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A rigorous derivation of the Gross-Pitaevskii hierarchy for weakly coupled two-dimensional bosons
Acta Mathematica Scientia 2010,30B(3):841–856 http://actams.wipm.ac.cn
A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS∗
)
Liu Chuangye (
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan 430071, China E-mail: [email protected]
Abstract In this article, we consider the dynamics of N two-dimensional boson systems interacting through a pair potential N −1 Va (xi − xj ) where Va (x) = a−2 V (x/a). It is well known that the Gross-Pitaevskii (GP) equation is a nonlinear Schr¨ odinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k-particle density matrices { k ut , k ≥ 1} solves the GP hierarchy. Denote by ψN,t the solution to the N -particle Schr¨ odinger equation. Under the assumption that a = N −ε for 0 < ε < 3/4, we prove that as N → ∞ the limit points of the k-particle density matrices of ψN,t are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by V (x) dx.
Æ
Ê
Key words Gross-Pitaevskii equation; Boson system; density matrix; BBGKY hierarchy 2000 MR Subject Classification
1
35Q40; 35Q55
Introduction
Motivated by recent experimental realizations of Bose-Einstein condensation the theory of dilute, inhomogeneous Bose systems was currently a subject of intensive studies [1]. The ground state of bosonic atoms in a trap was shown experimentally to display Bose-Einstein condensation (BEC). This fact was proved theoretically by Lieb [9, 10, 11] for bosons with twobody repulsive interaction potentials in the dilute limit, starting from the basic Schr¨ odinger equation. On the other hand, it is well known that the dynamics of Bose-Einstein condensates are well described by the the Gross-Pitaevskii equation. A rigorous derivation of this equation from the basic many-body Schr¨ odinger equation in an appropriate limit is not a simple matter, however, and has only been achieved recently in three spatial dimension [2, 3, 4, 5]. This article is concerned with the justification of the Gross-Pitaevskii equation in two spatial dimensions. In this case, several new issues arise and we refer to [11] for details. ∗ Received
Sepetember 11, 2007; revised April 7, 2008. This work is partially supported by NSFC (10571176)
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Consider N bosons in the two-dimension space. The bosons interact via a two body potential Va (x) = a−2 V (x/a). (1.1) We assume that the potential V is smooth, symmetric, and positive, with compact support. The parameter a determines the range and the strength of the potential: a and N will be coupled so that a → 0 as N → ∞. Thus the potential Va converges to Dirac δ-function. The N -body Hamiltonian for the N weakly coupled bosons is given by HN = −
N j=1
2
Δj +
1 Va (xi − xj ), N i
(1.2)
2
∂ ∂ 2 where Δj = Δxj = ∂α 2 + ∂β 2 for xj = (αj , βj ) ∈ R . The dynamics of the Bose system is j j governed by the N -body Schr¨ odinger equation
i∂t ψN,t = HN ψN,t .
(1.3)
Here, the wave function ψN,t lies in L2s (R2N ) which is the subspace of L2 (R2N ) consisting of functions symmetric with respect to permutation of the N particles. More generally, we can describe the N -body system by its density matrix γN,t . The density matrix is a positive self-adjoint operator γ acting on L2s (R2N ) with Trγ = 1. The density matrix corresponding to the wave function ψN,t is given by the one-dimensional orthogonal projection onto ψN,t , that is, γN,t = |ψN,t ψN,t |. Quantum mechanical states described by one-dimensional orthogonal projections are called pure states. In general, a density matrix (mixed states) is a weighted average of the one-dimensional orthogonal projections. The time evolution of the density matrix γN,t is then given by i∂t γN,t = [HN , γN,t ], (1.4) which is equivalent to the Schr¨odinger equation (1.3). As follows, we denote by x a general variable in R2 and by x = (x1 , · · · , xN ) a point in R2N . We will also use the notation xk = (x1 , · · · , xk ) ∈ R2k and xN −k = (xk+1 , · · · , xN ) ∈ R2(N −k) . (k) For k = 1, · · · , N − 1, the k-particle marginal distribution γN,t of γN,t is defined through its kernel by (k)
γN,t (xk , xk ) =
dxN −k γN,t (xk , xN −k ; xk , xN −k ),
(1.5)
where xk = (x1 , · · · , xk ) and γN,t (x; x ) denotes the kernel of the density matrix γN,t . From (k) TrγN,t = 1, it is concluded that TrγN,t = 1 for every k = 1, · · · , N − 1. In the sequel, we set (N )
(k)
γN,t = γN,t and γN,t = 0, k > N for convenience. From (1.4) and the symmetry of γN,t with respect to permutations of the N particles, we conclude that the evolution of the marginal distributions of γN,t is determined by the following hierarchy of N equations, commonly called the BBGKY hierarchy, (k)
i∂t γN,t (xk ; xk ) =
k (k) −Δxj + Δxj γN,t (xk ; xk ) j=1
+
1 N
1≤j
(k) Va (xj − xl ) − Va (xj − xl ) γN,t (xk ; xk )
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k k + 1− dxk+1 Va (xj − xk+1 ) − Va (xj − xk+1 ) N j=1 R (k+1)
×γN,t (xk , xk+1 ; xk , xk+1 )
(1.6)
for k = 1, · · · , N. Rewriting this hierarchy in integral form, we have (k)
(k)
γN,t (xk ; xk ) = γN,0 (xk ; xk ) − i −
i N
1≤j
t
0
k
t
0
j=1
(k) ds −Δxj + Δxj γN,s (xk ; xk )
(k) ds Va (xj − xl ) − Va (xj − xl ) γN,s (xk ; xk )
k k t ds dxk+1 Va (xj − xk+1 ) − Va (xj − xk+1 ) −i 1 − N j=1 0 R2 (k+1)
×γN,s (xk , xk+1 ; xk , xk+1 ).
(1.7)
Setting a = N −ε and letting N → ∞, one has that the BBGKY hierarchy converges formally to the following infinite hierarchy of equations (k)
(k)
γt (xk ; xk ) = γ0 (xk ; xk ) − i −ib
k t j=1
0
k j=1
ds R2
0
t
ds −Δxj + Δxj γs(k) (xk ; xk )
dxk+1 δ(xj − xk+1 ) − δ(xj − xk+1 )
×γs(k+1) (xk , xk+1 ; xk , xk+1 )
(1.8)
for k = 1, 2, · · · , where b = R2 V (x)tdx. Eq.(1.8) is said to be the infinite BBGKY hierarchy, or the Gross-Pitaevskii (GP) hierarchy. It turns out that (1.8) has a factorized solution. Specially, k the family of marginal distribution γ t (k) (xk , xk ) = φt (xj )φt (xj ) is a solution of (1.8) if and j=1
only if the function φt satisfies the nonlinear Schr¨ odinger equation i∂t φt = −Δφt + b|φt |2 φt .
(1.9)
This is the Gross-Pitaevskii (GP) equation [7, 8, 12], except that the coupling constant in front of the nonlinear interaction is given by b. The aim of this article is to prove the convergence of solutions of (1.7) to ones of (1.8). More
(k) N precisely, we will prove that for every 0 < ε < 3/4 and a = N −ε , the sequence ΓN,t = γN,t k=1
(k) has at least as N → ∞ one limit point Γ∞,t = γ∞,t k≥1 with respect to some weak topology, and that any weak limit point Γ∞,t satisfies the infinite hierarchy (1.8). The remainder of this article is divided into four sections. In Section 2 we prove some Sobolev-type inequalities and energy estimates. Some notations and the main result are presented in Section 3. Section 4 is devoted to proofs of two lemmas, which play a crucial role in the proof of the main result. Finally, in Section 5, the main result is proved.
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Energy Estimates We begin with the following Sobolev-type inequalities. Lemma 1 (i) Suppose that V ∈ Lp (R2 ), 1 < p ≤ ∞, and φ ∈ H 1 (R2 ). Then, 2 |φ(x)|2 |Va (x)|dx ≤ C(p)a p −2 ||V ||Lp (R2 ) ||φ||2H 1 (R2 ) .
(2.1)
R2
(ii) Let V ∈ L1 (R2 ) be a nonnegative function. Then, there is an absolute constant C > 0 such that Va (x − y)|φ(x, y)|2 dxdy ≤ C V L1 (R2 ) (1 − Δx )(1 − Δy )φ, φ (2.2) R2 ×R2
for all φ ∈ H 2 (R2 × R2 ). The following proposition plays a key role in the proof of the main result, which presents bounds for the L2 -norm of the derivatives of a wave function ψ in terms of the mean value of powers of the Hamiltonian HN in the state described by ψ. Proposition 1 Suppose that the potential V (x) is positive, smooth, compactly supported, and symmetric, that is, V (x) = V (−x). Set Va (x) = a12 V (x/a) and assume that a = N −ε , with 0 < ε < 3/4. Put N = H
N
Sj2 +
j=1
1 N
Va (xl − xm ) = HN + N.
1≤l
Here and in the sequel, Sj = (1 − Δj )1/2 for j = 1, · · · , N. Fix k ∈ N and 0 < C < 1. Then, there is N0 = N0 (k, C) such that N )k ψ) ≥ C k N k (ψ, S12 S22 · · · Sk2 ψ) (ψ, (H
(2.3)
for all N > N0 and all ψ ∈ D((HN )k ) (ψ is assumed to be symmetric with respect to any permutation of all its variables). Proof The proof of the proposition uses a two step induction over k. For k = 0 and k = 1, the claim is trivial because of the positivity of the potential and the symmetry of ψ. Now, we assume the proposition is true for all k ≤ n, and we prove it for k = n + 2. To this end, we apply the induction assumption and we find, for N > N0 (n, C), N (H N )n H N S 2 · · · S 2 H N ψ) ≥ C n N n (ψ, H N )n+2 ψ) = (ψ, H (ψ, (H 1 n N ψ). Set H (n) =
n j=1
Sj2 +
1 N
N − Vjm = H
N
(2.4)
Sj2 ,
j=n+1
1≤j
with Vjm = Va (xj − xm ). Then, we have
N ψ) ≥ N S12 · · · Sn2 H (ψ, H
(ψ, Sj21 S12 · · · Sn2 Sj22 ψ)
N ≥j1 ,j2 ≥n+1
+
(ψ, Sj2 S12 · · · Sn2 H (n) ψ) + c.c.
