (ri

rj ))

generated in the decomposition of HN . Such a technique was later used independently by Lewin, Nam, & Rougerie in [49], where they investigated the ground state problem in the focusing setting. The main portion of this paper is devoted to this problem in 2D. In particular, we prove Theorem 1.3. Consider the focusing many-body Hamiltonian HN =

k X j=1

4xj + ! 2 jxj j2 +

1 X 2 N V (N (xi N i<j

xj ));

in 2D. Assume ! > 0; < 16 ; and kV kL1 < C24 for some 2 (0; 1), then let c0 = min( 1p2 ; 12 ), gn we have 8k = 0; 1; :::, there is an N0 (k) > 0 such that D E 2 k 1 (1.16) HN + 1 > ck0 S (k) N L2 , N; N N for all N > N0 (k) and for all

N

2 L2s (R2N ). Here S

(k)

=

k Y

Sj

j=1

and Sj2 is the Hermite operator

4xj + ! 2 jxj j2 .

The di¢ culty of proving Theorem 1.3 is self-evident. In the 2D setting in which the kinetic energy, e¤ectively the H 1 norm, cannot control VN , e¤ectively a Dirac -mass8, not only Theorem 1.3 provides stability of matter, it also proves regularity. The key to the proof, as we will explain later, is to make use of a large N averaging e¤ect which is revealed via a clever application of a …nite dimensional quantum de Finette theorem in [49]. 7 In

fact, one of the authors of [27, 29, 30, 31, 32] remarked the a priori bound was the most delicate part in the defocusing case as well when the results were revisited in [6]. 8 Di¤erent from the limit NLS in which the L4 norm is easily controlled in H 1 , in the N -body setting, one has to control a trace with H 1 .

8

XUWEN CHEN AND JUSTIN HOLMER

1.1. Organization of the paper. As mentioned before, the main portion of this paper is devoted to proving Theorem 1.3. We do so in §2. We will …rst prove the k = 1 case: (1.17)

N;

N

1

HN + 1

N

> (1

) kS1

2 N kL2

which is Theorem 2.1 in §2.1. We remark that not only the proof of Theorem 2.1 departs totally from its analogues in the previous work, its underlying machinery is also signi…cantly di¤erent. Theorem 2.1 works because of a large N averaging e¤ect not observed before. To explain this fact, consider the general Hamiltonian (1.1) and let H+ij be de…ned as in (1.15), then by symmetry, N;

N

1

HN + 1

N xN

=h

N ; (2

+ H+12 )

N ixN

;

that is, (1.17) is equivalent to (1.18)

h

N ; (2

+ H+12 )

Ni

>C (

2

1

r1

+ W (r2 )) 2

N

L2

:

In all the defocusing work [1, 10, 15, 16, 17, 27, 29, 30, 32, 31, 44, 56], estimates like (1.18) are automatically true because V > 0. In the previous focusing work [19, 20], it takes substantial work to prove the similar estimates but they actually do not rely on the fact that N is a N -body bosonic wave function in the sense that they hold even if one replaces N by some f (x1 ; x2 ) in (1.18). However, Theorem 2.1 requires that N is a N -body bosonic wave function. In fact, when V < 0, in 2D, the quantity hf; (2 + H+12 ) f i is not even bounded below, because of the -function emerging from VN . Hence, we are observing a large N averaging e¤ect, or more precisely, "though VN gets more singular as N ! 1, larger N beats it.", as we will see in the proof.9 Moreover, this is the only energy estimate in the "nD to nD" 10 literature which requires the trapping ! 6= 0 at the moment. Based on the k = 1 case, we then prove the k > 1 case in §2.2 with a delicate computation using the 2-body operator. In §2.3, by giving a counter example, we show that with the current technique, one can not reach a higher . With Theorem 1.3 established, we prove Theorems 1.1 and 1.2 in §3. Though the technique in §3 is standard by now, this is the …rst time the derivation of the trapping case is written down without using the lens transform in [15, 16, 19] and it simpli…es the argument. 1.2. Acknowledgements. X.C. was supported in part by NSF grant DMS-1464869 and J.H. was supported in part by NSF grant DMS-1500106. 2. Stability of Matter / Energy Estimates for Focusing Quantum Many-body System In this section, we prove stability of matter / energy estimate (1.16).11 9 See

Remark 3. "nD to nD" means "deriving nD NLS from nD N -body dynamic". 11 For the defocusing case (V > 0) in which there is no need to worry about particles focusing to a point, it certainly makes sense to only call estimates like (1.16) "energy estimates". However, that is obviously not the case when V < 0. Moreover, (1.16) does have a similar form with the stability of matter estimates like (1.14). Hence we use the word "stability of matter / energy estimates" here. 10 Here,

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM

9

2.1. Stability of Matter when k = 1. Theorem 2.1 (Stability of Matter). Assume ! > 0; < 16 ; and kV kL1 < 2 (0; 1), then 8C0 > 0, there exists an N0 > 0 such that N;

N

1

HN + C0

N

> (1

2 N kL2

) kS1

2 4 Cgn

for some

,

for all N > N0 and for all N 2 L2s (R2N ). Here, N0 grows to in…nity as C0 approaches 0. In particular, the N -body system is stable provided N is larger than a threshold. Remark 1. In the previous focusing work [19, 20], there is a positive lower bound for the C0 while there is no such requirement in Theorem 2.1 as long as C0 > 0. To prove Theorem 2.1, we adopt the notation that: for any function f , write fN ij = N 2 f (N (xi

xj )):

The key of the proof of Theorem 2.1 is the following theorem. Theorem 2.2. De…ne Hij = Si2 + Sj2 +

(2.1)

Assume ! > 0; < 16 ; and kV kL1 < N0 > 0 such that h

N ; (2C0

for all N > N0 and for all

N

2 4 Cgn

+ H12 )

N

1 N

for some Ni

VN ij :

2 (0; 1), then 8C0 > 0, there exists an

> 2 (1

2 N kL2

) kS1

,

2 L2s (R2N ). Here, N0 grows to in…nity as C0 approaches 0.

Proof of Theorem 2.1 assuming Theorem 2.2. We decompose the Hamiltonian HN into X 1 (2.2) N 1 HN + C0 = (2C0 + Hij ) : 2N (N 1) i;j=1;:::;N i6=j

Hence N;

N

1

HN + C0

N

=

1 2N (N

X

h

N ; (2C0

+ Hij )

Ni

X

h

N ; (2C0

+ H12 )

Ni

1) i;j=1;:::;N i6=j

=

1 2N (N

1) i;j=1;:::;N i6=j

> (1

) kS1

2 N kL2

.

We then turn our attention onto the proof of Theorem 2.2. We will prove the following proposition.

10

XUWEN CHEN AND JUSTIN HOLMER

Proposition 2.1. Assume ! > 0; operator

< 16 ; and kV kL1
0, there exists an N0 > 0 such that

2C0 + Hij; > 0; 8N > N0 .

Here, N0 grows to in…nity as C0 approaches 0. Proof. See §2.1.1. In fact, assuming Proposition 2.1, then h

N ; (2C0

Ni

+ H12 )

= (1

)

N;

> 2 (1

S12 + S22 2 N kL2

) kS1

N

:

+h

N ; (2C0

+ H12; )

Ni

Hence we are left with the proof of Proposition 2.1. 2.1.1. Proof of Proposition 2.1. De…ne the Littlewood-Paley projectors (eigenspace projectors) by j P6M =

(0;M ]

j P>M =

(M;1)

(k) P6M

=

k Y

(Sj ) ; (Sj ) ;

j P6M ,

(k) P>M

j=1

We will need the following lemmas.

