ON THE UNIQUENESS OF SOLUTIONS TO THE GROSS-PITAEVSKII HIERARCHY SERGIU KLAINERMAN AND MATEI MACHEDON 7 0 0 2 Abstract. The purpose of this note is to give a new proof of uniqueness of the Gross- Pitaevskii hierarchy, first established in n [1], in a different space, based on space-time estimates similar in a J spirit to those of [2]. 3 1 v 6 0 0 1 1. Introduction 0 7 0 TheGross-Pitaevskiihierarchyreferstoasequenceoffunctionsγ(k)(t,x ,x′), / k k h k = 1,2,···,wheret ∈ R,x = (x ,x ,··· ,x ) ∈ R3k,x′ = (x′,x′,··· ,x′) ∈ k 1 2 k k 1 2 k p R3k which are symmetric, in the sense that - h t a γ(k)(t,x ,x′) = γ(k)(t,x′,x ) m k k k k : v and i X γ(k)(t,x ,···x ,x′ ,···x′ ) = γ(k)(t,x ,···x ,x′,···x′) (1) r σ(1) σ(k) σ(1) σ(k) 1 k 1 k a for any permutation σ, and satisfy the Gross-Pitaevskii infinite linear hierarchy of equations, k i∂t +∆xk −∆x′k γ(k) = Bj,k+1(γ(k+1)). (2) j=1 (cid:0) (cid:1) X with prescribed initial conditions γ(k)(0,x ,x′) = γ(k)(x ,x′). k k 0 k k Here ∆xk, ∆x′k refer to the standard Laplace operators with respect to the variables x ,x′ ∈ R3k and the operators B = B1 −B2 k k j,k+1 j,k+1 j,k+1 1 2 SERGIUKLAINERMANANDMATEI MACHEDON are defined according to, B1 (γ(k+1))(t,x ,x′) j,k+1 k k = δ(x −x )δ(x −x′ )γ(k+1)(t,x ,x′ )dx dx′ j k+1 j k+1 k+1 k+1 k+1 k+1 Z Z B2 (γ(k+1))(t,x ,x′) j,k+1 k k = δ(x′ −x )δ(x′ −x′ )γ(k+1)(t,x ,x′ )dx dx′ . j k+1 j k+1 k+1 k+1 k+1 k+1 Z Z In other words B1 , resp. B2 , acts on γ(k+1)(t,x ,x′ ) replac- j,k+1 j,k+1 k+1 k+1 ing both variables x and x′ by x , resp x′. We shall also make k+1 k+1 j j use of the operators, Bk+1 = B j,k+1 1≤j≤k X One can easily verify that a particular solution to (2) is given by, k γ(k)(t,x ,x′) = φ(t,x )φ(t,x′) (3) k k j j j=1 Y where each φ satisfies the non-linear Schrödinger equation in 3+1 di- mensions (i∂ +∆)φ = φ|φ|2, φ(0,x) = φ(x) (4) t In [1] L. Erdös, B. Schlein and H-T Yau provide a rigorous derivation of the cubic non-linear Schrödinger equation (4) from the quantum dynamics of many body systems. An important step in their program is to prove uniqueness to solutions of (2) corresponding to the special initial conditions k γ(k)(0,x ,x′) = γ(k)(x ,x′) = φ(x )φ(x′) (5) k k 0 k k j j j=1 Y with φ ∈ H1(R3). To stateprecisely the uniqueness result of [1], denote Sj = (1−∆xj)1/2, Sj′ = (1−∆x′j)1/2 and S(k) = kj=1Sj · kj=1Sj′. If the operator given by the integral kernel γ(k)(x ,x′) is positive (as an k Qk Q operator), then so is S(k)γ(k)(x ,x′), and the trace norm of S(k)γ(k) is k k kγ(k)k = S(k)γ(k)(x ,x′) dx . Hk k k x′k=xk k Z (cid:0) (cid:1)(cid:12) (cid:12) ON THE UNIQUENESS OF SOLUTIONS TO THE GROSS-PITAEVSKII HIERARCHY3 The