N ≥j≥n+1
=: I1 + I2
(2.5)
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because of H (n) S12 · · · Sn2 H (n) ≥ 0. Here and afterwards, c.c. denotes the complex conjugate of the term followed. For I1 and I2 , using the symmetry with respect to permutations, we have 2 2 2 Sn+2 ψ) + (N − n)(ψ, S14 S22 · · · Sn+1 ψ), I1 = (N − n)(N − n − 1)(ψ, S12 · · · Sn2 Sn+1
(2.6)
and 2 ψ) I2 ≥ 2n(N − n)(ψ, S14 S22 · · · Sn2 Sn+1 n(n + 1)(N − n) 2 (ψ, V12 S12 · · · Sn2 Sn+1 + ψ) + c.c. 2N (n + 1)(N − n − 1)(N − n) 2 (ψ, V1,n+2 S12 · · · Sn2 Sn+1 ψ) + c.c. . + N
(2.7)
Here, we have used the fact that 2 ψ) ≥ 0, (ψ, Vjm S12 . . . Sn+1
if j, m > n + 1, because of the positivity of the potential. Then, combining with (2.5), (2.6), and (2.7), we get N S12 · · · Sn2 H N ψ) (ψ, H 2 2 2 ≥ (N − n)(N − n − 1)(ψ, S12 · · · Sn2 Sn+1 Sn+2 ψ) + (2n + 1)(N − n)(ψ, S14 S22 · · · Sn2 Sn+1 ψ) n(n + 1)(N − n) 2 (ψ, V12 S12 · · · Sn2 Sn+1 + ψ) + c.c. 2N (n + 1)(N − n − 1)(N − n) 2 (ψ, V1,n+2 S12 · · · Sn2 Sn+1 ψ) + c.c. . (2.8) + N
Next, we consider the last two terms on the right-hand of (2.8). Setting ϕ = S3 . . . Sn+1 ψ, we have 2 (ψ, V12 S12 · · · Sn2 Sn+1 ψ) + c.c. ≥ 2(ϕ, V12 p22 ϕ) + (ϕ, V12 p21 p22 ϕ) + c.c. where p1 = −i∇1 , p2 = −i∇2 , and hence, 2 ψ) + c.c. (ψ, V12 S12 · · · Sn2 Sn+1
≥ 2(ϕ, −∇V12 ∇2 ϕ) + (∇2 ϕ, ∇V12 ∇1 ∇2 ϕ) + (ϕ, −∇V12 p21 ∇2 ϕ) + c.c., where ∇V12 = a−3 (∇V )((x1 − x2 )/a). Applying the Schwarz inequality and Lemma 1, we find that 2 (ψ, V12 S12 · · · Sn+1 ψ) + c.c. 2 p −3 (∇ ϕ, S 2 ∇ ϕ) ≥ −C α1 a−1 (ϕ, S12 S22 ϕ) + α−1 2 1 2 1 a 2 2 p −3 (∇ ∇ ϕ, S 2 ∇ ∇ ϕ) −C α2 a p −3 (∇2 ϕ, S12 ∇2 ϕ) + α−1 a 1 2 1 2 1 2 −1 −3 −1 2 2 2 2 −C α3 a (ϕ, S1 S2 ϕ) + α3 a (∇1 ∇2 ϕ, ∇1 ∇2 ϕ) .
(2.9)
Optimizing the choices of α1 , α2 , and α3 , we can conclude that
2 2 2 ψ) + c.c. ≥ −CN −β N 2 (ψ, S12 · · · Sn+1 ψ) + N (ψ, S14 · · · Sn+1 ψ) (ψ, V12 S12 · · · Sn+1
2 2 ψ) + N (ψ, S14 · · · Sn+1 ψ) . (2.10) ≥ −CN −β N 2 (ψ, S12 · · · Sn+2
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In contrast, letting φ = S2 · · · Sn+1 ψ, we conclude from Lemma 1 that 1
2 2 (ψ, V1,n+2 S12 · · · Sn+1 ψ) + c.c. = (φ, V1,n+2 S12 φ) + c.c. ≥ −Ca p −2 (ψ, S12 · · · Sn+2 ψ).
(2.11)
Inserting (2.10) and (2.11) into the right-hand side of (2.8), we obtain N S 2 · · · S 2 H (ψ, H 1 n N ψ) ≥ (N − n)(N − n − 1) 1 −
C 2− p1
−
C 2 (ψ, S12 · · · Sn+2 ψ) Nβ
Na C 2 +(2n + 1)(N − n) 1 − β (ψ, S14 S22 · · · Sn+1 ψ). N
(2.12) 1
For 0 < ε < 3/4 and 1 < p < 32 , one has β > 0, 1 − ε(2 − 1p ) > 0. Consequently, N a2− p 1 and N β 1. For any fixed C < 1 and n ∈ N , by (2.12) we can find N0 so that 2 N S12 · · · Sn2 H N ψ) ≥ C 2 N 2 (ψ, S12 · · · Sn+2 (ψ, H ψ),
(2.13)
for every N ≥ N0 . This, together with (2.4), completes the proof of the proposition. We have the energy estimates as follows. Corollary 1 Suppose that the initial density matrix γN,0 satisfies Tr(HN )k γN,0 ≤ C0k N k ,
(k) N for all k ≥ 1. Let γN,t be the solution of (1.4) and γN,t k=0 corresponding marginal distributions. Then, for any C > 2(C0 + 1) and any k ∈ N , there is N0 = N0 (k, C0 , C), such that (k) TrS1 · · · Sk γN,t Sk · · · S1 ≤ C k , (2.14) for all t ∈ R and all N ≥ N0 .
3
The Main Result (k)
Since our main result states properties of limit points of the sequence γN,t for N → ∞ , in order to formulate it, following [2], we specify a topology on the space of density matrices. Quantum mechanical states of an k-Boson system can be described by a density matrix γ (k) , where γ (k) is a positive, trace class operator with trace normalized to one. We can also identify γ (k) with its kernel and consider it as a function in L2 (R2k × R2k ). In fact, because γ (k) is a positive operator with trace equal to one, its Hilbert Schmidt norm is also bounded by one, that is, 1/2 ||γ (k) ||2 = dxk dxk |γ (k) (xk ; xk )|2 ≤ 1. For Γ = {γ (k) }k≥1 ∈
k≥1
R2k ×R2k
L2 (R2k × R2k ), we define the norm Γ H− =
∞
2−k γ (k) 2
(3.1)
k=1
and set
H− := Γ ∈ L2 (R2k × R2k ) : Γ H− < ∞ . k≥1
(3.2)
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H+ = Γ ∈ L2 (R2k × R2k ) : lim 2k γ (k) 2 = 0 ,
(3.3)
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Also, define
k→∞
k≥1
equipped with the norm Γ H+ = sup{2k γ (k) 2 }.
(3.4)
k≥1
(k) It is checked that H− is the dual space of H+ . Then, a sequence ΓN = γN k≥1 in H−
(k) converges to Γ∞ = γ∞ k≥1 in the weak∗ topology σ(H− , H+ ) if and only if lim
N →∞
∞ k=1
R2k ×R2k
(k) (k) dxk dxk J (k) (xk ; xk ) γN (xk ; xk ) − γ∞ (xk ; xk ) = 0
(3.5)
for every J = {J (k) }k≥1 ∈ H+ (for a bounded sequence this is actually equivalent to the convergence for each fixed k). We will denote by C([0, T ], H− ) the space of functions of t ∈ [0, T ] with values in H− , which are continuous with respect to the topology σ(H− , H+ ) on H− . As the space H+ is separable, we can fix a dense countable subset in the unit ball of H+ , denoted by {Ji }i≥1 . Define the metric ρ on H− by ∞ ∞ (k) = ρ(Γ, Γ) Ji , γ (k) − γ (3.6) 2−i (k) , i=1
k=1
= { for Γ = {γ (k) }k≥1 , Γ γ (k) }k≥1 ∈ H− . Here and afterwards, dxk dxk ψ(xk ; xk )ϕ(xk ; xk ), ψ, ϕ ∈ L2 (R2k × R2k ). ψ, ϕ = R2k ×R2k
Then, the topology induced by ρ(·, ·) and the weak∗ topology σ(H− , H+ ) are equivalent on the unit ball B of H− (see [13], Theorem 3.16). We equip C([0, T ], H− ) with the metric = sup ρ(Γ(t), Γ(t)). ρ(Γ, Γ) 0≤t≤T
(3.7)
We are now ready to state our main theorem. Theorem 1 Assume that the potential V (x) is positive, smooth, compactly supported, and symmetric, that is, V (x) = V (−x) and set Va (x) = a12 V (x/a). Suppose that a = N −ε for some 0 < ε < 3/4. Choose an initial density matrix γN,0 such that Tr(HN )k γN,0 ≤ C k N k
(k) N for some constant C and all k ≥ 1. Let ΓN,0 = γN,0 k=1 be the family of marginal distributions
(k) corresponding to the initial density matrix γN,0 . Fix now T > 0 and denote ΓN,t = γN,t with (k)
γN,t = 0 (k ≥ N + 1) for t ∈ [0, T ], the solution of the BBGKY hierarchy (1.7) corresponding to the initial data ΓN,0 . We have (i) The sequence {ΓN,t}N ≥1 is compact in C([0, T ], H− ) with respect to the metric ρ.
(k) (ii) Let Γ∞,t = γ∞,t k≥1 ∈ C([0, T ], H− ) be any limit point of ΓN,t with respect to the (k)
metric ρ. Then, for every k ≥ 1, γ∞,t ≥ 0 and there is a constant C > 0 such that (k)
TrS1 · · · Sk γ∞,t Sk · · · S1 ≤ C k ,
(3.8)
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for all t ≥ 0.
(iii) Assume that hr (x) = r−2 h(x/r) for any h ∈ C0∞ (R2 ) with R2 h dx = 1. Then, for any k ≥ 1 and t ∈ [0, T ], the limit lim dxk+1 dxk+1 hr (xk+1 − xk+1 )hr (xk+1 − xj ) r,r →0
R2 ×R2 (k+1) ×γ∞,t (xk+1 ; xk+1 )
(k+1)
γ∞,t (xk , xj ; xk , xj )
exists in the following sense: lim r,r →0
R2(k+1) ×R2(k+1)
(3.9)
dxk+1 dxk+1 J (k) (xk ; xk ) (k+1)
×hr (xk+1 − xk+1 )hr (xk+1 − xj )γ∞,t (xk+1 ; xk+1 )
(3.10)
exists for any J (k) ∈ L2 (R2k × R2k ) satisfying |||J (k) |||j < ∞. Here, |||J (k) |||j = sup x1 3 · · · xk 3 x1 3 · · · xk 3 xk ,xk
× |J (k) (xk ; xk )| + |∇xj J (k) (xk ; xk )| + |∇xj J (k) (xk ; xk )| .
J
(k)
(3.11)
(iv) Γ∞,t satisfies the infinite Gross-Pitaevskii (1.8) in the following sense: For any ∈ W 2,2 (R2k × R2k ) with |||J (k) |||j < ∞ for all 1 ≤ j ≤ k, we have
(k)
dxk dxk J (k) (xk ; xk )γ∞,t (xk ; xk )
R2k ×R2k
(k)
dxk dxk J (k) (xk ; xk )γ∞,0 (xk ; xk )
= R2k ×R2k
−i
k j=1
−ib
ds R2k ×R2k
0
k j=1
t
0
(k) dxk dxk γ∞,s (xk ; xk )(−Δxj + Δxj )J (k) (xk ; xk )
t
ds R2k ×R2k
dxk dxk dxk+1 J (k) (xk ; xk )
(k+1) × δ(xj − xk+1 ) − δ(xj − xk+1 ) γ∞,s (xk , xk+1 ; xk , xk+1 ). (k+1)
Here, the action of the δ-functions on γ∞,s of the δ-function in (iii).
(3.12)
is well defined by (3.9) through a regularization
The proof of Theorem 1 will be presented in Section 5.
4
Two Lemmas
In this section, we prove two lemmas. Let 1(|y| ≤ r) be the characteristic function of the set {y ∈ R2 : |y| ≤ r}. Lemma 2 Suppose that δα is a function on R2 satisfying 0 ≤ δα (x) ≤ Cα−2 1(|x| ≤ α)
for some constant C > 0 and δα (x)dx = 1 (for example, δα (x) = α−2 h(x/α) for a bounded probability density h supported in the unit ball of R2 ). Then, if γ (k+1) (xk+1 ; xk+1 ) is the kernel
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function of a density matrix on L2 (R2(k+1) ), for any 1 ≤ j ≤ k, we have dxk+1 dxk+1 J (k) (xk ; xk )γ (k+1) (xk+1 ; xk+1 ) 2(k+1) 2(k+1) R ×R × δα1 (xk+1 − xk+1 ) − δ(xk+1 − xk+1 ) δα2 (xj − xk+1 ) ≤ C k α1 |||J (k) |||j Tr|Sj Sk+1 γ (k+1) Sk+1 Sj |.