=

k Y

j P>M

j=1

12 Lemma q 2.1. Let Hij; be de…ned as in Proposition 2.1, then, for all " 2 (0; 1), as long as 3kV k1 N M> , we have 2 "

H12; > P6M H12; P6M (2)

(2)

(2)

(2)

2"2 P6M jVN 12 j P6M :

Proof. We write (2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

Sj2 = P6M + P>M Sj2 P6M + P>M = P6M Sj2 P6M + P>M Sj2 P>M because (2)

(2)

(2)

(2)

P>M Sj2 P6M = P6M Sj2 P>M = 0: We then write VN 12 =

(2)

(2)

(2)

(2)

P6M + P>M VN 12 P6M + P>M (2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

= P6M VN 12 P6M + P>M VN 12 P>M + P>M VN 12 P6M + P6M VN 12 P>M We estimate the high-high terms by D (2) (2) N ; P>M VN 12 P>M 12 This

N

lemma is essentially [49, Lemma 3.6].

E

>

2

(2)

N 2 kV k1 P>M

N

L2

:

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM

11

and the high-low and the low-high terms by Cauchy-Schwarz, D E E D 1 D (2) (2) (2) (2) (2) (2) 2 > P>M N ; jVN 12 j P>M N " P6M N ; jVN 12 j P6M N ; P>M VN 12 P6M N 2 " D E 2 N 2 kV k1 (2) (2) (2) 2 " P ; jV j P > P N 12 6M N 6M N : >M N "2 L2 Hence

N

E

> P6M H12; P6M + P>M C0 P>M + P>M S12 P>M + P>M S22 P>M 2 (2) (2) (2) (2) P>M kV k1 (1 + 2 )N 2 P>M 2"2 P6M jVN 12 j P6M : " q 3kV k1 N Whenever M > , we have 2 " (2)

H12;

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

P>M S12 P>M + P>M S22 P>M

> P>M 2 M 2 P>M (2)

(2)

(2)

(2)

P>M kV k1 (1 +

(2)

P>M kV k1 (1 +

(2)

2 (2) )N 2 P>M "2

2 (2) )N 2 P>M > 0: 2 "

Hence H12; > P6M H12; P6M (2)

(2)

(2)

(2)

2"2 P6M jVN 12 j P6M

as claimed. Lemma 2.2 (Finite dimensional quantum de Finetti [23, Theorem II.8] or [49, Lemma n oN (k) 3.4]). 13Assume is the marginal density generated by a N -body wave function N k=1

2 2N ). Then there is a probability measure d N 2 Ls (R 2 2 P6M (Ls (R )) such that Z (2) (2) (2) 2 Tr P6M N P6M S(P6M (L2s (R2 )))

N

2

supported on the unit sphere of

d

N(

) 6

8DM N

where DM is the dimension of P6M (L2s (R2 )). Remark 2. Lemma 2.2 is the only place in which this paper needs ! > 0. It is a major open problem to prove Lemma 2.2 without assuming a …nite dimensional Hilbert space. Lemma 2.3. If kV kL1 < C24 , then there exists " which depends solely on kV kL1 such that, gn for all 2 L2 (R2 ) with k kL2 = 1, we have E" ( ) =

(x1 ) (x2 ) > 0

" (x1 ) (x2 ); H12;

where (2.3) 13 To

" H12; = S12 + S22 +

N

1 N

VN 12

2"2 jVN 12 j

be precise, this version we are using is [49, Lemma 3.4]. If one uses [23, Theorem II.8] to prove it, one will have a 16 instead of a 8. The optimal coe¢ cient is important in the literature of de Finetti theorems, but it does not matter for our application here.

12

XUWEN CHEN AND JUSTIN HOLMER

Proof. We …rst compute directly that Z Z N 1 2 E" ( ) = 2 jS j dx + VN 12 j (x1 ) (x2 )j2 dx1 dx2 N Z 2"2 jVN 12 j j (x1 ) (x2 )j2 dx1 dx2 :

Apply Cauchy-Schwarz,

>2

Z

jS j2 dx

Use Young’s convolution inequality, Z > 2 jS j2 dx Z = 2 jS j2 dx With estimate (1.7), we get to E" ( ) > 2 Hence, when kV kL1
0:

With Lemmas 2.1 to 2.3, we now prove Proposition 2.1. Proof of Proposition 2.1. The trick is to notice the equaltiy E D (2) (2) (2) " " P6M P6M N ; H12; P6M N = Tr H12;

(2) (2) N P6M

" where H12; is de…ned in (2.3). It helps because

h

N ; (2C0

> 2C0 +

provided that M > Rewrite

q

3kV k1 N 2 "

(2)

+

+ H12; )

(2) P6M

Ni

" N ; H12;

2 " = 2C0 + Tr H12; P6M

" Tr H12; P6M Z = Tr

"

D

(2) P6M

(2) (2) N P6M

N

E

, by Lemma 2.1.

(2) (2) N P6M

S(P6M (L2s (R2 )))

(2) " Tr H12; P6M

" H12;

(2) (2) N P6M

2

2

Z

d

N(

)

S(P6M (L2s (R2 )))

" H12;

2

2

d

N(

#

)

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM

13

We can use the inequality Tr AB 6 kAkop Tr jBj to get to h

N ; (2C0

> 2C0 +

Z

Ni

+ H12; )

E" ( )d

S(P6M (L2s (R2 )))

" H12;

op

Tr

(2) P6M

(2) (2) N P6M

N(

)

Z

2

2

d

N(

S(P6M (L2s (R2 )))

)

Now …x " as in Lemma 2.3, apply Lemma 2.3 on the second term and Lemma 2.2 on the third term, it becomes h

N ; (2C0

+ H12; )

Ni

> 2C0

" H12;

8DM : N

op

On the one hand, with frequency smaller than M , the Hermite operator in 2D has at most M 4 eigenfunctions, that is CN 4 2 DM 6 M 2 6 : "4 On the other hand, " H12;

op

6 2 M 2 + (1 + 2"2 ) kV kL1 N 2 6

CN 2 : "2

Thus we conclude that h

N ; (2C0

+ H12; )

Ni

> 2C0

CN 6 >0 N

provided that N is large enough and < 16 . Thence we have completed the proof of Proposition 2.1, concluded Theorem 2.2, and obtained Theorem 2.1. Remark 3. The above proof is exactly what we meant by saying "though VN gets more singular as N ! 1, but larger N beats it." in the introduction. 2.2. High Energy Estimates when k > 1. Assuming (1.16) holds for k, we now prove it for k + 2. Using the induction hypothesis, we arrive at (2.4)

1 ck+2 0

h

N ; (N

We decompose N

1

HN + 1)k+2

Ni

>

1 (k) hS (N c20

1

HN + 1)

N; S

(k)

(N

1

HN + 1)

1

HN + 1 like in (2.2), but this time we separate the sum as X X 1 1 N 1 HN + 1 = (2 + Hij ) + (2 + Hij ) : N (N 1) 1 i<j N N (N 1) 1 i<j N i k

i>k

Then (2.4) unfold into three terms if we combine the two crossing terms, namely 1 ck+2 0

N ; (N

1

HN + 1)k+2

N

> M + EC + EP

N i:

14

XUWEN CHEN AND JUSTIN HOLMER

where the main term M is 1 M= 2 2 c0 N (N 1)2 the cross error term E is 1 EC = 2 2 c0 N (N 1)2

X

S (k) (2 + Hi1 j1 )

(k)

(2 + Hi2 j2 )

N

;

1 i1 <j1 N 1 i2 <j2 N such that i1 >k; i2 >k

X

2 Re S (k) (2 + Hi1 j1 )

N; S

(k)

(2 + Hi2 j2 )

N

;

1 i1 <j1 N 1 i2 <j2 N such that i1 k; i2 >k

and the nonnegative error term EP is X 1 EP = 2 2 c0 N (N 1)2 1 i <j N 1 1 1 i2 <j2 such that i1

1 = 2 2 c0 N (N

N; S

1)2

*

X

S (k) (2 + Hi1 j1 )

N;

S (k) (2 + Hi2 j2 )

N

N k; i2 k

S (k) (2 + Hi1 j1 )

X

N;

1 i<j N i6k

S (k) (2 + Hi2 j2 )

1 i<j N i6k

N

+

> 0:

Here, we distinguish the terms by the cardinality of the sums. Implicitly, we always have P N >> k, hence the main contribution comes from the sum k 0, we drop it and (2.4) becomes (2.5)

1 ck+2 0

N ; (N

1

HN + 1)k+2

N

> M + EC :

The strategy is to …rst extract the desired kinetic energy part from the main term M in §2.2.1 then prove that the cross error term EC can be absorbed into M for large N in §2.2.2. During the course of the proof, we will need the following lemma. Lemma 2.4 ([19, Lemma A.2]). 1 i; j 2, then A1 B1 A2 B2 .