authors of [1] prove uniqueness of solutions to (2) in the set of symmetric, positive operators γ satisfying, for some C > 0, k sup kγ(k)(t,·,·)k ≤ Ck (6) Hk 0≤t≤T In that work, the equations (2) are obtained as a limit of the BBGKY hierarchy (see [1]), and it is proved that solutions to BBGKY with initial conditions (5) converge, in a weak sense, to a solution of (2) in the space (6). The purpose of this note is to give a new proof of uniqueness of the Gross-Pitaevskii hierarchy (2), inadifferent space, motivated, inpart, by space-time type estimates, similar in spirit to those of [2]. Our norms will be kR(k)γ(k)(t,·,·)k (7) L2(R3k×R3k) Here, Rj = (−∆xj)1/2, Rj′ = (−∆x′j)1/2 and R(k) = kj=1Rj · kj=1Rj′. Notice that for a symmetric, smooth kernel γ, for which the associated Q Q linear operator is positive we have kR(k)γ(k)(t,·,·)k L2(R3k×R3k) ≤ kS(k)γ(k)(t,·,·)k L2(R3k×R3k) ≤ kγ(k)(t,·,·)k Hk since |S(k)γ(k)(x,x′)|2 ≤ S(k)γ(k)(x,x)S(k)γ(k)(x′,x′). This is similar to the condition a a −|a |2 ≥ 0 which is satisfied by all n×n positive ii jj ij semi-definite Hermitian matrices. Our main result is the following: Theorem 1.1 (Main Theorem). Consider solutions γ(k)(t,x ,x′) of k k the Gross-Pitaevskii hierarchy (2), with zero initial conditions, which verify the estimates, T kR(k)B γ(k)(t,·,·)k dt ≤ Ck (8) j,k L2(R3k×R3k) Z0 for some C > 0 and all 1 ≤ j < k. Then kR(k)γ(k)(t,·,·)k = L2(R3k×R3k) 0 for all k and all t. 4 SERGIUKLAINERMANANDMATEI MACHEDON Therefore,solutionsto(2)verifyingtheinitialconditions(5),areunique in the space-time norm (8). We plan to address the connection with solutions of BBGKY in a future paper. The following remark is however reassuring. Remark 1.2. The sequence γ(k), given by (3) with φ an arbitrary solution of (4) with H1 data, verifies (8) for every T > 0 sufficiently small. Moreover, if the H1 norm of the initial data is sufficiently small then (8) is verified for all values of T > 0. Proof. Observe thatR(k)B γ(k)(t,x ,...,x ;x′ ...,x′)canbewritten 1,k 1 k 1 k in the form, R(k)B γ(k)(t,·,·) = R |φ(t,x )|2φ(t,x ) R (φ(t,x )···R (φ(t,x ) 1,k 1 1 1 2 2 k k · R′(φ(t,x′)···R′(φ(t,x′) 1(cid:0) 1 k (cid:1) k Therefore, in [0,T]×R3k ×R3k we derive kR(k)Bj,kγ(k)kL1tL2 ≤ k|R1(|φ|2φ)kL1tL2x ·kR2φkL∞t L2x···kRkφkL∞t L2x · kR1′φkL∞t L2xkR2′φkL∞t L2x ···kRk′φkL∞t L2x ≤ Ck∇(|φ|2φ)k ×k∇φk2k−1 L1tL2x L∞t L2x where the norm on the left is in [0,T]×R3k×R3k and all norms on the right hand side are taken relative to the space-time domain [0,T]×R3. In view of the standard energy identity for the nonlinear equation (4) we have apriori bounds for supt∈[0,T]k∇φ(t)kL2(R3). Therefore we only need to