(4.1)
Similarly, for every 1 ≤ j ≤ k, we have dxk+1 dxk+1 J (k) (xk ; xk )γ (k+1) (xk+1 ; xk+1 ) 2(k+1) 2(k+1) R ×R × δα (xj − xk+1 ) − δ(xj − xk+1 ) δ(xk+1 − xk+1 ) ≤ C k α1/p |||J (k) |||j Tr|Sj Sk+1 γ (k+1) Sk+1 Sj | and
(4.2)
dxk+1 dxk+1 J (k) (xk ; xk )γ (k+1) (xk+1 ; xk+1 ) × δα (xj − xk+1 ) − δ(xj − xk+1 ) R2(k+1) ×R2(k+1)
≤ C k α1/p |||J (k) |||j Tr|Sj Sk+1 γ (k+1) Sk+1 Sj |,
(4.3)
where constant C is only dependent on 1 < p < 2. The inequalities still hold when xj is replaced by xj in (4.1), (4.2), and (4.3), respectively. Proof (4.1) will be proved only, since (4.2) and (4.3) can be proved similarly. As γ (k+1) is a density matrix, which can be written as a convex combination of pure states, it suffices to prove the results for the case of pure states γ (k+1) (xk+1 ; xk+1 ) = f (xk+1 )f (xk+1 ), where f ∈ L2 (Rk+1 ) so that f 2 = 1. Denote by Π1 the left-hand side of (4.1), we have Π1 ≤ dxk dxk+1 |J (k) (xk ; xk )|δα2 (xj − xk+1 )|f (xk+1 )| (4.4) × dxk+1 δα1 (xk+1 − xk+1 ) f (xk , xk+1 ) − f (xk , xk+1 ) . A standard Poincar´e-type inequality yields dxk+1 δα1 (xk+1 − xk+1 ) f (xk , xk+1 ) − f (xk , xk+1 ) |∇k+1 f (xk , xk+1 + y)| dy, ≤C |y| |y|≤α1 for any xk and xk+1 . Inserting this inequality on the right-hand side of (4.4) and applying the Schwarz inequality, we obtain 1(|y| ≤ α1 ) Π1 ≤ C dxk dxk+1 dy δα2 (xj − xk+1 )|J (k) (xk ; xk )| |y| × |f (xk+1 )|2 + |∇k+1 f (xk , xk+1 + y)|2 dxk+1 δα2 (xj − xk+1 )|f (xk , xk+1 )|2 ≤ Cα1 sup dxk |J (k) (xk ; xk )| xk
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dxk δα2 (xj − xk+1 )|J (k) (xk ; xk )|
sup
xk ,xk+1
dxk dxk+1 dy
1(|y| ≤ α1 ) |∇k+1 f (xk , xk+1 + y)|2 . |y|
In the first term we apply Lemma 1 and in the second term we shift the xk+1 variable, and then we compute the y-integral. Because sup dxk |J (k) (xk ; xk )| ≤ C k |||J (k) |||j xk
and
dxk δα2 (xj − xk+1 )|J (k) (xk ; xk )| ≤ C k |||J (k) |||j
sup
xk ,xk+1
for a universal constant C > 0, we conclude Π1 ≤ C k α1 |||J (k) |||j Tr(1 − Δj )(1 − Δk+1 )γ (k+1) . This proves (4.1). Recall that J (k) , γ (k) =
dxk dxk J (k) (xk ; xk )γ (k) (xk ; xk ).
We have Lemma 3 Fix k ≥ 1 and |||J (k) |||j < ∞ for all 1 ≤ j ≤ k. For α > 0, set δα (x) = (4πα2 )−1 1(|x| ≤ α). Then, k t (k) (k) (k) ds dxk dxk J (k) (xk ; xk )(−Δxj + Δxj )γN,s (xk ; xk ) J (k) , γN,t = J (k) , γN,0 − i −ib
k j=1
t
0
j=1
dxk dxk dxk+1 J (k) (xk ; xk )
ds
0
(k+1) × δα (xj − xk+1 ) − δα (xj − xk+1 ) γN,s (xk , xk+1 ; xk , xk+1 ) 1 (2) sup Tr|S1 S2 γN,s S2 S1 |. +tO k(α1/p + a1/p ) + k 2 N a1/2 s∈[0,t] W
Proof We start with the BBGKY Hierarchy (1.7). (R2k × R2k ), we obtain
(4.5)
After multiplying it by J (k) ∈
1,2
k (k) (k) (k) (k) J , γN,t = J , γN,0 − i
j=1
t
ds 0
i (k) ds dxk dxk J (k) (xk ; xk )γN,s (xk ; xk ) N 1≤j
(k)
dxk dxk J (k) (xk ; xk )(−Δxj + Δxj )γN,s (xk ; xk )
t
(4.6)
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By Lemma 1, we have (Va )1/2 (xj − xl )Sj−1 ≤ Ca−1/2 , Then,
(Va )1/2 (xj − xl )Sj−1 Sl−1 ≤ C.
(k) dxk dxk J (k) (xk ; xk )Va (xj − xl )γN,s (xk ; xk ) ≤ Sj−1 J (k) Sj Sj−1 Va (xj − xl )Sj−1 Sl−1 Sl−1 TrSj Sl γN,s Sj Sl (k)
(2)
≤ Ck a−1/2 TrS1 S2 γN,s S2 S1 , where J (k) (xk ; xk ) = J (k) (xk ; xk ). Here, we have used the fact that Sj−1 J (k) Sj ≤ J (k) Sj and J (k) Sj 2 ≤ J (k) Sj2 (J (k) )∗ ≤ dxk dxk |J (k) (xk ; xk )|2 + |∇j J (k) (xk ; xk )|2 ≤ J (k) 2W 1,2 (R2k ×R2k ) .
(4.7)
Similarly, we have (k) (2) dxk dxk J (k) (xk ; xk )Va (xj − xl )γN,s (xk ; xk ) ≤ Ck a−1/2 TrS1 S2 γN,s S2 S1 , and so, we find k t 1 (k) ds dxk dxk J (k) (xk ; xk )γN,s (xk ; xk ) Va (xj − xl ) − Va (xj − xl ) N 0 j
Ck t (2) sup TrS1 S2 γN,s S2 S1 . ≤ N a1/2 s∈[0,t]
(4.8)
Analogously, we have k t k (k+1) ds dxk dxk dxk+1 J (k) (xk ; xk )γN,s (xk , xk+1 ; xk , xk+1 ) N j=1 0 × Va (xj − xk+1 ) − Va (xj − xk+1 ) ≤
Ck t (2) sup TrS1 S2 γN,s S2 S1 . N a1/2 s∈[0,t]
(4.9)
In contrast, by (4.3) of Lemma 2, we have (k+1) dxk dxk dxk+1 J (k) (xk ; xk )γN,s (xk , xk+1 ; xk , xk+1 ) Va (xj − xk+1 ) − bδα (xj − xk+1 ) (2) ≤ C a1/p + α1/p TrS1 S2 γN,s S2 S1 , (4.10) for some constant C which only depends on J (k) , but is independent of N, α and s ∈ [0, t]. Similarly, we have (k+1) dxk dxk dxk+1 J (k) (xk ; xk )γN,s (xk , xk+1 ; xk , xk+1 ) Va (xj − xk+1 ) − bδα (xj − xk+1 ) (2) ≤ C a1/p + α1/p TrS1 S2 γN,s S2 S1 . (4.11) Therefore, inserting (4.8), (4.9), (4.10), and (4.11) into (4.6), we obtain (4.5).
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Proof of Theorem 1
The proof is divided into three subsections. Compactness of the Sequence ΓN,t The aim of this subsection is to prove Theorem 1 (i), that is, {ΓN,t }N ≥1 is compact in C([0, T ], H− ). By the Arzela-Ascoli theorem, it suffices to prove that {ΓN,t }N ≥1 is equicontinuous in t with respect to the metric ρ.
(k) First of all, we note that the sequence ΓN,t = γN,t k≥1 is uniformly bounded in H− . In
5.1
(k)
fact, since TrγN,t = 1, we have ρ(0, ΓN,t) ≤ ΓN,t H− =
N
(k)
2−k γN,t 2 ≤
k=1
N
(k)
2−k TrγN,t ≤ 1.
(5.1)
k=1
To check the equi-continuity of {ΓN,t}N ≥1 , we need the following lemma (e.g., Lemma 9.2 in 3). (k) Lemma 4 The sequence ΓN,t = {γN,t}k≥1 (N = 1, 2, . . .) satisfying (5.1) is equicontinuous in t with respect to the metric ρ if and only if for every fixed k ≥ 1, for arbitrary J (k) ∈ L2 (R2k × R2k ) with |||J (k) |||j < ∞ (1 ≤ j ≤ k) and for every ε > 0, there exists δ > 0 such that (k) (k) (k) (5.2) J , γN,t − γN,s < ε whenever |t − s| ≤ δ. Proof of Theorem 1 (i) We choose k ≥ 1, J (k) ∈ L2 (R2k × R2k ) with |||J (k) |||j < ∞ (1 ≤ j ≤ k), and ε > 0. By Lemma 3, we have (k) (k) (k) J , γN,t − γN,s k t k t (k) (k) dτ J , (−Δxj + Δxj )γN,τ + b dτ dxk dxk dxk+1 J (k) (xk ; xk ) ≤ j=1
s
j=1
s
(k+1) × δα (xj − xk+1 ) − δα (xj − xk+1 ) γN,τ (xk , xk+1 ; xk , xk+1 ) 1 (2) sup Tr|S1 S2 γN,τ S2 S1 |, +|t − s|O k(α1/p + a1/p ) + k 2 N a1/2 τ ∈[0,t]
(5.3)
whenever t > s. Note that (k) (k) (k) (k) J , (−Δxj + Δxj )γN,τ ≤ C J (k) Sj + J Sj TrSj γN,τ Sj ≤ C, where J
(k)
(xk ; xk ) = J (k) (xk ; xk ), because of Corollary 1 and (4.7). Consequently, k j=1
s
t
(k) dτ J (k) , (−Δxj + Δxj )γN,τ ≤ Ck,J (k) |t − s|.
(5.4)
In contrast, because J (k) ||L2 (R2 ×R2 ) ≤ |||J (k) |||j and the norm of δα is of order α−2 , by Schwarz inequality, we have b
k t dτ dxk dxk dxk+1 J (k) (xk ; xk ) j=1
s
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853
(k+1) × δα (xj − xk+1 ) − δα (xj − xk+1 ) γN,τ (xk , xk+1 ; xk , xk+1 ) ≤ Cα−2
k j=1
s
t
(k+1)
dτ J (k) L2 (R2k ×R2k ) TrγN,τ
≤ Cα,k,J (k) |t − s|.