If A1

A2

0, B1

B2

0 and Ai Bj = Bj Ai for all

2.2.1. Handling the Main Term. Commute (1 + Hi1 j1 ) and (1 + Hi2 j2 ) with S (k) in M , X 1 (2.6) M= 2 2 hS (k) N ; (2 + Hi1 j1 ) (2 + Hi2 j2 ) S (k) N i 2 c0 N (N 1) 1 i <j N 1

1

1 i2 <j2 N such that i1 >k; i2 >k

We decompose the sum into three pieces M = M1 + M2 + M3 where M1 consists of the terms with fi1 ; j1 g \ fi2 ; j2 g = ?; M2 consists of the terms with jfi1 ; j1 g \ fi2 ; j2 gj = 1;

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM

15

and M3 consists of the terms with jfi1 ; j1 g \ fi2 ; j2 gj = 2: By symmetry of

N,

we have

1 h 2 + H(k+1)(k+2) S (k) N ; 2 + H(k+3)(k+4) S (k) N i 4c20 1 N 1 h 2 + H(k+1)(k+2) S (k) N ; 2 + H(k+2)(k+3) S (k) = 2c20 1 = N 2 h 2 + H(k+1)(k+2) S (k) N ; 2 + H(k+1)(k+2) S (k) 2c20

M1 = M2 M3

Ni Ni

We drop M3 since it is nonnegative. Thus (2.6) becomes M > M1 + M2 . By the fact that 2 + H(k+1)(k+2) ; 2 + H(k+3)(k+4) = 0; we deduce M1 >

4(1 )2 (k) hS 4c20

2 2 (k) N ; Sk+1 Sk+2 S

Ni

using Theorem 2.2 and Lemma 2.4. Recall c0 = min( 1p2 ; 12 ), hence M1 > 2hS (k+2)

(2.7)

N; S

(k+2)

Ni

= 2 S (k+2)

2 N

L2

:

We now deal with M2 . We expand M2 = M21 + M22 + M23 where M21 M22 M23

N 1 2 2 2 2 h 2 + Sk+1 + Sk+2 S (k) N ; 2 + Sk+2 + Sk+3 S (k) N i; = 2 2c0 N 1 2 2 = Reh 2 + Sk+1 + Sk+2 S (k) N ; VN (k+2)(k+3) S (k) N i; c20 N 1 = hVN (k+1)(k+2) S (k) N ; VN (k+2)(k+3) S (k) N i: 2c20

4 We keep only the Sk+2 terms inside M21 , which carries as many derivatives as in (2.7) and hence is the second main contribution. That is (2.8) N 1 2 2 4 hSk+2 Sk+2 S (k) N ; S (k) N i > 2N 1 hSk+1 S (k) N ; S (k) N i = 2N 1 kS1 S (k+1) N k2L2 . M21 > 2 2c0

16

XUWEN CHEN AND JUSTIN HOLMER

For M22 , we …rst rearrange the derivatives 2N 1 (k) (k) hS N ; VN (k+2)(k+3) S Ni c20 N 1 + 2 hS (k+1) N ; VN (k+2)(k+3) S (k+1) N i c0 N 1 + 2 hSk+2 S (k) N ; VN (k+2)(k+3) Sk+2 S (k) N i c0 N 1 + 2 RehSk+2 S (k) N ; (rV )N (k+2)(k+3) S (k) c0

M22 =

Ni

Notice that, in the above, we have used the fact that r is the only thing inside Sj that needs the Leibniz’s rule.14 Do Hölder, jM22 j . N

1

VN (k+2)(k+3)

+N

1

+N

1

VN (k+2)(k+3)

S (k)

2 N

L2 L1 x

k+3

k+3

S (k+1)

L1+ xk+3

VN (k+2)(k+3) 1

+N

L1x

Sk+2 S (k)

L1+ xk+3

(rV )N (k+2)(k+3)

2

L1+ xk+3

L2 L1 xk+3

N

2 L2 L1 xk+3

N

Sk+2 S (k)

L2 L1 xk+3

N

S (k)

N

L2 L1 xk+3

,

Apply Sobolev, (2.9)

jM22 j . N

1

S (k+2)

+N

1+

6 CN

1+

2 N

+N

L2

1+

2

S (k+2)

N

L2

+N

1+

S (k+2)

2 N

L2

2

S (k+2)

N

S (k+2)

L2 2

N

L2

;

which is easily absorbed into the positive contributions. Alert reader should notice the loss 1 due to the failure of the 2D endpoint Sobolev: 21 1 = 12 . Do the same thing for M23 , (2.10) jM23 j . N

1

6 CN

VN (k+1)(k+2) 1+

S (k+2)

L1+ xk+1 2

N

L2

VN (k+2)(k+3)

L1+ xk+3

S (k)

N

1 L2 L1 xk+1 Lxk+3

:

Collecting (2.7)-(2.10), we arrive at the following estimate for M : (2.11)

M> 2

CN

1+

kS (k+2) k2L2 + N

1

kS1 S (k+1) k2L2 :

2.2.2. Handling the Cross Error Term. Next we turn our attention to estimating EC . We will prove that (2.12)

EC >

C max(N 2

3 + 2

;N

1+

) kS (k+2)

2 N kL2

+N

That is, EC is an absorbable error if added into (2.11). 14 This

is a fact proved and used by many authors. See, for example, [62].

1

kS1 S (k+1)

2 N kL2

:

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM

17

We assume k > 1, since EC = 0 when k = 0. We decompose the sum into three parts (2.13)

EC = E1 + E2 + E3

where E1 contains the terms with j1 6 k, E2 contains the terms with j1 > k and j1 2 fi2 ; j2 g, and E3 contains those terms with j1 > k, j1 6= i2 and j1 6= j2 . Since Hij = Hji , by symmetry of N , we have E1 = k 2 N

2 2

E2 = kN 1

E3 = N

hS (k) (2 + H12 )

hS

hS

(k)

(k)

N; S

2 + H1(k+1)

(k)

N; S

2 + H1(k+1)

N; S

2 + H(k+1)(k+2) (k)

(k)

2 + H(k+1)(k+2)

Ni

Ni

Ni

2 + H(k+2)(k+3)

We …rst address E1 . We commute (2 + H12 ) with S (k) and obtain E1 = E11 + E12 + E13 ; where E11 = N

2

h(2 + H12 ) S (k)

E12 = N

2

hS1 [S2 ; H12 ]

E13 = N

2

S (k) S1 S2

S (k) h[S1 ; H12 ] S1

2 + H(k+1)(k+2) S (k)

N;

N;

2 + H(k+1)(k+2) S (k)