provide a uniform bound for the norm k∇(|φ|2φ)k . We L1L2 t x shall show below that this is possible for all values of T > 0 provided that the H1 norm of φ(0) is sufficiently small. The case of arbitrary size for kφ(0)k and sufficiently small T is easier and can be proved H1 in a similar manner. WeshallrelyonthefollowingStrichartzestimate(see[3])forthelinear, inhomogeneous, Schrödinger equation i∂ φ+∆φ = f in [0,T]×R3, t kφkL2tL6x ≤ C kfkL1tL2x +kφkL∞t L2x (9) (cid:0) (cid:1) We start by using Hölder inequality, in [0,T]×R3, k∇(|φ|2φ)k ≤ Ck∇φk kφ2k ≤ Ck∇φk kφk2 L1tL2x L2tL6x L2tL3x L2tL6x L4tL6x ON THE UNIQUENESS OF SOLUTIONS TO THE GROSS-PITAEVSKII HIERARCHY5 Using (9) for f = |φ|2φ we derive, k∇φkL2tL6x ≤ C k∇(|φ|2φ)kL1tL2x +k∇φkL∞t L2x (cid:0) (cid:1) Denoting, A(T) = k|φ|2φk L1L2([0,T]×R3) t x B(T) = k∇(|φ|2φ)k L1L2([0,T]×R3) t x we derive, B(T) ≤ C B(T)+k∇φ(0)k kφk2 L2 L4L6 t x ≤ C(cid:0)B(T)+k∇φ(0)kL2(cid:1)kφkL∞t L6xkφkL2tL6x ≤ C A(T)+kφ(0)k B(T)+k∇φ(0)k k∇φ(0)k (cid:0) L2 (cid:1) L2 L2 x On the other han(cid:0)d, using (9) again(cid:1)(cid:0), (cid:1) A(T) ≤ C kφ3k +kφ(0)k L1L2 L2 t x ≤ C(A(T)3 +kφ(0)k (cid:0) L2 (cid:1) Observe thislast inequality implies that, forsuffic(cid:1)iently small kφ(0)k , L2 A(T)remainsuniformlyboundedforallvaluesofT. Thus, forallvalues of T, with another value of C, B(T) ≤ C B(T)+k∇φ(0)k k∇φ(0)k L2 L2 x from which we get a un(cid:0)iform bound for B(T(cid:1)) provided that k∇φ(0)k L2 is also sufficiently small. (cid:3) TheproofofTheorem (1.1)isbased ontwo ingredients. Oneis express- ingγ(k) intermsofthefuture iteratesγ(k+1)···, γ(k+n) usingDuhamel’s formula. Since B(k+1) = k B is a sum of k terms, the iterated j=1 j,k+1 Duhamel’s formula involves k(k+1)···(k+n−1) terms. These have P to be grouped into much fewer O(Cn) sets of terms. This part of our paper follows in the spirit of the Feynman path combinatorial argu- ments of [1]. The second ingredient is the main novelty of our work. We derive a space-time estimate, reminiscent of the bilinear estimates of [2]. 6 SERGIUKLAINERMANANDMATEI MACHEDON Theorem 1.3. Let γ(k+1)(t,x ,x′ ) verify the homogeneous equa- k+1 k+1 tion, i∂t +∆k±+1 γ(k+1) = 0, ∆(±k+1) = ∆xk+1 −∆x′k+1 (10) (cid:0)γ(k+1)(0,x (cid:1) ,x′ ) = γ(k+1)(x ,x′ ). k+1 k+1 0 k+1 k+1 Then there exists a constant C, independent of j,k, such that kR(k)B (γ(k+1))k (11) j,k+1 L2(R×R3k×R3k) ≤ CkR(k+1)γ(k+1)k 0 L2(R3(k+1)×R3(k+1)) 2. Proof of the estimate Without loss of generality, we may take j = 1 in B . It also suffices j,k+1 to estimate the term in B1 , the term in B2 can be treated in the j,k+1 j,k+1 same manner. Let γ(k+1) be as in (10). Then the Fourier transform of γ(k+1) with respect to the variables (t,x ,x′) is given by the formula, k k δ(τ +|ξ |2 −|ξ′|2)γˆ(ξ,ξ′) k k where τ corresponds to the time t and ξ = (ξ ,ξ ,...,ξ ), ξ′ = 1 2 k k k (ξ′,ξ′,...,ξ′) correspond to the space variables x = (x ,x ,...,x ) 1 2 k k 1 2 k and x′ = (x′,x′,...,x′). We also write ξ = (ξ ,ξ ), ξ′ = k 1 2 k k+1 k k+1 k+1 (ξ′,ξ′ ) and, k k+1 |ξ |2 = |ξ |2 +...