(5.5)
Inserting (5.4) and (5.5) into (5.3), we obtain (k) (k) (k) J , γN,t − γN,s ≤ C|t − s| where C depends only on α, J (k) , k, but independent of N, t, s. By Lemma 4, the equi-continuity of {ΓN,t}N ≥1 in t is proved. 5.2 A Priori Bounds on Γ∞,t In this subsection, we prove Theorem 1 (ii). To this end, we define a new topology in the space of density matrices following [2]. Denote by L1 (H) and K(H) the spaces of the trace class operators and the space of compact operators on a Hilbert space H, respectively. Set Hk = L2 (R2k ). We define (5.6) Wk = W (k) = S1−1 · · · Sk−1 γ (k) S1−1 · · · Sk−1 : γ (k) ∈ L1 (Hk ) , where Sj = (1 − Δj )1/2 , equipped with W (k) Wk = Tr|S1 · · · Sk W (k) S1 · · · Sk |. We define moreover the space A(k) = A(k) = S1 · · · Sk K (k) S1 · · · Sk : K (k) ∈ K(Hk )
(5.7)
equipped with the norm ||A(k) ||A(k) = ||S1−1 · · · Sk−1 A(k) S1−1 · · · Sk−1 ||, where · denotes the operator norm. It is checked that ∗ (k) A , · A(k) = Wk , · Wk
(5.8)
by the fact that K(Hk )∗ = L1 (Hk ). The identification of Wk as the dual space of A(k) implies the existence of a weak star topology σ(Wk , A(k) ) on Wk .
(k) Proof of Theorem 1 (ii) Let Γ∞,t = γ∞,t k≥1 be a limit point of the sequence
(k) ΓN,t = γN,t k≥1 in the space C([0, T ], H− ) with respect to the metric ρ. By passing to a subsequence we can assume that ΓN,t → Γ∞,t as N → ∞. This implies that, for every fixed t ∈ [0, T ] and for every k ≥ 1, we have, for N → ∞, (k)
(k)
γN,t → γ∞,t
(5.9)
with respect to the weak topology of L2 (R2k × R2k ). This follows because ||ΓN,t ||H− ≤ 1 and in the unit ball, the metric ρ is equivalent to the weak∗ topology of H− . Convergence with respect to the weak∗ topology of H− implies then weak convergence in every k-particle sector (k) L2 (R2k × R2k ). It is concluded from (5.9) that γ∞,t determines a positive operator on L2 (R2k ).
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By Corollary 1, for every k ≥ 1 there exists a constant C > 0 such that (k)
(k)
||γN,t ||Wk = TrS1 · · · Sk γN,t S1 · · · Sk ≤ C k
(5.10) (k)
for every t ∈ [0, T ] and N ≥ 1. By the Banach-Alaoglu Theorem, the sequence γN,t is compact in Wk with respect to the weak∗ topology. In particular, there exists a subsequence Nj → ∞ (k) (k) (k) and γ ∞,t ∈ Wk such that γNj ,t → γ ∞,t and (k)
(k)
γ∞,t Wk = Tr|S1 · · · Sk γ∞,t S1 · · · Sk | ≤ C k .
(5.11)
(k)
So, the sequence γNj ,t satisfies, for j → ∞, (k)
(k)
w.r.t. the weak topology of L2 (R2k × R2k ) and
(k)
(k)
w.r.t. the weak∗ topology of Wk .
γNj ,t → γ∞,t γNj ,t → γ∞,t
(5.12)
If J (k) ∈ L2 (R2k × R2k ), then the operator with kernel given by J (k) (which will be still denoted by J (k) ) is Hilbert-Schmidt and thus compact; in particular, J (k) ∈ A(k) . Thus, if we use (5.12), it is verified that (k) (k) dxk dxk J (k) (xk ; xk )γ∞,t (xk ; xk ) = dxk dxk J (k) (xk ; xk ) γ∞,t (xk ; xk ) (k)
(k)
for every J (k) ∈ L2 (R2k × R2k ). This implies that γ∞,t = γ ∞,t as an element of L2 (R2k × R2k ). Thus, by (5.11) we have, for k ≥ 1, (k)
TrS1 · · · Sk γ∞,t S1 · · · Sk ≤ C k (k)
for every t ∈ [0, T ]. More precisely, we should say that there is a version of γ∞,t ∈ L2 (R2k ×R2k ), (k) which satisfies this bound (the version used here is exactly the density matrix γ ∞,t ). 5.3 Convergence to the Infinite BBGKY Hierarcy Finally, in this subsection we prove the last two parts of Theorem 1. Proof of Theorem 1 (iii) From (4.1) and (4.2) of Lemma 2, it is concluded that (k+1) dxk+1 dxk+1 J (k) (xk ; xk )γ∞,t (xk+1 ; xk+1 ) × hr1 (xk+1 − xk+1 )hr1 (xj − xk+1 ) − hr2 (xk+1 − xk+1 )hr2 (xj − xk+1 ) (k+1) (5.13) ≤ C k |||J (k) |||j r1 + r2 + (r1 )1/p + (r2 )1/p TrSj Sk+1 γ∞,t Sk+1 Sj . By (3.8), we have (k+1)
(k+1)
TrSj Sk+1 γ∞,t Sk+1 Sj ≤ TrS1 · · · Sk+1 γ∞,t Sk+1 · · · S1 ≤ C k+1 . This together with (5.13) implies that the sequence (k+1) dxk+1 dxk+1 J (k) (xk ; xk )γ∞,t (xk+1 ; xk+1 ) × hr (xk+1 − xk+1 )hr (xj − xk+1 ) has the Cauchy property for r, r → 0 and thus is convergent, if |||J (k) |||j < ∞ for any 1 ≤ j ≤ k.
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Proof of Theorem 1 (iv) By Lemma 3 we have, for every J (k) ∈ L2 (R2k × R2k ) with |||J |||j < ∞ for 1 ≤ j ≤ k and for N large enough, (k)
k (k) (k) (k) (k) J , γN,t = J , γN,0 − i
−ib
k t j=1
ds
0
j=1
t
ds 0
(k)
dxk dxk J (k) (xk ; xk )(−Δxj + Δxj )γN,s (xk ; xk )
dxk dxk dxk+1 J (k) (xk ; xk )
(k+1) × δα (xj − xk+1 ) − δα (xj − xk+1 ) γN,s (xk , xk+1 ; xk , xk+1 ) +tO(α1/p ) + to(1),
(5.14)
where o(1) → 0 for N → ∞. By passing to a subsequence, we can assume that ΓN,t → Γ∞,t =
(k) γ∞,t k≥1 ∈ C([0, T ], H− ) with respect to the metric ρ. As shown in (5.9), for every fixed k ≥ 1 (k)
(k)
and t ∈ [0, T ], γN,t → γ∞,t in the weak topology of L2 (R2k × R2k ). Hence, (k) (k) (k) J , γN,t − γ∞,t → 0,
(k) (k) (k) J , γN,0 − γ∞,0 → 0
(5.15)
for N → ∞. Because Δxj J (k) (xk ; xk ) and Δxj J (k) (xk ; xk ) are both elements of L2 (R2k × R2k ), this implies that k
(k) (k) dxk dxk (−Δxj + Δxj )J (k) (xk ; xk ) γN,s (xk ; xk ) − γ∞,s (xk ; xk ) → 0
j=1
as N → ∞, uniformly for s ∈ [0, T ]. By the Lebesgue domination theorem, we have, for every fixed t ∈ [0, T ], k j=1
=
ds
0
k j=1
t
0
t
ds
(k) (k) dxk dxk J (k) (xk ; xk ) × (−Δxj + Δxj ) γN,s (xk ; xk ) − γ∞,s (xk ; xk ) dxk dxk −Δxj J (k) (xk ; xk ) + Δxj J (k) (xk ; xk )
(k) (k) (xk ; xk ) → 0 × γN,s (xk ; xk ) − γ∞,s
(5.16)
as N → ∞. Finally, we consider the last term of the right-hand side of (5.15). By (4.2) of Lemma 2, we have (k+1) dxk dxk+1 J (k) (xk ; xk ) δα (xj − xk+1 ) − δα (xj − xk+1 ) γN,s (xk , xk+1 ; xk , xk+1 ) = dxk+1 dxk+1 J (k) (xk ; xk ) δα (xj − xk+1 ) − δα (xj − xk+1 ) (k+1)
×δη (xk+1 − xk+1 )γN,s (xk+1 ; xk+1 ) + O(η),
(5.17)
where O(η) is independent of α, N and s. At this point we can take the limit N → ∞ with fixed α and η. As J (k) ∈ L2 (R2k × R2k ), it is concluded that, for fixed α, η > 0, J (k) (xk ; xk )δα (xj −
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xk+1 )δη (xk+1 − xk+1 ) ∈ L2 (R2(k+1) × R2(k+1) ). Hence, dxk+1 dxk+1 J (k) (xk ; xk ) δα (xj − xk+1 ) − δα (xj − xk+1 ) (k+1) (k+1) (xk+1 ; xk+1 ) → 0 ×δη (xk+1 − xk+1 ) γN,s (xk+1 ; xk+1 ) − γ∞,s
Vol.30 Ser.B
(5.18)
as N → ∞, uniformly in s ∈ [0, T ]. By (5.15), (5.16), (5.17), and (5.18), it follows from (5.14) that k (k) (k) (k) (k) J , γ∞,t = J , γ∞,0 − i
−ib
k t j=1
0
ds
j=1
0
t
(k) ds J (k) , (−Δj + Δj )γ∞,s
dxk dxk dxk+1 dxk+1 J (k) (xk ; xk )
× δα (xj − xk+1 ) − δα (xj − xk+1 ) δη (xk+1 − xk+1 ) (k+1) ×γ∞,s (xk , xk+1 ; xk , xk+1 ) + O(α1/p + η).
(5.19)
for any fixed t and k. Finally, we apply (4.1) and (4.2) of Lemma 2 to replace δη (xk+1 −xk+1 ) by δ(xk+1 −xk+1 ), and δα (xj −xk+1 ) (or δα (xj −xk+1 )) by δ(xj −xk+1 )(respectively, δ(xj −xk+1 )). The error here is of order α1/p + η. Hence, letting η → 0 and α → 0, we obtain (3.12). Acknowledgments We are grateful to Prof.Zeqian Chen and Dr.Chengjun He for helpful discussions and useful suggestions. Also, Prof.Chen has made some improvements of the draft. References [1] Dalfovo F, Giorgini S, Pitaevskii L P et al. Theory of Bose-Einstein condensation in trapped gases. Reviews of Modern Phys, 1999, 71: 463–512 [2] Elgart A, Erd¨ os L, Schlein B et al. Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons. Arch. Rat. Mech. Anal, 2006, 179(2): 265–283 [3] Erd¨ os L, Schlein B, Yau H T. Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate. Commun Pure Appl Math, 2006, 59(12): 1659–1741 [4] Erd¨ os L, Schlein B, Yau H T. Derivation of the cubic non-linear Schr¨ odinger equation from quantum dynamics of many-body systems. Invent Math, 2007, 167: 515–614 [5] Erd¨ os L, Schlein B, Yau H T. Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. arXiv:math-ph/0606017 v3 [6] Erd¨ os L, Yau H T. Derivation of the nonlinear Schr¨ odinger equation from a many body Coulomb system. Adv Theor Math Phys, 2001, 5(6): 1169–1205 [7] Gross E P. Structure of a quantized vortex in boson systems. Nuovo Cimento, 1961, 20: 454–466 [8] Gross E P. Hydrodynamics of a superfluid condensate. J Math Phys, 1963, 4: 195–207 [9] Lieb E H, Seiringer R. Proof of Bose-Einstein condensation for dilute trapped gases. Phys Rev Lett, 2002, 88: 170409 [10] Lieb E H, Seiringer R, Yngvason J. Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional. Phys Rev A, 2000, 61: 043602 [11] Lieb E H, Seiringer R, Yngvason J. A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas. Comm Math Phys, 2001, 224: 17–31 [12] Pitaevskii L P. Vortex lines in an imperfect Bose gas. Sov Phys JETP, 1961, 13: 451–454 [13] Rudin W. Functional Analysis. 2nd ed. Beijing: China Machine Press, 2004: 70–70
A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS∗
)
Liu Chuangye (
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan 430071, China E-mail: [email protected]
Abstract In this article, we consider the dynamics of N two-dimensional boson systems interacting through a pair potential N −1 Va (xi − xj ) where Va (x) = a−2 V (x/a). It is well known that the Gross-Pitaevskii (GP) equation is a nonlinear Schr¨ odinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k-particle density matrices { k ut , k ≥ 1} solves the GP hierarchy. Denote by ψN,t the solution to the N -particle Schr¨ odinger equation. Under the assumption that a = N −ε for 0 < ε < 3/4, we prove that as N → ∞ the limit points of the k-particle density matrices of ψN,t are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by V (x) dx.