2 + H(k+1)(k+2) S (k)

N;

Ni Ni

Ni

By Theorem 2.2 and Lemma 2.4, E11 > 0 and we drop it. For E12 , since [S2 ; H12 ] = N (rV )N 12 , expanding 2 + H(k+1)(k+2) gives E12 =

2N N N

S (k) ; S1 S (k) N i S1 S2 N S (k) 2 2 2 h(rV )N 12 ; (Sk+1 + Sk+2 )S1 S (k) N i S1 S2 N S (k) 2 (k) h(rV )N 12 N ; VN (k+1)(k+2) S1 S Ni S1 S2 2

h(rV )N 12

Use Holder, jE12 j . N +N +N

3 2

S (k) S1 S2

k(rV )N 12 kL2+ x 1

3 2

k(rV )N 12 kL2+ x 1

3 2

N

S (k+1) S1 S2

S1 S (k)

L2 L1 x1 N

N

1 2

L1+ xk+1

N

L2

S1 S (k+1)

L2 L1 x1

k(rV )N 12 kL2+ VN (k+1)(k+2) x 1

1 2

N

S (k) S1 S2

N

N

L2

N 1 L2 L1 x1 Lxk+1

1 2

S1 S (k)

N

L2 L1 xk+1

18

XUWEN CHEN AND JUSTIN HOLMER

Use Sobolev and notice that k(rV )N 12 kL2+ x 3 + 2

jE12 j . N 2

(2.14)

S (k

1) N

+N 2

3 + 2

S (k)

2

3 + 2

(k)

+N

S

3 + 2

. N2

+

N

1

S (k)

L2

N

L2 2

N

L2

1 2

S1 S (k)

N

L2

1 2

S1 S (k)

N

L2

N

L2

N

in 2D, we have

N N

1 2

+N

S1 S 1

(k+1) N

S1 S (k+1)

L2 2 N L2

:

Now, for E13 , notice that [S1 ; H12 ] = N (rV )N 12 , writting out 2 + H(k+1)(k+2) gives, S (k) S1 S (k) 2 h(rV )N 12 S1 S (k) 2 h(rV )N 12 S1

2

E13 = 2N +N +N

h(rV )N 12

N; S

(k)

Ni

2 N ; (Sk+1

2 + Sk+2 )S (k)

N ; VN (k+1)(k+2) S

(k)

Ni

N i:

Thus jE13 j . N +N +N

2

S (k) S1

k(rV )N 12 kL2+ x 1

2

N

L2 L1 x1

S (k+1) S1

k(rV )N 12 kL2+ x 1

2

S (k)

N

N

L2

S (k+1) L2 L1 x1

k(rV )N 12 kL2+ VN (k+1)(k+2) x

S (k) S1

L1+ xk+1

1

N

N

L2

S (k) L2 L1 x1

L1 xk+1

N

L2 L1 xk+1

Hence, with the Sobolev estimates, (2.15)

jE13 j

. N2

+N . N2

2

2+ 2+ 2+

S (k) S

N

S (k)

L2

(k+1)

S (k+1)

N L2 2 N

L2

N

S

L2 (k+1)

+ N2 N

2+

S (k+1)

N

L2

S (k+1)

L2

:

Hence, combining with (2.14), we have acquired (2.16)

E1 >

CN 2

3 + 2

S (k+1)

2 N

L2

+N

1

S1 S (k+1)

2 N

L2

since E11 > 0. Next, we deal with E2 . We remind the readers that E2 = kN

2

hS (k) 2 + H1(k+1)

N; S

(k)

Commuting 2 + H1(k+1) to the front, we write E2 = E21 + E22

2 + H(k+1)(k+2)

N i:

N

L2

:

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM

19

where E21 = N

2

h 2 + H1(k+1) S (k)

E22 = N

2

h[S1 ; H1(k+1) ]

S (k) S1

N; S N; S

(k)

(k)

N i;

2 + H(k+1)(k+2)

N i:

2 + H(k+1)(k+2)

For E21 , expanding 2 + Hij yields E21 = E211 + E212 + E213 + E214 where E211 = N

2

E212 = N

2

E213 = N

2

E214 = N

2

2 h 2 + S12 + Sk+1 S (k)

N; S

2 h 2 + S12 + Sk+1 S (k)

hVN 1(k+1) S (k)

N; S

hVN 1(k+1) S (k)

(k)

2 2 2 + Sk+1 + Sk+2

N ; VN (k+1)(k+2) S

(k)

(k)

2 2 2 + Sk+1 + Sk+2

N ; VN (k+1)(k+2) S

(k)

N i;

N i;

N i;

N i:

Note that E211 > 0, so we can discard it. Expand E212 , E212 = 2N

2

(k) N ; VN (k+1)(k+2) S Ni 2 (k) (k) hS1 S N ; VN (k+1)(k+2) S1 S

+N

hS (k) 2

+N

2

+N

hS

(k+1)

hS (k+1)

N ; (rV )N (k+1)(k+2)

N ; VN (k+1)(k+2) S

S

Ni (k)

(k+1)

Ni

Ni

Apply Hölder, jE212 j

. N

2

+N

VN (k+1)(k+2) 1

+N

VN (k+1)(k+2) 2

+N

L1x

2

N

L2 L1 x

k+1

L1+ xk+1

(rV )N (k+1)(k+2) VN (k+1)(k+2)

2

S (k)

k+1

N

1

L1+ xk+2

S1 S

S (k+1)

N

L2 L1 xk+1

N

L2 L1 xk+2

S (k)

N

L2 L1 xk+2

2

S (k+1)

L1+ xk+2

2

(k)

N

L2 L1 xk+2

With Sobolev, we see (2.17)

jE212 j . N

2

S (k+2) 2+

+N +N . N where we used max(N

2+

;N

2+

1+

1+

2 N

L2

S (k+2)

+N

N

L2

N

S (k+2)

L2 2

N

L2

1+

1

N

S (k+1)

S1 S (k+1) N

2 N

L2

L2

2

S (k+2)

)=N

1+

+N

1

S1 S (k+1)

2 N

for our problem in which

L2

< 1.

20

XUWEN CHEN AND JUSTIN HOLMER

For E213 , E213 = N

2

hVN 1(k+1) S (k) 2

= 2N +N

N; S

2 2 + Sk+2 2 + Sk+1

(k)

Ni

hVN 1(k+1) S (k) 2

h(rV

(k) 2 hVN 1(k+1) S (k) Sk+2 N ; S (k) Sk+2 N i N; S Ni + N )N 1(k+1) S (k) N ; S (k+1) N i + N 2 hVN 1(k+1) S (k+1) N ; S (k+1) N i

Apply Hölder, jE213 j . N

2

+N

VN 1(k+1) 2

1

+N +N

VN 1(k+1)

(rV )N 1(k+1)

L2 L1 xk+1

N

2

Sk+2 S (k)

L1+ xk+1

(rV )N 1(k+1)

2

2

S (k)

L1+ xk+1

L1+ x1

1

N

N

S (k) N

S (k+1)

L2 L1 x1

N 2

S (k+1)

L1 x1

L2 L1 xk+1 N

L2 L1 x1

:

L2

Utilize Sobolev, jE213 j . N

(2.18)

2+

1+

+N +N 2 . N

2

S (k+1) 1

N

2+

+N

(k) N 2

N

L2

S (k+2) S1 S

L2

2 N

L2

(k+1) N

L2

:

S1 S (k+1)

1

N

L2

S1 S

S (k+1)

2

1+

N

2 N

L2

2

+ S (k+2)

N

L2

Then, for E214 jE214 j =

2

N

hVN 1(k+1) S (k)