+|ξ |2 +|ξ |2 = |ξ |2 +|ξ |2 1 k k+1 k+1 k+1 k |ξ′ |2 = |ξ′|2 +...+|ξ′|2 +|ξ′ |2 k+1 1 k k+1 The Fourier transform of B1 (γ(k+1)), with respect to the same vari- 1,k+1 ables (t,x ,x′), is given by, k k δ(···)γˆ(ξ −ξ −ξ′ ,ξ ,··· ,ξ ,ξ′ )dξ dξ′ (12) 1 k+1 k+1 2 k+1 k+1 k+1 k+1 Z Z where, δ(···) = δ(τ +|ξ −ξ −ξ′ |2 +|ξ |2 −|ξ |2 −|ξ′ |2) 1 k+1 k+1 k+1 1 k+1 and γ denotes the initial condition γ(k+1). By Plancherel’s theorem, 0 estimate (11) is equivalent to the following estimate, kI [f]k ≤ Ckfˆk , (13) k L2(R×Rk×Rk) L2(Rk+1×Rk+1) ON THE UNIQUENESS OF SOLUTIONS TO THE GROSS-PITAEVSKII HIERARCHY7 applied to f = R(k+1)γ, where, |ξ |fˆ(ξ −ξ −ξ′ ,ξ ,··· ,ξ ,ξ′ ) I [f](τ,ξ ,ξ′) = δ(...) 1 1 k+1 k+1 2 k+1 k+1 dξ dξ′ . k k k |ξ −ξ −ξ′ ||ξ ||ξ′ | k+1 k+1 Z Z 1 k+1 k+1 k+1 k+1 ApplyingtheCauchy-Schwarzinequalitywithmeasures, weeasilycheck that, |ξ |2 |I [f]|2 ≤ δ(...) 1 dξ dξ′ k |ξ −ξ −ξ′ |2|ξ |2|ξ′ |2 k+1 k+1 Z Z 1 k+1 k+1 k+1 k+1 · δ(...)|fˆ(ξ −ξ −ξ′ ,ξ ,··· ,ξ ,ξ′ )|2dξ dξ′ 1 k+1 k+1 2 k+1 k+1 k+1 k+1 Z Z If we can show that the supremum over τ, ξ ···ξ ,ξ′ ···ξ′ of the first 1 k 1 k integral above is bounded by a constant C2, we infer that, kI [f]k2 ≤ C2 δ(...)|fˆ(ξ −ξ −ξ′ ,ξ ,··· ,ξ ,ξ′ )|2dξ dξ′ dτ k L2 1 k+1 k+1 2 k+1 k+1 k+1 k+1 Z Z Z ≤ C2kfˆk2 L2(Rk+1×Rk+1) Thus, kI [f]k2 ≤ C2kfˆk2 k L2(R×Rk×Rk) L2(Rk+1×Rk+1) as desired. Thus we have reduced matters to the following, Proposition 2.1. There exists a constant C such that δ(τ +|ξ −ξ −ξ′ |2 +|ξ |2 −|ξ′ |2) 1 k+1 k+1 k+1 k+1 Z |ξ |2 1 dξ dξ′ ≤ C |ξ −ξ −ξ′ |2|ξ |2|ξ′ |2 k+1 k+1 1 k+1 k+1 k+1 k+1 uniformly in τ,ξ . 1 The proof is based on the following lemmas Lemma 2.2. Let P be a 2 dimensional plane or sphere in R3 with the usual induced surface measure dS. Let 0 < a < 2, 0 < b < 2, a+b > 2. Let ξ ∈ R3. Then there exists C independent of ξ and P such that 1 C dS(η) ≤ |ξ −η|a|η|b |ξ|a+b−2 ZP 8 SERGIUKLAINERMANANDMATEI MACHEDON Proof. If P = R2 and ξ ∈ R2 this is well known. The same proof works in our case, by breaking up the integral I ≤ I +I +I over the 1 2 3 overlapping regions: Region 1: |ξ| < |η| < 2|ξ|. In this region |ξ −η| < 3|ξ| and 2 1 1 I ≤C dS(η) 1 |ξ|b |ξ −η|a ZP∩{|ξ−η|<3|ξ|} 1 1 1 ≤C dS(η) |ξ|b |ξ −η|a i=−∞ZP∩{3i−1|ξ|<|ξ−η|<3i|ξ|} X 1 1 1 C ≤C (3i|ξ|)2 = |ξ|b |3iξ|a |ξ|a+b−2 i=−∞ X where we have used the obvious fact that the area of P ∩ {3i−1|ξ| < |ξ −η| < 3i|ξ|} is ≤ C(3i|ξ|)2. Region 2: |ξ| < |ξ −η| < 2|ξ|. In this region |η| < 3|ξ| and 2 1 1 I ≤C dS(η) 2 |ξ|a |η|b ZP∩{|η|<3|ξ|} C ≤ (14) |ξ|a+b−2 in complete analogy with region 1. Region 3: |η| > 2|ξ| or |ξ − η| > 2|ξ|. In this region, |η| > |ξ| and 2|ξ −η| ≥ |ξ −η|+|ξ| ≥ |η|, thus 1 I ≤C dS(η) 3 |η|a+b ZP∩{|η|>|ξ|} ∞ 1 ≤C dS(η) |η|a+b i=1 ZP∩{2i−1|ξ|<|η|<2i|ξ|} X ∞ 1 C ≤C (2i|ξ|)2 = |2iξ|a+b |ξ|a+b−2 i=1 X (15) (cid:3) We also have ON THE UNIQUENESS OF SOLUTIONS TO THE GROSS-PITAEVSKII HIERARCHY9 Lemma 2.3. Let P, ξ as in Lemma (2.2) and let ǫ = 1 . Then 10 1 C dS(η) ≤ |ξ −η|2−ǫ|ξ −η||η|2−ǫ |ξ|3−2ǫ ZP 2 Proof. Consider separately the regions |η| > |ξ|, and |ξ −η| > |ξ|, and 2 2 (cid:3) apply the previous lemma. We are ready to prove the main estimate of Proposition (2.1) Proof. Changing k +1 to 2, we have to show |ξ |2 I = δ(τ +|ξ −ξ −ξ′|2 +|ξ |2 −|ξ′|2) 1 dξ dξ′ ≤ C 1 2 2 2 2 |ξ −ξ −ξ′|2|ξ |2|ξ′|2 2 2 Z 1 2 2 2 2 The integral is symmetric in ξ − ξ − ξ′ and ξ so we can integrate, 1 2 2 2 without loss of generality, over |ξ −ξ −ξ′| > |ξ |. 1 2 2 2 Case 1. Consider the integral I restricted to the region |ξ′| > |ξ | and 1 2 2 integrate ξ′ first. 2 Notice δ(τ +|ξ −ξ −ξ′|2 +|ξ |2 −|ξ′|2)dξ′ 1 2 2 2 2 2 = δ(τ +|ξ −ξ |2 −2(ξ −ξ )·ξ′ +|ξ |2)dξ′ 1 2 1 2 2 2 2 dS(ξ′) = 2 (16) 2|ξ −ξ | 1 2 where dS is surface measure on a plane P in R3, i.e. the plane ξ′·ω = λ with ω ∈ S2 and λ = τ+|ξ1−ξ2|2+|ξ2|2. 2|ξ1−ξ2| In this region 10 SERGIUKLAINERMANANDMATEI MACHEDON dξ dS(ξ′) I ≤ |ξ |2 2 2 1 1 |ξ |2|ξ −ξ | |ξ −ξ −ξ′|2|ξ′|2 ZR3 2 1 2 ZP 1 2 2 2 dξ dS(ξ′) ≤ |ξ |2 2 2 1 |ξ |2+2ǫ|ξ −ξ | |ξ −ξ −ξ′|2−ǫ|ξ′|2−ǫ ZR3 2 1 2 ZP 1 2 2 2 dξ ≤ C|ξ |2 2 1 |ξ |2+2ǫ|ξ −ξ |3−2ǫ R3 2 1 2 Z ≤ C (17) Case 2. Consider the integral I restricted to the region |ξ′| < |ξ | and 2 2 2 integrate ξ first. 2 Notice δ(τ +|ξ −ξ −ξ′|2 +|ξ |2 −|ξ′|2)dξ 1 2 2 2 2 2 ξ −ξ′ ξ −ξ′ ξ −ξ′ ξ −ξ′ = δ τ +| 1 2 −(ξ − 1 2)|2 +|(ξ − 1 2)+ 1 2|2 −|ξ′|2 dξ 2 2 2 2 2 2 2 2 (cid:18) (cid:19) |ξ −ξ′|2 ξ −ξ′ = δ τ + 1 2 +2|ξ − 1 2|2 −|ξ′|2 dξ (18) 2 2 2 2 2 (cid:18) (cid:19) dS(ξ ) 2 = 4|ξ − ξ1−ξ2′| 2 2 wheredS issurfacemeasureonasphereP,i.e. thesphereinξ centered 2 at 1(ξ −ξ′) and radius 1 |ξ′|2 −τ − |ξ1−ξ2′|2 . 2 1 2 2 2 2 (cid:0) (cid:1) Thus dξ′ dS(ξ ) I ≤ |ξ |2 2 2 2 1 ZR3 |ξ2′|2 ZP |ξ2|2|ξ2− ξ1−2ξ2′||ξ1 −ξ2 −ξ2′|2 dξ′ dS(ξ ) ≤ |ξ |2 2 2 1 ZR3 |ξ2′|2+2ǫ ZP |ξ2|2−ǫ|ξ2− ξ1−2ξ2′||ξ1 −ξ2 −ξ2′|2−ǫ dξ′ ≤ |ξ |2 2 ≤ C 1 |ξ′|2+2ǫ|ξ −ξ′|3−2ǫ ZR3 2 1 2 (cid:3)

On the uniqueness of solutions to the Gross-Pitaevskii hierarchy

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