Æ
Ê
Key words Gross-Pitaevskii equation; Boson system; density matrix; BBGKY hierarchy 2000 MR Subject Classification
1
35Q40; 35Q55
Introduction
Motivated by recent experimental realizations of Bose-Einstein condensation the theory of dilute, inhomogeneous Bose systems was currently a subject of intensive studies [1]. The ground state of bosonic atoms in a trap was shown experimentally to display Bose-Einstein condensation (BEC). This fact was proved theoretically by Lieb [9, 10, 11] for bosons with twobody repulsive interaction potentials in the dilute limit, starting from the basic Schr¨ odinger equation. On the other hand, it is well known that the dynamics of Bose-Einstein condensates are well described by the the Gross-Pitaevskii equation. A rigorous derivation of this equation from the basic many-body Schr¨ odinger equation in an appropriate limit is not a simple matter, however, and has only been achieved recently in three spatial dimension [2, 3, 4, 5]. This article is concerned with the justification of the Gross-Pitaevskii equation in two spatial dimensions. In this case, several new issues arise and we refer to [11] for details. ∗ Received
Sepetember 11, 2007; revised April 7, 2008. This work is partially supported by NSFC (10571176)
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Consider N bosons in the two-dimension space. The bosons interact via a two body potential Va (x) = a−2 V (x/a). (1.1) We assume that the potential V is smooth, symmetric, and positive, with compact support. The parameter a determines the range and the strength of the potential: a and N will be coupled so that a → 0 as N → ∞. Thus the potential Va converges to Dirac δ-function. The N -body Hamiltonian for the N weakly coupled bosons is given by HN = −
N j=1
2
Δj +
1 Va (xi − xj ), N i
(1.2)
2
∂ ∂ 2 where Δj = Δxj = ∂α 2 + ∂β 2 for xj = (αj , βj ) ∈ R . The dynamics of the Bose system is j j governed by the N -body Schr¨ odinger equation
i∂t ψN,t = HN ψN,t .
(1.3)
Here, the wave function ψN,t lies in L2s (R2N ) which is the subspace of L2 (R2N ) consisting of functions symmetric with respect to permutation of the N particles. More generally, we can describe the N -body system by its density matrix γN,t . The density matrix is a positive self-adjoint operator γ acting on L2s (R2N ) with Trγ = 1. The density matrix corresponding to the wave function ψN,t is given by the one-dimensional orthogonal projection onto ψN,t , that is, γN,t = |ψN,t ψN,t |. Quantum mechanical states described by one-dimensional orthogonal projections are called pure states. In general, a density matrix (mixed states) is a weighted average of the one-dimensional orthogonal projections. The time evolution of the density matrix γN,t is then given by i∂t γN,t = [HN , γN,t ], (1.4) which is equivalent to the Schr¨odinger equation (1.3). As follows, we denote by x a general variable in R2 and by x = (x1 , · · · , xN ) a point in R2N . We will also use the notation xk = (x1 , · · · , xk ) ∈ R2k and xN −k = (xk+1 , · · · , xN ) ∈ R2(N −k) . (k) For k = 1, · · · , N − 1, the k-particle marginal distribution γN,t of γN,t is defined through its kernel by (k)
γN,t (xk , xk ) =
dxN −k γN,t (xk , xN −k ; xk , xN −k ),
(1.5)
where xk = (x1 , · · · , xk ) and γN,t (x; x ) denotes the kernel of the density matrix γN,t . From (k) TrγN,t = 1, it is concluded that TrγN,t = 1 for every k = 1, · · · , N − 1. In the sequel, we set (N )
(k)
γN,t = γN,t and γN,t = 0, k > N for convenience. From (1.4) and the symmetry of γN,t with respect to permutations of the N particles, we conclude that the evolution of the marginal distributions of γN,t is determined by the following hierarchy of N equations, commonly called the BBGKY hierarchy, (k)
i∂t γN,t (xk ; xk ) =
k (k) −Δxj + Δxj γN,t (xk ; xk ) j=1
+
1 N
1≤j
(k) Va (xj − xl ) − Va (xj − xl ) γN,t (xk ; xk )
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k k + 1− dxk+1 Va (xj − xk+1 ) − Va (xj − xk+1 ) N j=1 R (k+1)
×γN,t (xk , xk+1 ; xk , xk+1 )
(1.6)
for k = 1, · · · , N. Rewriting this hierarchy in integral form, we have (k)
(k)
γN,t (xk ; xk ) = γN,0 (xk ; xk ) − i −
i N
1≤j
t
0
k
t
0
j=1
(k) ds −Δxj + Δxj γN,s (xk ; xk )
(k) ds Va (xj − xl ) − Va (xj − xl ) γN,s (xk ; xk )
k k t ds dxk+1 Va (xj − xk+1 ) − Va (xj − xk+1 ) −i 1 − N j=1 0 R2 (k+1)
×γN,s (xk , xk+1 ; xk , xk+1 ).
(1.7)
Setting a = N −ε and letting N → ∞, one has that the BBGKY hierarchy converges formally to the following infinite hierarchy of equations (k)
(k)
γt (xk ; xk ) = γ0 (xk ; xk ) − i −ib
k t j=1
0
k j=1
ds R2
0
t
ds −Δxj + Δxj γs(k) (xk ; xk )
dxk+1 δ(xj − xk+1 ) − δ(xj − xk+1 )
×γs(k+1) (xk , xk+1 ; xk , xk+1 )
(1.8)
for k = 1, 2, · · · , where b = R2 V (x)tdx. Eq.(1.8) is said to be the infinite BBGKY hierarchy, or the Gross-Pitaevskii (GP) hierarchy. It turns out that (1.8) has a factorized solution. Specially, k the family of marginal distribution γ t (k) (xk , xk ) = φt (xj )φt (xj ) is a solution of (1.8) if and j=1
only if the function φt satisfies the nonlinear Schr¨ odinger equation i∂t φt = −Δφt + b|φt |2 φt .
(1.9)
This is the Gross-Pitaevskii (GP) equation [7, 8, 12], except that the coupling constant in front of the nonlinear interaction is given by b. The aim of this article is to prove the convergence of solutions of (1.7) to ones of (1.8). More
(k) N precisely, we will prove that for every 0 < ε < 3/4 and a = N −ε , the sequence ΓN,t = γN,t k=1
(k) has at least as N → ∞ one limit point Γ∞,t = γ∞,t k≥1 with respect to some weak topology, and that any weak limit point Γ∞,t satisfies the infinite hierarchy (1.8). The remainder of this article is divided into four sections. In Section 2 we prove some Sobolev-type inequalities and energy estimates. Some notations and the main result are presented in Section 3. Section 4 is devoted to proofs of two lemmas, which play a crucial role in the proof of the main result. Finally, in Section 5, the main result is proved.
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Energy Estimates We begin with the following Sobolev-type inequalities. Lemma 1 (i) Suppose that V ∈ Lp (R2 ), 1 < p ≤ ∞, and φ ∈ H 1 (R2 ). Then, 2 |φ(x)|2 |Va (x)|dx ≤ C(p)a p −2 ||V ||Lp (R2 ) ||φ||2H 1 (R2 ) .
(2.1)
R2
(ii) Let V ∈ L1 (R2 ) be a nonnegative function. Then, there is an absolute constant C > 0 such that Va (x − y)|φ(x, y)|2 dxdy ≤ C V L1 (R2 ) (1 − Δx )(1 − Δy )φ, φ (2.2) R2 ×R2
for all φ ∈ H 2 (R2 × R2 ). The following proposition plays a key role in the proof of the main result, which presents bounds for the L2 -norm of the derivatives of a wave function ψ in terms of the mean value of powers of the Hamiltonian HN in the state described by ψ. Proposition 1 Suppose that the potential V (x) is positive, smooth, compactly supported, and symmetric, that is, V (x) = V (−x). Set Va (x) = a12 V (x/a) and assume that a = N −ε , with 0 < ε < 3/4. Put N = H
N
Sj2 +
j=1
1 N
Va (xl − xm ) = HN + N.
1≤l
Here and in the sequel, Sj = (1 − Δj )1/2 for j = 1, · · · , N. Fix k ∈ N and 0 < C < 1. Then, there is N0 = N0 (k, C) such that N )k ψ) ≥ C k N k (ψ, S12 S22 · · · Sk2 ψ) (ψ, (H
(2.3)
for all N > N0 and all ψ ∈ D((HN )k ) (ψ is assumed to be symmetric with respect to any permutation of all its variables). Proof The proof of the proposition uses a two step induction over k. For k = 0 and k = 1, the claim is trivial because of the positivity of the potential and the symmetry of ψ. Now, we assume the proposition is true for all k ≤ n, and we prove it for k = n + 2. To this end, we apply the induction assumption and we find, for N > N0 (n, C), N (H N )n H N S 2 · · · S 2 H N ψ) ≥ C n N n (ψ, H N )n+2 ψ) = (ψ, H (ψ, (H 1 n N ψ). Set H (n) =
n j=1
Sj2 +
1 N
N − Vjm = H
N
(2.4)
Sj2 ,
j=n+1
1≤j
with Vjm = Va (xj − xm ). Then, we have
N ψ) ≥ N S12 · · · Sn2 H (ψ, H
(ψ, Sj21 S12 · · · Sn2 Sj22 ψ)
N ≥j1 ,j2 ≥n+1
+
(ψ, Sj2 S12 · · · Sn2 H (n) ψ) + c.c.