. N

2

. N

1+

VN 1(k+1) N

1

N ; VN (k+1)(k+2) S

VN (k+1)(k+2)

L1+ x1

S1 S (k+1)

2 N

L2

(k)

L1+ xk+2

Ni

S (k)

2 N

1 L2 L1 x1 Lxk+2

:

Together with (2.17)-(2.18), we have the estimate for E21 ; (2.19)

E21 >

CN

1+

N

1

S1 S (k+1)

2 N

L2

+ S (k+2)

2 N

L2

;

because E211 > 0. We now turn to E22 which is E22 = N

2

h[S1 ; H1(k+1) ]

S (k) S1

N; S

(k)

2 + H(k+1)(k+2)

N i:

Substitute [S1 ; H1(k+1) ] = N (rV )N 1(k+1) and expand 2 + H(k+1)(k+2) to obtain E22 = E221 + E221 + E223

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM

21

where S (k) S1 S (k) 2 h(rV )N 1(k+1) S1 S (k) 2 h(rV )N 1(k+1) S1 2

E221 = N E222 = N E223 = N

h(rV )N 1(k+1)

N; S

(k)

2 Sk+1

Ni

N; S

(k)

2 Sk+2

Ni

N; S

(k)

VN (k+1)(k+2)

Ni

For E221 , we …rst Hölder at x1 as follows: jE221 j . N

2

k(rV )N 1(k+1) kL2+ k x 1

S (k) S1

N kL2 L1 x1

kS (k+1) Sk+1

N kL2 ;

then Soblev to obtain (2.20)

3 + 2

jE221 j . N 2

3 + 2

. N2

kS (k)

kS (k)

N kL2 N 2 N kL2

1 2

kS1 S (k+1) 1

+N

N kL2

kS1 S (k+1)

2 N kL2

:

Use Hölder in xk+1 for E222 , we get (2.21) jE222 j . N . N

2

k k(rV )N 1(k+1) kL1+ x k+1

2+

kS (k+2)

S (k) Sk+2 S1

kS N kL2 L1 xk+1

(k)

Sk+2

N kL2 L1 xk+1

2 N kL2 :

Then, argue in the same way for E223 , jE223 j . N k

2

k(rV )N 1(k+1) kL1+ VN (k+1)(k+2) x 1

S (k) S1

. N

3 + 2

. N

3 + 2

1 kS N kL2 L1 x1 Lxk+2

kS (k+1) kS

(k+1)

N kL2 N 2 N kL2

1 2

(k)

1 N kL2 L1 x1 Lxk+2

kS1 S (k+1)

+N

L1+ xk+2

1

kS1 S

N kL2

(k+1)

2 N kL2

Together with (2.20) and (2.21), we have the estimate for E22 , (2.22)

jE22 j . N 2

3 + 2

kS (k)

2 N kL2

1

+N

kS1 S (k+1)

2 N kL2

:

This completes the treatment of E2 . Speci…cally, (2.19) and (2.22) give (2.23)

E2 >

C max(N 2

3 + 2

;N

1+

) kS (k)

2 N kL2

+N

1

kS1 S (k+1)

Finally, we treat E3 which is E3 = N

1

hS (k) 2 + H1(k+1)

N; S

(k)

Commute 2 + H1(k+1) and S (k) , E3 = E31 + E32 ;

2 + H(k+2)(k+3)

N i:

2 N kL2

:

22

XUWEN CHEN AND JUSTIN HOLMER

where E31 = N

1

h 2 + H1(k+1) S (k)

N; S

E32 = N

1

h S1 ; H1(k+1)

N; S

S (k) S1

(k)

2 + H(k+2)(k+3)

N i;

(k)

2 + H(k+2)(k+3)

N i:

We …rst discard E31 because E31 > 0 by Theorem 2.2 and Lemma 2.4. For E32 , we plug in S1 ; H1(k+1) = N (rV )N 1(k+1) and expand 2 + H(k+2)(k+3) to obtain S (k) 2 2 + Sk+3 ; S (k) 2 + Sk+2 S1 N S (k) +N 1 h(rV )N 1(k+1) ; S (k) VN (k+2)(k+3) N i S1 N S (k) = 2N 1 h(rV )N 1(k+1) ; S (k) N i S1 N S (k) Sk+2 N ; S (k) Sk+2 N i +2N 1 h(rV )N 1(k+1) S1 S (k) +N 1 h(rV )N 1(k+1) ; S (k) VN (k+2)(k+3) N i: S1 N 1

E32 = N

h(rV )N 1(k+1)

Ni

First Hölder again jE32 j . N

1

(rV )N 1(k+1)

S (k) k 2 1 kS (k) N kL2 L1 xk+1 S1 N L Lxk+1 S (k) kS (k) Sk+2 N kL2 L1 k Sk+2 N kL2 L1 1+ xk+1 xk+1 Lxk+1 S1 S (k) 1 V k kS (k) N (k+2)(k+3) L1+ N kL2 L1 xk+1 Lxk+2 L1+ xk+2 xk+1 S1

L1+ xk+1

+N

1

(rV )N 1(k+1)

+N

1

(rV )N 1(k+1)

k

1 ; N kL2 L1 xk+1 Lxk+2

then Sobolev gives jE32 j . N

1+

kS (k)

. N

1+

kS (k+2)

+N

(k+1) N kL2 kS N kL2 + N 1+ kS (k+1) N kL2 kS (k+2) N kL2

1+

kS (k+1)

N kL2 kS

(k+2)

N kL2

2 N kL2 .

That is E3 >

(2.24)

CN

1+

kS (k+2)

2 N kL2 :

Putting (2.16), (2.23) and (2.24) in one line, we obtain the estimate for the cross error term EC >

C max(N 2

which is exactly (2.12).

3 + 2

;N

1+

) kS (k+2)

2 N kL2

+N

1

kS1 S (k+1)

2 N kL2

;

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM

23

Finally, combining (2.11)and (2.12), we have 1 h N ; (N 1 HN + 1)k+2 N i k+2 c0 >

2

3 + 2

C max(N 2

> kS (k+2) k2L2 + N

1

;N

1+

kS (k+2) k2L2 + N

)

1

kS1 S (k+1) k2L2

kS1 S (k+1) k2L2

for N larger than some threshold, as originally claimed. Whence, we have proved (1.16) for all k and established Theorem 1.3. 2.3. Remark on higher . It is easy to see from §2.2 that Theorem 1.3 will hold up to < 3=4 as long as Theorem 2.1 works for higher . It is certainly of mathematical and physical interest to push for a higher in Theroem 1.3. On the one hand, higher makes the convergence VN ! b0 as N ! 1 faster and hence is more singular, di¢ cult, and interesting to deal with. On the other hand, larger means stronger and more localized interaction. Examing the proof of Theorem 2.1, one immediately notice the obstacles lie in Lemmas 2.1 and 2.2. While it is extremely di¢ cult to improve Lemma 2.2, one would certainly wonder how to improve the crude estimate, Lemma 2.1. However, it turns out that the crude estimate is actually optimal in the sense that it fails if M 6 C N " for some > 0. (See Lemma 2.5 below.) Thus, there is no obvious way to improve the current result and reach a higher . Lemma 2.5. Suppose that V 2 S(R2 ) with V^ ( ) = 1 for j j 4. Suppose that Mj = Mj (N ), M2 j = 1; 2 are dyads with 0 M1 M2 N and limN !1 M = 1. There does not exist 1 a constant C independent of N such that the following estimate holds: for all symmetric (x1 ; x2 ), Z (2) (2.25) jVN (x1 x2 )jjPM1 M2 (x1 ; x2 )j2 dx1 dx2 Ckr1 k2L2 Before proceeding with the proof, we make a few remarks. First, the assumption V^ ( ) = 1 for j j 4 can be eliminated, but we add it since it simpli…es the proof and still covers a wide class of Schwartz class potentials. Second, we note the estimate (2.25) is in fact true when M2 =M1 remains bounded as N ! 1. This follows readily from scaling and the Bernstein inequality: if M is a single dyadic interval, then kPM kL1 M kPM kL2 . Moreover, the core of Lemma 2.1 is e¤ectively the estimate Z (2) (2.26) jVN (x1 x2 )jjP>M1 (x1 ; x2 )j2 dx1 dx2 Ckr1 k2L2 for M1 > N .