N ≥j≥n+1
=: I1 + I2
(2.5)
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because of H (n) S12 · · · Sn2 H (n) ≥ 0. Here and afterwards, c.c. denotes the complex conjugate of the term followed. For I1 and I2 , using the symmetry with respect to permutations, we have 2 2 2 Sn+2 ψ) + (N − n)(ψ, S14 S22 · · · Sn+1 ψ), I1 = (N − n)(N − n − 1)(ψ, S12 · · · Sn2 Sn+1
(2.6)
and 2 ψ) I2 ≥ 2n(N − n)(ψ, S14 S22 · · · Sn2 Sn+1 n(n + 1)(N − n) 2 (ψ, V12 S12 · · · Sn2 Sn+1 + ψ) + c.c. 2N (n + 1)(N − n − 1)(N − n) 2 (ψ, V1,n+2 S12 · · · Sn2 Sn+1 ψ) + c.c. . + N
(2.7)
Here, we have used the fact that 2 ψ) ≥ 0, (ψ, Vjm S12 . . . Sn+1
if j, m > n + 1, because of the positivity of the potential. Then, combining with (2.5), (2.6), and (2.7), we get N S12 · · · Sn2 H N ψ) (ψ, H 2 2 2 ≥ (N − n)(N − n − 1)(ψ, S12 · · · Sn2 Sn+1 Sn+2 ψ) + (2n + 1)(N − n)(ψ, S14 S22 · · · Sn2 Sn+1 ψ) n(n + 1)(N − n) 2 (ψ, V12 S12 · · · Sn2 Sn+1 + ψ) + c.c. 2N (n + 1)(N − n − 1)(N − n) 2 (ψ, V1,n+2 S12 · · · Sn2 Sn+1 ψ) + c.c. . (2.8) + N
Next, we consider the last two terms on the right-hand of (2.8). Setting ϕ = S3 . . . Sn+1 ψ, we have 2 (ψ, V12 S12 · · · Sn2 Sn+1 ψ) + c.c. ≥ 2(ϕ, V12 p22 ϕ) + (ϕ, V12 p21 p22 ϕ) + c.c. where p1 = −i∇1 , p2 = −i∇2 , and hence, 2 ψ) + c.c. (ψ, V12 S12 · · · Sn2 Sn+1
≥ 2(ϕ, −∇V12 ∇2 ϕ) + (∇2 ϕ, ∇V12 ∇1 ∇2 ϕ) + (ϕ, −∇V12 p21 ∇2 ϕ) + c.c., where ∇V12 = a−3 (∇V )((x1 − x2 )/a). Applying the Schwarz inequality and Lemma 1, we find that 2 (ψ, V12 S12 · · · Sn+1 ψ) + c.c. 2 p −3 (∇ ϕ, S 2 ∇ ϕ) ≥ −C α1 a−1 (ϕ, S12 S22 ϕ) + α−1 2 1 2 1 a 2 2 p −3 (∇ ∇ ϕ, S 2 ∇ ∇ ϕ) −C α2 a p −3 (∇2 ϕ, S12 ∇2 ϕ) + α−1 a 1 2 1 2 1 2 −1 −3 −1 2 2 2 2 −C α3 a (ϕ, S1 S2 ϕ) + α3 a (∇1 ∇2 ϕ, ∇1 ∇2 ϕ) .
(2.9)
Optimizing the choices of α1 , α2 , and α3 , we can conclude that
2 2 2 ψ) + c.c. ≥ −CN −β N 2 (ψ, S12 · · · Sn+1 ψ) + N (ψ, S14 · · · Sn+1 ψ) (ψ, V12 S12 · · · Sn+1
2 2 ψ) + N (ψ, S14 · · · Sn+1 ψ) . (2.10) ≥ −CN −β N 2 (ψ, S12 · · · Sn+2
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In contrast, letting φ = S2 · · · Sn+1 ψ, we conclude from Lemma 1 that 1
2 2 (ψ, V1,n+2 S12 · · · Sn+1 ψ) + c.c. = (φ, V1,n+2 S12 φ) + c.c. ≥ −Ca p −2 (ψ, S12 · · · Sn+2 ψ).
(2.11)
Inserting (2.10) and (2.11) into the right-hand side of (2.8), we obtain N S 2 · · · S 2 H (ψ, H 1 n N ψ) ≥ (N − n)(N − n − 1) 1 −
C 2− p1
−
C 2 (ψ, S12 · · · Sn+2 ψ) Nβ
Na C 2 +(2n + 1)(N − n) 1 − β (ψ, S14 S22 · · · Sn+1 ψ). N
(2.12) 1
For 0 < ε < 3/4 and 1 < p < 32 , one has β > 0, 1 − ε(2 − 1p ) > 0. Consequently, N a2− p 1 and N β 1. For any fixed C < 1 and n ∈ N , by (2.12) we can find N0 so that 2 N S12 · · · Sn2 H N ψ) ≥ C 2 N 2 (ψ, S12 · · · Sn+2 (ψ, H ψ),
(2.13)
for every N ≥ N0 . This, together with (2.4), completes the proof of the proposition. We have the energy estimates as follows. Corollary 1 Suppose that the initial density matrix γN,0 satisfies Tr(HN )k γN,0 ≤ C0k N k ,
(k) N for all k ≥ 1. Let γN,t be the solution of (1.4) and γN,t k=0 corresponding marginal distributions. Then, for any C > 2(C0 + 1) and any k ∈ N , there is N0 = N0 (k, C0 , C), such that (k) TrS1 · · · Sk γN,t Sk · · · S1 ≤ C k , (2.14) for all t ∈ R and all N ≥ N0 .
3
The Main Result (k)
Since our main result states properties of limit points of the sequence γN,t for N → ∞ , in order to formulate it, following [2], we specify a topology on the space of density matrices. Quantum mechanical states of an k-Boson system can be described by a density matrix γ (k) , where γ (k) is a positive, trace class operator with trace normalized to one. We can also identify γ (k) with its kernel and consider it as a function in L2 (R2k × R2k ). In fact, because γ (k) is a positive operator with trace equal to one, its Hilbert Schmidt norm is also bounded by one, that is, 1/2 ||γ (k) ||2 = dxk dxk |γ (k) (xk ; xk )|2 ≤ 1. For Γ = {γ (k) }k≥1 ∈
k≥1
R2k ×R2k
L2 (R2k × R2k ), we define the norm Γ H− =
∞
2−k γ (k) 2
(3.1)
k=1
and set
H− := Γ ∈ L2 (R2k × R2k ) : Γ H− < ∞ . k≥1
(3.2)
Liu: DERIVATION OF GROSS-PETAEVSKII HIERARCHY
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H+ = Γ ∈ L2 (R2k × R2k ) : lim 2k γ (k) 2 = 0 ,
(3.3)
No.3
Also, define
k→∞
k≥1
equipped with the norm Γ H+ = sup{2k γ (k) 2 }.
(3.4)
k≥1
(k) It is checked that H− is the dual space of H+ . Then, a sequence ΓN = γN k≥1 in H−
(k) converges to Γ∞ = γ∞ k≥1 in the weak∗ topology σ(H− , H+ ) if and only if lim
N →∞
∞ k=1
R2k ×R2k
(k) (k) dxk dxk J (k) (xk ; xk ) γN (xk ; xk ) − γ∞ (xk ; xk ) = 0
(3.5)
for every J = {J (k) }k≥1 ∈ H+ (for a bounded sequence this is actually equivalent to the convergence for each fixed k). We will denote by C([0, T ], H− ) the space of functions of t ∈ [0, T ] with values in H− , which are continuous with respect to the topology σ(H− , H+ ) on H− . As the space H+ is separable, we can fix a dense countable subset in the unit ball of H+ , denoted by {Ji }i≥1 . Define the metric ρ on H− by ∞ ∞ (k) = ρ(Γ, Γ) Ji , γ (k) − γ (3.6) 2−i (k) , i=1
k=1
= { for Γ = {γ (k) }k≥1 , Γ γ (k) }k≥1 ∈ H− . Here and afterwards, dxk dxk ψ(xk ; xk )ϕ(xk ; xk ), ψ, ϕ ∈ L2 (R2k × R2k ). ψ, ϕ = R2k ×R2k
Then, the topology induced by ρ(·, ·) and the weak∗ topology σ(H− , H+ ) are equivalent on the unit ball B of H− (see [13], Theorem 3.16). We equip C([0, T ], H− ) with the metric = sup ρ(Γ(t), Γ(t)). ρ(Γ, Γ) 0≤t≤T
(3.7)
We are now ready to state our main theorem. Theorem 1 Assume that the potential V (x) is positive, smooth, compactly supported, and symmetric, that is, V (x) = V (−x) and set Va (x) = a12 V (x/a). Suppose that a = N −ε for some 0 < ε < 3/4. Choose an initial density matrix γN,0 such that Tr(HN )k γN,0 ≤ C k N k
(k) N for some constant C and all k ≥ 1. Let ΓN,0 = γN,0 k=1 be the family of marginal distributions
(k) corresponding to the initial density matrix γN,0 . Fix now T > 0 and denote ΓN,t = γN,t with (k)
γN,t = 0 (k ≥ N + 1) for t ∈ [0, T ], the solution of the BBGKY hierarchy (1.7) corresponding to the initial data ΓN,0 . We have (i) The sequence {ΓN,t}N ≥1 is compact in C([0, T ], H− ) with respect to the metric ρ.
(k) (ii) Let Γ∞,t = γ∞,t k≥1 ∈ C([0, T ], H− ) be any limit point of ΓN,t with respect to the (k)
metric ρ. Then, for every k ≥ 1, γ∞,t ≥ 0 and there is a constant C > 0 such that (k)
TrS1 · · · Sk γ∞,t Sk · · · S1 ≤ C k ,
(3.8)
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for all t ≥ 0.
(iii) Assume that hr (x) = r−2 h(x/r) for any h ∈ C0∞ (R2 ) with R2 h dx = 1. Then, for any k ≥ 1 and t ∈ [0, T ], the limit lim dxk+1 dxk+1 hr (xk+1 − xk+1 )hr (xk+1 − xj ) r,r →0
R2 ×R2 (k+1) ×γ∞,t (xk+1 ; xk+1 )
(k+1)
γ∞,t (xk , xj ; xk , xj )
exists in the following sense: lim r,r →0
R2(k+1) ×R2(k+1)
(3.9)
dxk+1 dxk+1 J (k) (xk ; xk ) (k+1)
×hr (xk+1 − xk+1 )hr (xk+1 − xj )γ∞,t (xk+1 ; xk+1 )
(3.10)
exists for any J (k) ∈ L2 (R2k × R2k ) satisfying |||J (k) |||j < ∞. Here, |||J (k) |||j = sup x1 3 · · · xk 3 x1 3 · · · xk 3 xk ,xk
× |J (k) (xk ; xk )| + |∇xj J (k) (xk ; xk )| + |∇xj J (k) (xk ; xk )| .
J
(k)
(3.11)
(iv) Γ∞,t satisfies the infinite Gross-Pitaevskii (1.8) in the following sense: For any ∈ W 2,2 (R2k × R2k ) with |||J (k) |||j < ∞ for all 1 ≤ j ≤ k, we have
(k)
dxk dxk J (k) (xk ; xk )γ∞,t (xk ; xk )
R2k ×R2k
(k)
dxk dxk J (k) (xk ; xk )γ∞,0 (xk ; xk )
= R2k ×R2k
−i
k j=1
−ib
ds R2k ×R2k
0
k j=1
t
0
(k) dxk dxk γ∞,s (xk ; xk )(−Δxj + Δxj )J (k) (xk ; xk )
t
ds R2k ×R2k
dxk dxk dxk+1 J (k) (xk ; xk )
(k+1) × δ(xj − xk+1 ) − δ(xj − xk+1 ) γ∞,s (xk , xk+1 ; xk , xk+1 ). (k+1)
Here, the action of the δ-functions on γ∞,s of the δ-function in (iii).
(3.12)
is well defined by (3.9) through a regularization
The proof of Theorem 1 will be presented in Section 5.
4
Two Lemmas
In this section, we prove two lemmas. Let 1(|y| ≤ r) be the characteristic function of the set {y ∈ R2 : |y| ≤ r}. Lemma 2 Suppose that δα is a function on R2 satisfying 0 ≤ δα (x) ≤ Cα−2 1(|x| ≤ α)
for some constant C > 0 and δα (x)dx = 1 (for example, δα (x) = α−2 h(x/α) for a bounded probability density h supported in the unit ball of R2 ). Then, if γ (k+1) (xk+1 ; xk+1 ) is the kernel
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function of a density matrix on L2 (R2(k+1) ), for any 1 ≤ j ≤ k, we have dxk+1 dxk+1 J (k) (xk ; xk )γ (k+1) (xk+1 ; xk+1 ) 2(k+1) 2(k+1) R ×R × δα1 (xk+1 − xk+1 ) − δ(xk+1 − xk+1 ) δα2 (xj − xk+1 ) ≤ C k α1 |||J (k) |||j Tr|Sj Sk+1 γ (k+1) Sk+1 Sj |.