Lemma 2.5 shows that Lemma 2.1 cannot be improved in the sense that one cannot select M2 = N and M1 N (for example M1 = N for any > 0) and expect (2.26) to hold. Proof. Replacing xj by equivalent to Z

jVN~ (x1

xj 1=2

M1

1=2

M2

x2 )jP

~= and N

1=2

M1

(2) M1 M2

1=2

M2 M1

1=2

N 1=2 M2

, we obtain that the estimate (2.25) is

(x1 ; x2 )j2 dx1 dx2

Ckr1 k2L2

24

XUWEN CHEN AND JUSTIN HOLMER 1=2

1=2

2 ~ and limN !1 M2 = 1 implies that limN !1 M2 N implies M N M1 M1 M1 ~ 1 and hence limN !1 N = 1. Thus, it su¢ ces to assume that in (2.25), we in fact have limN !1 M1 = 0 and limN !1 M2 = 1. For any functions W , 1 , 2 , consider Z def I = W (x1 x2 ) 1 (x1 ; x2 ) 2 (x1 ; x2 ) dx1 dx2 x1 ;x2 Z ^ ( ) ^ 1 ( 1 ; 2 ) 2 (x1 ; x2 ) dx1 dx2 ei(x1 x2 ) eix1 1 eix2 2 W =

Notice that M2

x1 ;x2 ; ;

= = = =

Z Z

Z

Z

1; 2

^( ) W ;

1; 2

;

1;

Z 1( 1; 2)

e

ix1 (

1+

) e ix2 (

2

)

2 (x1 ; x2 ) dx1

d

2

dx2 d d

1

dx2

x1 ;x2

^( ) W

1( 1; 2)

2( 1

+ ;

)d d

2

1

2

^ ( ) ^ 1( W ;

1; 2

;

1;

1

^ (2 ) ^ 1 ( W

2 1

;

2

;

+ ) ^ 2( 2

2

+ ) ^ 2(

1

+ ; 2

2

1

+ ;

2

2

) d 1d 2d

) d 1d 2d

2

Let def

JV = and

Z

def

J = =

jVN (x1

Z

Z

x2 )jjPM1

(x1

M2

x2 )jPM1

jPM1

M2

(x1 ; x2 )j2 dx1 dx2

(x1 ; x2 )j2 dx1 dx2

M2

(x; x)j2 dx

We show that JV = J . To obtain I = JV and j = PM1 M2 . Then

J , in the expression for I, we take W = VN

^ (2 ) = V^ ( 2 ) W N

1

^ (2 ) = 0 for j j 2N . On the other hand, the frequency restrictions on so W j 1 j M2 N and j 1 + j M2 N . It follows that j2 j = j(

1

+ )

(

1

)j

j

1

+ j+j

1

j

j

imply that

2N

Consequently I = 0, completing the proof of the claim. We argue by contradiction assuming that (2.25) holds with C independent of N . Since JV = J , Z J = jPM1 M2 (x; x)j2 dx Ckrx1 k2L2

=

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM

25

with a constant C independent of N , where M1 ! 0 and M2 ! 1 as N ! 1. By Fatou’s lemma, Z (2.27) j (x; x)j2 dx Ckrx1 k2L2 which is the (false) 2D endpoint trace estimate. A counterexample can be constructed as follows. Let be a smooth function with ( x) = (x), (x) = 1 for jxj 41 and (x) = 0 for jxj 21 . Then x2 ) (x1 ) (x2 ) ln( ln jx1

(x1 ; x2 ) = (x1

x2 j)

is a symmetric function for which the left side of (2.27) is in…nite but the right side is …nite. More properly written, we can introduce a smooth function (x1 ; x2 ) = (x1 Then

x2 ) (x1 ) (x2 ) ln( ln(jx1

Z

j

(x; x)j2 dx

x2 j + ))

1

ln ln

while krx1 kL2 is bounded independently of as ! 0. Sending choice of C in (2.27) can be beat, giving us the contradiction.

! 0 shows that any

3. Derivation of the 2D Focusing NLS 3.1. Proof of Theorem 1.2. We start by introducing an appropriate topology on the density matrices as was previously done in [27, 28, 29, 30, 31, 32, 44, 10, 15, 16, 17, 18]. Denote the spaces of compact operators and trace class operators on L2 R2k as Kk and L1k , respectively. Then (Kk )0 = L1k . By the fact that Kk is separable, we select a dense countable (k) (k) subset fJi gi>1 Kk in the unit ball of Kk (so kJi kop 6 1 where k kop is the operator norm). For (k) ; ~ (k) 2 L1k , we then de…ne a metric dk on L1k by dk (

(k)

;~

(k)

)=

1 X

2

i

(k)

(k)

Tr Ji

~ (k)

:

i=1

A uniformly bounded sequence topology if and only if

(k) N

2 L1k converges to

lim dk (

N !1

(k) N ;

(k)

(k)

2 L1k with respect to the weak*

) = 0:

For …xed T > 0, let C ([0; T ] ; L1k ) be the space of functions of t 2 [0; T ] with values in L1k which are continuous with respect to the metric dk : On C ([0; T ] ; L1k ) ; we de…ne the metric d^k (

(k)

( ) ; ~ (k) ( )) = sup dk (

(k)

(t) ; ~ (k) (t));

t2[0;T ]

and denote by prod the topology on the space k>1 C ([0; T ] ; L1k ) given by the product of topologies generated by the metrics d^k on C ([0; T ] ; L1k ).

26

XUWEN CHEN AND JUSTIN HOLMER

By Theorem 1.3, we have, 8k = 0; 1; :::; Tr S (k)

(k) (k) N (t)S

S (k) D 6 Ck D k = C =

=

2 N (t) L2 N (t); N

1

N (0);

1

k X k h j

N

N (0);

j=0

k X k j k 6 1C j

HN + 1

k

HN + 1 1 Nk

j

E (t) N E N (0)

k

HNk

j

N (0)i

j

j=0

6 Ck

provided that N > N0 (k). That is the energy estimate: sup Tr S (k)

(3.1)

t

(k) (k) N (t)S

6 Ck:

With estimate (3.1), one can go through Lemmas 3.1, 3.2, and 3.3 to conclude that, as trace class operators: (k) N (t)

k

* j (t)i h (t)j

weak*.

By the argument on [16, p.398-399]15, we can upgrade the above weak* convergence to strong and hence …nish the proof of Theorem 1.2. Lemma 3.1 (Compactness). For all …nite T > 0, the sequence oN n M (k) C [0; T ] ; L1k ; (t) = N N k=1

k>1

which satis…es the 2D focusing BBGKY hierarchy (3.2)

i@t

(k) N

=

k h X j=1

+

N N

4xj +! 2 jxj j2 ; k kX j=1

(k) N

h Trk+1 VN (xj

i

+

1 N

X h

VN (xi

16i<j6k

xk+1 );

(k+1) N

i

xj );

(k) N

i

;

where V < 0, subject to energy condition (3.1) is compact with respect to the product topology (k) N (t) = ; (k) is a symmetric nonnegative trace class prod . For any limit point k=1 operator with trace bounded by 1; and it veri…es the energy bound (3.3)

sup Tr S (k) t2[0;T ]

15 The

(k)

S (k) 6 C k :

proof [16, p.398-399] is actually for more general datum.

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM (k) 1 k=1

Lemma 3.2 (Convergence). Let (t) =

be a limit point of

the sequence in Theorem 3.1, with respect to the product topology to the focusing GP hierarchy (3.4)

i@t

(k)

=

k X

2

2

(k)

4xj +! jxj j ;

j=1

(3.5)

(t) = U

(k)

(k)

(t)

(0) + ib0

k

k Z X j=1

Trk+1

N (t)

prod ,

(xj

=

n

(k) N

oN

;

k=1

then (t) is a solution

(k+1)

xk+1 );

;

j=1

subject to initial data (k) (0) = j 0 i h 0 j written in integral form, is (k)

b0

k X

27

with coupling constant b0 =

R

jV (x)j dx. which,

t

U (k) (t

s) Trk+1

(xj

xk+1 ) ;

(k+1)

(s) ds:

0

where U (k) (t) = eit(

4xj +! 2 jxj j2 )

e

it

2

4x0 +! 2 jx0j j j

:

1

Lemma 3.3. 16If (t) = (k) k=1 is a solution to (3.4) subject to the following two conditions: (a) (t) is sequence of normalized symmetry nonnegative trace class opertors which is a limit point of some N -body marginals with respect to the product topology prod or satisifes Trk+1 (k+1) = (k) . (b) For some > 23 , we have the regularity estimate sup Tr S (k)

(k)

S (k)

t2[0;T ]

6 Ck;

then (t) is also the only solution of (3.4) subject to (a) and (b). In particular, if (t) checks (a) and (b) of this lemma and (k) (0) = j 0 i h 0 j satis…es

2

2

4x + ! jxj

1 2

0

L2 (R)

k

where

0

< 1, then

(k)

(t) = j (t)i h (t)j

k

where (t) solves the 2D focusing cubic NLS (1.11). This is because j (t)i h (t)j solution to (3.4) subject to (a) and (b) of this lemma.

k

is a

To prove Lemma 3.1 and 3.2, we need the following lemma. R Lemma 3.4 ([44, Lemma A.2]). Let f 2 L1 (R2 ) such that R2 hri jf (r)j dr < 1 and R f (r) dr = 1 but we allow that f not be nonnegative everywhere. De…ne f (r) = 2 f r : R2 16 One

can also use the Strichartz type uniqueness theorems [15, Theorem 3] or [44, Theorem 7.1] here.

28

XUWEN CHEN AND JUSTIN HOLMER

Then, for every

2 (0; 1) , there exists C > 0 s.t.

Tr J (k) (f (rj rk+1 ) Z 6 C jf (r)j jrj dr 1

4xj ) 2 J (k) (1

(1

6 C

Tr 1 Z

(rj

4 xj

(k+1)

(k+1)

4 xj )

+ (1 op

1 2

1

J (k) (1

4 xj ) 2

op

(k+1) 1

Sj J (k) Sj

jf (r)j jrj dr

for all nonnegative

1 2

4 xj )

4xk+1

1

rk+1 ))

2 L1 L2 R2k+2

op

+ Sj 1 J (k) Sj

Tr Sj Sk+1

op

(k+1)

Sj Sk+1

:

Proof of Compactness. By [32, Lemma 6.2], this is equivalent to the statement that for every test function f (k) from a dense subset of Kk and for every " > 0, there exists (f (k) ; ") such that for all t1 ; t2 2 [0; T ] with jt1 t2 j 6 , we have (k) N (t1 )

sup Tr f (k) N

Tr f (k)

6 ":

(k) N (t2 )

We select the test functions f (k) 2 Kk which satisfy Si Sj f (k) Si 1 Sj

1 op

+ Si 1 Sj 1 f (k) Si Sj

Let 0 6 t1 6 t2 6 T , we take advantage of the @t fundamental theorem of calculus to get to Tr f (k) k Z X

6

j=1

1 + N +

(k) N (t2 )

Tr f (k)

t2

Tr f

(k)

t1

X Z

16i<j6k

N N

t2

(k) N

t2

t1

< 1;

in the hierarchy (3.2) and use the

(k) N (t1 )

i (s) ds

h Tr f (k) VN (xi

t1

k Z kX j=1

h Sj2 ;

(k) N

op

xj ) ;

h Tr f (k) VN (xj

(k) N

i (s) ds

xk+1 ) ;

(k+1) N

i (s) ds:

We estimate each term as follow. The …rst term can be easily estimated Z t2 h i (k) (k) 2 Tr f Sj ; N (s) ds t1 t2

= 6

Z

t1 Z t2

Tr Sj 1 f (k) Sj Sj Sj 1 f (k) Sj

t1

6 Cf C jt2

t1 j :

op

(k) N

(s) Sj

+ Sj f (k) Sj

Tr Sj f (k) Sj 1 Sj 1 op

Tr Sj

(k) N

(k) N

(s) Sj ds

(s) Sj ds

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM

29

For the second and the third terms, we use the fact that conjugation preserves traces and the Sobolev inequality 1 Sij 1 Sk+1 VN (xi

(3.6) to deduce 1 N

X Z

Z

Ck N Z t2

Z

h

VN (xi

(k) N

xj ) ;

op

t2

j Tr Si Sj f (k) Si 1 Sj 1 Si Sj Si 1 Sj 1 f (k) Si Sj

Tr Si Sj

(k) N

op

(k) N

6 C kVN kL1 = C kV kL1

i (s) ds

j Tr Si 1 Sj 1 f (k) Si Sj Si 1 Sj 1 VN (xi

t1

2

6

t2

t1

k2 + N

Tr f

(k)

t1

16i<j6k

k2 6 N

t2

1 xj ) Sj 1 Sk+1

xj ) Si 1 Sj 1 Si Sj

(s) Si Sj Si 1 Sj 1 VN (xi

+ Si Sj f (k) Si 1 Sj

1 op

(k) N

(s) Si Sj jds

xj ) Si 1 Sj 1 jds

Si 1 Sj 1 VN (xi

xj ) Si 1 Sj

1

(s) Si Sj ds

t1 2

k Cf C 2 jt2 N

6

t1 j ;

and k Z kX

N

N Z t2

6 k

t1

+k

Z

j=1

t1

xk+1 ) ;

(k+1) N

1 (k) 1 j Tr Sj 1 Sk+1 f Sj Sk+1 Sj 1 Sk+1 VN (xj t2

t1

6 Ck Z t2

h Tr f (k) VN (xj

t2

1 j Tr Sj Sk+1 f (k) Sj 1 Sk+1 Sj Sk+1

Sj 1 f (k) Sj Tr Sj Sk+1

op

(k+1) N

+ Sj f (k) Sj

1 op

(k+1) N

i (s) ds 1 xk+1 ) Sj 1 Sk+1 Sj Sk+1

1 (s) Sj Sk+1 Sj 1 Sk+1 VN (xj

1 Sij 1 Sk+1 VN (xi

(k+1) N

(s) Sj Sk+1 jds

1 xk+1 ) Sj 1 Sk+1 jds

1 xj ) Sj 1 Sk+1

(s) Sj Sk+1 ds

t1

6 kCf C 2 jt2

t1 j :

That is Tr f (k)

(k) N (t2 )

Tr f (k)

(k) N (t1 )

which is enough to end the proof of Theorem 3.1.