(4.1)
Similarly, for every 1 ≤ j ≤ k, we have dxk+1 dxk+1 J (k) (xk ; xk )γ (k+1) (xk+1 ; xk+1 ) 2(k+1) 2(k+1) R ×R × δα (xj − xk+1 ) − δ(xj − xk+1 ) δ(xk+1 − xk+1 ) ≤ C k α1/p |||J (k) |||j Tr|Sj Sk+1 γ (k+1) Sk+1 Sj | and
(4.2)
dxk+1 dxk+1 J (k) (xk ; xk )γ (k+1) (xk+1 ; xk+1 ) × δα (xj − xk+1 ) − δ(xj − xk+1 ) R2(k+1) ×R2(k+1)
≤ C k α1/p |||J (k) |||j Tr|Sj Sk+1 γ (k+1) Sk+1 Sj |,
(4.3)
where constant C is only dependent on 1 < p < 2. The inequalities still hold when xj is replaced by xj in (4.1), (4.2), and (4.3), respectively. Proof (4.1) will be proved only, since (4.2) and (4.3) can be proved similarly. As γ (k+1) is a density matrix, which can be written as a convex combination of pure states, it suffices to prove the results for the case of pure states γ (k+1) (xk+1 ; xk+1 ) = f (xk+1 )f (xk+1 ), where f ∈ L2 (Rk+1 ) so that f 2 = 1. Denote by Π1 the left-hand side of (4.1), we have Π1 ≤ dxk dxk+1 |J (k) (xk ; xk )|δα2 (xj − xk+1 )|f (xk+1 )| (4.4) × dxk+1 δα1 (xk+1 − xk+1 ) f (xk , xk+1 ) − f (xk , xk+1 ) . A standard Poincar´e-type inequality yields dxk+1 δα1 (xk+1 − xk+1 ) f (xk , xk+1 ) − f (xk , xk+1 ) |∇k+1 f (xk , xk+1 + y)| dy, ≤C |y| |y|≤α1 for any xk and xk+1 . Inserting this inequality on the right-hand side of (4.4) and applying the Schwarz inequality, we obtain 1(|y| ≤ α1 ) Π1 ≤ C dxk dxk+1 dy δα2 (xj − xk+1 )|J (k) (xk ; xk )| |y| × |f (xk+1 )|2 + |∇k+1 f (xk , xk+1 + y)|2 dxk+1 δα2 (xj − xk+1 )|f (xk , xk+1 )|2 ≤ Cα1 sup dxk |J (k) (xk ; xk )| xk
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dxk δα2 (xj − xk+1 )|J (k) (xk ; xk )|
sup
xk ,xk+1
dxk dxk+1 dy
1(|y| ≤ α1 ) |∇k+1 f (xk , xk+1 + y)|2 . |y|
In the first term we apply Lemma 1 and in the second term we shift the xk+1 variable, and then we compute the y-integral. Because sup dxk |J (k) (xk ; xk )| ≤ C k |||J (k) |||j xk
and
dxk δα2 (xj − xk+1 )|J (k) (xk ; xk )| ≤ C k |||J (k) |||j
sup
xk ,xk+1
for a universal constant C > 0, we conclude Π1 ≤ C k α1 |||J (k) |||j Tr(1 − Δj )(1 − Δk+1 )γ (k+1) . This proves (4.1). Recall that J (k) , γ (k) =
dxk dxk J (k) (xk ; xk )γ (k) (xk ; xk ).
We have Lemma 3 Fix k ≥ 1 and |||J (k) |||j < ∞ for all 1 ≤ j ≤ k. For α > 0, set δα (x) = (4πα2 )−1 1(|x| ≤ α). Then, k t (k) (k) (k) ds dxk dxk J (k) (xk ; xk )(−Δxj + Δxj )γN,s (xk ; xk ) J (k) , γN,t = J (k) , γN,0 − i −ib
k j=1
t
0
j=1
dxk dxk dxk+1 J (k) (xk ; xk )
ds
0
(k+1) × δα (xj − xk+1 ) − δα (xj − xk+1 ) γN,s (xk , xk+1 ; xk , xk+1 ) 1 (2) sup Tr|S1 S2 γN,s S2 S1 |. +tO k(α1/p + a1/p ) + k 2 N a1/2 s∈[0,t] W
Proof We start with the BBGKY Hierarchy (1.7). (R2k × R2k ), we obtain
(4.5)
After multiplying it by J (k) ∈
1,2
k (k) (k) (k) (k) J , γN,t = J , γN,0 − i
j=1
t
ds 0
i (k) ds dxk dxk J (k) (xk ; xk )γN,s (xk ; xk ) N 1≤j
(k)
dxk dxk J (k) (xk ; xk )(−Δxj + Δxj )γN,s (xk ; xk )
t
(4.6)
No.3
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By Lemma 1, we have (Va )1/2 (xj − xl )Sj−1 ≤ Ca−1/2 , Then,
(Va )1/2 (xj − xl )Sj−1 Sl−1 ≤ C.
(k) dxk dxk J (k) (xk ; xk )Va (xj − xl )γN,s (xk ; xk ) ≤ Sj−1 J (k) Sj Sj−1 Va (xj − xl )Sj−1 Sl−1 Sl−1 TrSj Sl γN,s Sj Sl (k)
(2)
≤ Ck a−1/2 TrS1 S2 γN,s S2 S1 , where J (k) (xk ; xk ) = J (k) (xk ; xk ). Here, we have used the fact that Sj−1 J (k) Sj ≤ J (k) Sj and J (k) Sj 2 ≤ J (k) Sj2 (J (k) )∗ ≤ dxk dxk |J (k) (xk ; xk )|2 + |∇j J (k) (xk ; xk )|2 ≤ J (k) 2W 1,2 (R2k ×R2k ) .
(4.7)
Similarly, we have (k) (2) dxk dxk J (k) (xk ; xk )Va (xj − xl )γN,s (xk ; xk ) ≤ Ck a−1/2 TrS1 S2 γN,s S2 S1 , and so, we find k t 1 (k) ds dxk dxk J (k) (xk ; xk )γN,s (xk ; xk ) Va (xj − xl ) − Va (xj − xl ) N 0 j
Ck t (2) sup TrS1 S2 γN,s S2 S1 . ≤ N a1/2 s∈[0,t]
(4.8)
Analogously, we have k t k (k+1) ds dxk dxk dxk+1 J (k) (xk ; xk )γN,s (xk , xk+1 ; xk , xk+1 ) N j=1 0 × Va (xj − xk+1 ) − Va (xj − xk+1 ) ≤
Ck t (2) sup TrS1 S2 γN,s S2 S1 . N a1/2 s∈[0,t]
(4.9)
In contrast, by (4.3) of Lemma 2, we have (k+1) dxk dxk dxk+1 J (k) (xk ; xk )γN,s (xk , xk+1 ; xk , xk+1 ) Va (xj − xk+1 ) − bδα (xj − xk+1 ) (2) ≤ C a1/p + α1/p TrS1 S2 γN,s S2 S1 , (4.10) for some constant C which only depends on J (k) , but is independent of N, α and s ∈ [0, t]. Similarly, we have (k+1) dxk dxk dxk+1 J (k) (xk ; xk )γN,s (xk , xk+1 ; xk , xk+1 ) Va (xj − xk+1 ) − bδα (xj − xk+1 ) (2) ≤ C a1/p + α1/p TrS1 S2 γN,s S2 S1 . (4.11) Therefore, inserting (4.8), (4.9), (4.10), and (4.11) into (4.6), we obtain (4.5).
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Proof of Theorem 1
The proof is divided into three subsections. Compactness of the Sequence ΓN,t The aim of this subsection is to prove Theorem 1 (i), that is, {ΓN,t }N ≥1 is compact in C([0, T ], H− ). By the Arzela-Ascoli theorem, it suffices to prove that {ΓN,t }N ≥1 is equicontinuous in t with respect to the metric ρ.
(k) First of all, we note that the sequence ΓN,t = γN,t k≥1 is uniformly bounded in H− . In
5.1
(k)
fact, since TrγN,t = 1, we have ρ(0, ΓN,t) ≤ ΓN,t H− =
N
(k)
2−k γN,t 2 ≤
k=1
N
(k)
2−k TrγN,t ≤ 1.
(5.1)
k=1
To check the equi-continuity of {ΓN,t}N ≥1 , we need the following lemma (e.g., Lemma 9.2 in 3). (k) Lemma 4 The sequence ΓN,t = {γN,t}k≥1 (N = 1, 2, . . .) satisfying (5.1) is equicontinuous in t with respect to the metric ρ if and only if for every fixed k ≥ 1, for arbitrary J (k) ∈ L2 (R2k × R2k ) with |||J (k) |||j < ∞ (1 ≤ j ≤ k) and for every ε > 0, there exists δ > 0 such that (k) (k) (k) (5.2) J , γN,t − γN,s < ε whenever |t − s| ≤ δ. Proof of Theorem 1 (i) We choose k ≥ 1, J (k) ∈ L2 (R2k × R2k ) with |||J (k) |||j < ∞ (1 ≤ j ≤ k), and ε > 0. By Lemma 3, we have (k) (k) (k) J , γN,t − γN,s k t k t (k) (k) dτ J , (−Δxj + Δxj )γN,τ + b dτ dxk dxk dxk+1 J (k) (xk ; xk ) ≤ j=1
s
j=1
s
(k+1) × δα (xj − xk+1 ) − δα (xj − xk+1 ) γN,τ (xk , xk+1 ; xk , xk+1 ) 1 (2) sup Tr|S1 S2 γN,τ S2 S1 |, +|t − s|O k(α1/p + a1/p ) + k 2 N a1/2 τ ∈[0,t]
(5.3)
whenever t > s. Note that (k) (k) (k) (k) J , (−Δxj + Δxj )γN,τ ≤ C J (k) Sj + J Sj TrSj γN,τ Sj ≤ C, where J
(k)
(xk ; xk ) = J (k) (xk ; xk ), because of Corollary 1 and (4.7). Consequently, k j=1
s
t
(k) dτ J (k) , (−Δxj + Δxj )γN,τ ≤ Ck,J (k) |t − s|.
(5.4)
In contrast, because J (k) ||L2 (R2 ×R2 ) ≤ |||J (k) |||j and the norm of δα is of order α−2 , by Schwarz inequality, we have b
k t dτ dxk dxk dxk+1 J (k) (xk ; xk ) j=1
s
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(k+1) × δα (xj − xk+1 ) − δα (xj − xk+1 ) γN,τ (xk , xk+1 ; xk , xk+1 ) ≤ Cα−2
k j=1
s
t
(k+1)
dτ J (k) L2 (R2k ×R2k ) TrγN,τ
≤ Cα,k,J (k) |t − s|.