6 Cf;k jt2

t1 j ;

Proof of Convergence. By Theorem 3.1, passing to subsequences if necessary, we have (3.7)

lim sup Tr f (k)

N !1 t2[0;T ]

(k) N

(k)

= 0, 8f (k) 2 Kk :

We test (3.5) against the test functions f (k) in Theorem 3.1. We prove that the limit point veri…es (3.8)

Tr f (k)

(k)

(0) = Tr f (k) j 0 i h 0 j

k

;

30

XUWEN CHEN AND JUSTIN HOLMER

and Tr f (k)

(3.9)

(k)

= Tr f (k) U (k) (t) (k) (0) k Z t X +ib0 Tr f (k) U (k) (t

s)

(xj

xk+1 ) ;

(k+1)

(s) ds:

0

j=1

Rewrite the BBGKY hierarchy (3.2) as the following Tr f (k)

(k) N

(k)

= Tr f (k) U (k) (t) N (0) Z t i X Tr f (k) U (k) (t + N 16i<j6k 0 +i

k Z kX

N

= I+

N i N

j=1

X

s)

h

t

Tr f (k) U (k) (t

s)

0

II + i 1

16i<j6k

k N

k X

VN (xi h

VN (xj

xj );

i

(k) N (s)

xk+1 );

ds i

(k+1) (s) N

ds

III:

j=1

R Notice that b0 = VN (x)dx, we have put a minus sign in front of VN to match 3.9. Immediately following (3.7), we have (k)

lim Tr f (k) uN

N !1

lim Tr f (k) U (k) (t)

N !1

(k) N

= Tr f (k) u(k) ;

(0) = Tr f (k) U (k) (t)f (k) (0) : (k)

By the well-known argument on [50, p.64], we know N (0) ! j 0 i h 0 j k strongly as trace (1) operators because N (0) ! j 0 i h 0 j strongly as trace operators. So we have checked relation (3.8) and the left hand side and the …rst term on the right hand side of (3.9) for (t). We now prove II k = lim III = 0; N !1 N N !1 N

(3.10)

lim

and (3.11)

lim III =

N !1

Z

t

Tr J (k) U (k) (t

s)

(xj

xk+1 ) ;

(k+1)

(s) ds:

0

In the proof of Theorem 3.1, we have already shown that jIIj and jIIIj are uniformly bounded for every …nite time, thus (3.10) has been checked. So we are left to prove 3.11. To use Lemma 3.4, we take a probability measure 2 L1 (R2 ), de…ne (y) = 12 y : Adopt the

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM (k) t

= f (k) U (k) (t

notation fs

Tr f (k) U (k) (t 6 Tr fs

(k) t

s), we have

s)

VN (xj

( VN (xj

xk+1 )

(s)

(xj

xk+1 ))

(k+1) N

(s)

(xj

xk+1 )

(

xk+1 )

(k+1) N

(k) t

(k) t

b0 (xj

xk+1 ))

xk+1 )

+ b0 Tr fs

(s)

b0 (xj

( (xj

+ b0 Tr fs

(k+1) N

xk+1 )

(k) t

+ b0 Tr fs

31

(xj

xk+1 )

(k+1) N

(k+1)

(s)

(xj

xk+1 ))

(k+1)

(s)

(s) (k+1)

(s)

= IV + V + V I + V II: Lemma 3.4 and the energy condition (3.1) gives C (k+1) Sj 1 f (k) Sj op + Sj f (k) Sj 1 op Tr Sj Sk+1 N Sj Sk+1 N Cf 6 ! 0 as N ! 1, uniformly for s 2 [0; T ] with T < 1: N Similarly, we obtain V; V II 6 Cf ! 0 as ! 0. For VI, IV

6

G 6 b0 Tr fs

(k) t (k) t

+b0 Tr fs

(xj

1 1 + "Sk+1 "Sk+1 xk+1 ) 1 + "Sk+1

(k+1) N

xk+1 )

(xj

(s)

(k+1) N

(s)

(k+1)

(s)

(k+1)

(s)

:

The …rst term in the above inequality tends to zero as N ! 1 for every " > 0, since we (k) have assumed (3.7) and fs t (xj xk+1 ) 1+"S1 k+1 is a compact operator. Due to the energy bounds (3.1) and (3.3), the second term tends to zero as " ! 0, uniformly in N . Combining the estimates for IV V II, we have justi…ed limit (3.11) and thus limit (3.9). Hence, we have proved Theorem 3.2. Proof of Uniqueness. The proof is essentially already in [8] and [22]. One merely needs to set 2 2 A =0, switch the Strichartz estimate for eit4 to the ones for eit(4 ! jxj ) in [22] and notice that kf kH . kS f kL2 for > 0. We skip the details. 3.2. Proof of Theorem 1.1. Assuming Theorem 1.2, we now prove Theorem 1.1. If N (0) satis…es (a), (b), and (c) in Theorem 1.1, then N (0) checks the requirements of the following lemma. Lemma 3.5. Assume N (0) satis…es (a), (b), and (c) in Theorem 1.1. Let 2 C01 (R) be a cut-o¤ such that 0 6 6 1, (s) = 1 for 0 6 s 6 1 and (s) = 0 for s > 2: For > 0; we de…ne an approximation N (0) of N (0) by (3.12)

N (0)

=

( HN =N ) k ( HN =N )

This approximation has the following properties:

(0) : N (0)k

N

32

XUWEN CHEN AND JUSTIN HOLMER

(i)

N (0)

veri…es the energy condition h

k N (0); HN

2k N k

N (0)i 6

k

:

(ii) sup k

N (0)

N

(iii) For small enough

> 0, lim Tr

N !1

N (0)

N (0)kL2

1 2

6C

:

is asymptotically factorized as well

;(1) 0 N (0; x1 ; x1 )

0 0 (x1 ) 0 (x1 )

;(1)

where N (0) is the marginal density associated with assumption (b) in Theorem 1.1.

= 0; and

N (0)

0

is the same as in

Proof. (i) and (ii) follows from [19, Lemma B.1] and [20, Lemma B.1]. (iii) follows from the proof of [30, Proposition 5.1 (iii)]. Notice that for two dimension, we get a N instead of a 3 N 2 in [30, (5.20)] and hence we get a N 1 in the estimate of [30, (5.18)] which goes to zero for 2 (0; 1). Thus we can de…ne an approximation N (0) of N (0) as in (3.12). Via (i) and (iii) of Lemma 3.5, N (0) veri…es the requirements of Theorem 1.2 for small enough > 0: ;(k) Therefore, for N (t) ; the marginal density associated with eitHN N (0), Theorem 1.2 gives the convergence: (k) N (t)

(3.13)

! j (t)i h (t)j

k

strongly, 8k; t

as trace class operators, for all small enough > 0. (k) For N (t) in Theorem 1.2, we notice that, for any test function f (k) 2 Kk and any t 2 R, we have

6

Tr f (k)

(k) N

(t)

Tr f (k)

(k) N

(t)

j (t)i h (t)j ;(k) N

;(k) N

+ Tr f (k)

k

(t) k

j (t)i h (t)j

(t)

= A + B: Convergence (3.13) then takes care of B. To handle A, part (ii) of Lemma 3.5 yields eitHN

N (0)

eitHN

N (0) L2

=k

N (0)

N (0)kL2

6C

which implies (k) N

A = Tr f (k) Since

;(k) N

(t)

6 C f (k)

(t)

op

> 0 is arbitrary, we deduce that lim Tr f (k)

N !1

(k) N

(t)

j (t)i h (t)j

k

i.e. (k) N

(t) * j (t)i h (t)j

k

weak*

= 0;

1 2

:

1 2

2D CUBIC FOCUSING NLS FROM 2D N -BODY QUANTUM

33 (k)

as trace class operators. Notice that the limit has the same trace norm as N (t) for every N , the Grümm’s convergence theorem then upgrades the above weak* convergence to strong. Thence, we have concluded Theorem 1.1 via Theorem 1.2.

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