(5.5)
Inserting (5.4) and (5.5) into (5.3), we obtain (k) (k) (k) J , γN,t − γN,s ≤ C|t − s| where C depends only on α, J (k) , k, but independent of N, t, s. By Lemma 4, the equi-continuity of {ΓN,t}N ≥1 in t is proved. 5.2 A Priori Bounds on Γ∞,t In this subsection, we prove Theorem 1 (ii). To this end, we define a new topology in the space of density matrices following [2]. Denote by L1 (H) and K(H) the spaces of the trace class operators and the space of compact operators on a Hilbert space H, respectively. Set Hk = L2 (R2k ). We define (5.6) Wk = W (k) = S1−1 · · · Sk−1 γ (k) S1−1 · · · Sk−1 : γ (k) ∈ L1 (Hk ) , where Sj = (1 − Δj )1/2 , equipped with W (k) Wk = Tr|S1 · · · Sk W (k) S1 · · · Sk |. We define moreover the space A(k) = A(k) = S1 · · · Sk K (k) S1 · · · Sk : K (k) ∈ K(Hk )
(5.7)
equipped with the norm ||A(k) ||A(k) = ||S1−1 · · · Sk−1 A(k) S1−1 · · · Sk−1 ||, where · denotes the operator norm. It is checked that ∗ (k) A , · A(k) = Wk , · Wk
(5.8)
by the fact that K(Hk )∗ = L1 (Hk ). The identification of Wk as the dual space of A(k) implies the existence of a weak star topology σ(Wk , A(k) ) on Wk .
(k) Proof of Theorem 1 (ii) Let Γ∞,t = γ∞,t k≥1 be a limit point of the sequence
(k) ΓN,t = γN,t k≥1 in the space C([0, T ], H− ) with respect to the metric ρ. By passing to a subsequence we can assume that ΓN,t → Γ∞,t as N → ∞. This implies that, for every fixed t ∈ [0, T ] and for every k ≥ 1, we have, for N → ∞, (k)
(k)
γN,t → γ∞,t
(5.9)
with respect to the weak topology of L2 (R2k × R2k ). This follows because ||ΓN,t ||H− ≤ 1 and in the unit ball, the metric ρ is equivalent to the weak∗ topology of H− . Convergence with respect to the weak∗ topology of H− implies then weak convergence in every k-particle sector (k) L2 (R2k × R2k ). It is concluded from (5.9) that γ∞,t determines a positive operator on L2 (R2k ).
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By Corollary 1, for every k ≥ 1 there exists a constant C > 0 such that (k)
(k)
||γN,t ||Wk = TrS1 · · · Sk γN,t S1 · · · Sk ≤ C k
(5.10) (k)
for every t ∈ [0, T ] and N ≥ 1. By the Banach-Alaoglu Theorem, the sequence γN,t is compact in Wk with respect to the weak∗ topology. In particular, there exists a subsequence Nj → ∞ (k) (k) (k) and γ ∞,t ∈ Wk such that γNj ,t → γ ∞,t and (k)
(k)
γ∞,t Wk = Tr|S1 · · · Sk γ∞,t S1 · · · Sk | ≤ C k .
(5.11)
(k)
So, the sequence γNj ,t satisfies, for j → ∞, (k)
(k)
w.r.t. the weak topology of L2 (R2k × R2k ) and
(k)
(k)
w.r.t. the weak∗ topology of Wk .
γNj ,t → γ∞,t γNj ,t → γ∞,t
(5.12)
If J (k) ∈ L2 (R2k × R2k ), then the operator with kernel given by J (k) (which will be still denoted by J (k) ) is Hilbert-Schmidt and thus compact; in particular, J (k) ∈ A(k) . Thus, if we use (5.12), it is verified that (k) (k) dxk dxk J (k) (xk ; xk )γ∞,t (xk ; xk ) = dxk dxk J (k) (xk ; xk ) γ∞,t (xk ; xk ) (k)
(k)
for every J (k) ∈ L2 (R2k × R2k ). This implies that γ∞,t = γ ∞,t as an element of L2 (R2k × R2k ). Thus, by (5.11) we have, for k ≥ 1, (k)
TrS1 · · · Sk γ∞,t S1 · · · Sk ≤ C k (k)
for every t ∈ [0, T ]. More precisely, we should say that there is a version of γ∞,t ∈ L2 (R2k ×R2k ), (k) which satisfies this bound (the version used here is exactly the density matrix γ ∞,t ). 5.3 Convergence to the Infinite BBGKY Hierarcy Finally, in this subsection we prove the last two parts of Theorem 1. Proof of Theorem 1 (iii) From (4.1) and (4.2) of Lemma 2, it is concluded that (k+1) dxk+1 dxk+1 J (k) (xk ; xk )γ∞,t (xk+1 ; xk+1 ) × hr1 (xk+1 − xk+1 )hr1 (xj − xk+1 ) − hr2 (xk+1 − xk+1 )hr2 (xj − xk+1 ) (k+1) (5.13) ≤ C k |||J (k) |||j r1 + r2 + (r1 )1/p + (r2 )1/p TrSj Sk+1 γ∞,t Sk+1 Sj . By (3.8), we have (k+1)
(k+1)
TrSj Sk+1 γ∞,t Sk+1 Sj ≤ TrS1 · · · Sk+1 γ∞,t Sk+1 · · · S1 ≤ C k+1 . This together with (5.13) implies that the sequence (k+1) dxk+1 dxk+1 J (k) (xk ; xk )γ∞,t (xk+1 ; xk+1 ) × hr (xk+1 − xk+1 )hr (xj − xk+1 ) has the Cauchy property for r, r → 0 and thus is convergent, if |||J (k) |||j < ∞ for any 1 ≤ j ≤ k.
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Proof of Theorem 1 (iv) By Lemma 3 we have, for every J (k) ∈ L2 (R2k × R2k ) with |||J |||j < ∞ for 1 ≤ j ≤ k and for N large enough, (k)
k (k) (k) (k) (k) J , γN,t = J , γN,0 − i
−ib
k t j=1
ds
0
j=1
t
ds 0
(k)
dxk dxk J (k) (xk ; xk )(−Δxj + Δxj )γN,s (xk ; xk )
dxk dxk dxk+1 J (k) (xk ; xk )
(k+1) × δα (xj − xk+1 ) − δα (xj − xk+1 ) γN,s (xk , xk+1 ; xk , xk+1 ) +tO(α1/p ) + to(1),
(5.14)
where o(1) → 0 for N → ∞. By passing to a subsequence, we can assume that ΓN,t → Γ∞,t =
(k) γ∞,t k≥1 ∈ C([0, T ], H− ) with respect to the metric ρ. As shown in (5.9), for every fixed k ≥ 1 (k)
(k)
and t ∈ [0, T ], γN,t → γ∞,t in the weak topology of L2 (R2k × R2k ). Hence, (k) (k) (k) J , γN,t − γ∞,t → 0,
(k) (k) (k) J , γN,0 − γ∞,0 → 0
(5.15)
for N → ∞. Because Δxj J (k) (xk ; xk ) and Δxj J (k) (xk ; xk ) are both elements of L2 (R2k × R2k ), this implies that k
(k) (k) dxk dxk (−Δxj + Δxj )J (k) (xk ; xk ) γN,s (xk ; xk ) − γ∞,s (xk ; xk ) → 0
j=1
as N → ∞, uniformly for s ∈ [0, T ]. By the Lebesgue domination theorem, we have, for every fixed t ∈ [0, T ], k j=1
=
ds
0
k j=1
t
0
t
ds
(k) (k) dxk dxk J (k) (xk ; xk ) × (−Δxj + Δxj ) γN,s (xk ; xk ) − γ∞,s (xk ; xk ) dxk dxk −Δxj J (k) (xk ; xk ) + Δxj J (k) (xk ; xk )
(k) (k) (xk ; xk ) → 0 × γN,s (xk ; xk ) − γ∞,s
(5.16)
as N → ∞. Finally, we consider the last term of the right-hand side of (5.15). By (4.2) of Lemma 2, we have (k+1) dxk dxk+1 J (k) (xk ; xk ) δα (xj − xk+1 ) − δα (xj − xk+1 ) γN,s (xk , xk+1 ; xk , xk+1 ) = dxk+1 dxk+1 J (k) (xk ; xk ) δα (xj − xk+1 ) − δα (xj − xk+1 ) (k+1)
×δη (xk+1 − xk+1 )γN,s (xk+1 ; xk+1 ) + O(η),
(5.17)
where O(η) is independent of α, N and s. At this point we can take the limit N → ∞ with fixed α and η. As J (k) ∈ L2 (R2k × R2k ), it is concluded that, for fixed α, η > 0, J (k) (xk ; xk )δα (xj −
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xk+1 )δη (xk+1 − xk+1 ) ∈ L2 (R2(k+1) × R2(k+1) ). Hence, dxk+1 dxk+1 J (k) (xk ; xk ) δα (xj − xk+1 ) − δα (xj − xk+1 ) (k+1) (k+1) (xk+1 ; xk+1 ) → 0 ×δη (xk+1 − xk+1 ) γN,s (xk+1 ; xk+1 ) − γ∞,s
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(5.18)
as N → ∞, uniformly in s ∈ [0, T ]. By (5.15), (5.16), (5.17), and (5.18), it follows from (5.14) that k (k) (k) (k) (k) J , γ∞,t = J , γ∞,0 − i
−ib
k t j=1
0
ds
j=1
0
t
(k) ds J (k) , (−Δj + Δj )γ∞,s
dxk dxk dxk+1 dxk+1 J (k) (xk ; xk )
× δα (xj − xk+1 ) − δα (xj − xk+1 ) δη (xk+1 − xk+1 ) (k+1) ×γ∞,s (xk , xk+1 ; xk , xk+1 ) + O(α1/p + η).
(5.19)
for any fixed t and k. Finally, we apply (4.1) and (4.2) of Lemma 2 to replace δη (xk+1 −xk+1 ) by δ(xk+1 −xk+1 ), and δα (xj −xk+1 ) (or δα (xj −xk+1 )) by δ(xj −xk+1 )(respectively, δ(xj −xk+1 )). The error here is of order α1/p + η. Hence, letting η → 0 and α → 0, we obtain (3.12). Acknowledgments We are grateful to Prof.Zeqian Chen and Dr.Chengjun He for helpful discussions and useful suggestions. Also, Prof.Chen has made some improvements of the draft. References [1] Dalfovo F, Giorgini S, Pitaevskii L P et al. Theory of Bose-Einstein condensation in trapped gases. Reviews of Modern Phys, 1999, 71: 463–512 [2] Elgart A, Erd¨ os L, Schlein B et al. Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons. Arch. Rat. Mech. Anal, 2006, 179(2): 265–283 [3] Erd¨ os L, Schlein B, Yau H T. Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate. Commun Pure Appl Math, 2006, 59(12): 1659–1741 [4] Erd¨ os L, Schlein B, Yau H T. Derivation of the cubic non-linear Schr¨ odinger equation from quantum dynamics of many-body systems. Invent Math, 2007, 167: 515–614 [5] Erd¨ os L, Schlein B, Yau H T. Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. arXiv:math-ph/0606017 v3 [6] Erd¨ os L, Yau H T. Derivation of the nonlinear Schr¨ odinger equation from a many body Coulomb system. Adv Theor Math Phys, 2001, 5(6): 1169–1205 [7] Gross E P. Structure of a quantized vortex in boson systems. Nuovo Cimento, 1961, 20: 454–466 [8] Gross E P. Hydrodynamics of a superfluid condensate. J Math Phys, 1963, 4: 195–207 [9] Lieb E H, Seiringer R. Proof of Bose-Einstein condensation for dilute trapped gases. Phys Rev Lett, 2002, 88: 170409 [10] Lieb E H, Seiringer R, Yngvason J. Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional. Phys Rev A, 2000, 61: 043602 [11] Lieb E H, Seiringer R, Yngvason J. A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas. Comm Math Phys, 2001, 224: 17–31 [12] Pitaevskii L P. Vortex lines in an imperfect Bose gas. Sov Phys JETP, 1961, 13: 451–454 [13] Rudin W. Functional Analysis. 2nd ed. Beijing: China Machine Press, 2004: 70–70