On the Cauchy problem for Gross-Pitaevskii hierarchies

0.17 MB·English

by  Zeqian Chen

ON THE CAUCHY PROBLEM FOR
GROSS-PITAEVSKII HIERARCHIES
1 ZEQIANCHEN ANDCHUANGYELIU
1
0
2

n
Abstract. ThepurposeofthispaperistoinvestigatetheCauchyprob-
a
lemfortheGross-PitaevskiiinfinitelinearhierarchyofequationsonRn,
J
n ≥ 1. We prove local existence and uniqueness of solutions in certain
7
SobolevtypespacesHα ofsequencesofmarginaldensityoperatorswith
ξ
α> n/2. In particular, we give a clear discussion of all cases α>n/2,
]
h whichcoversthelocalwell-posednessproblemforGross-Pitaevskiihier-
p archy in this situation.
-
h
at 1. Introduction
m
Motivated by recent experimental realizations of Bose-Einstein condensa-
[
tion the theory of dilute, inhomogeneous Bose systems is currently a subject

2 of intensive studies in physics [5]. The ground state of bosonic atoms in a
v
trap has been shown experimentally to display Bose-Einstein condensation
7
2 (BEC). This fact is proved theoretically by Lieb et al [15, 16, 17] for bosons
8 with two-body repulsive interaction potentials in the dilute limit, starting
3
from the basic Schro¨dinger equation. On the other hand, it is well known
.
0 that the dynamics of Bose-Einstein condensates are well described by the
1
Gross-Pitaevskii equation [12,13,18]. Arigorousderivation of thisequation
0
1 from the basic many-body Schro¨dinger equation in an appropriate limit is
: not a simple matter, however, and has only achieved recently in three spa-
v
i tial dimensions by Elgart, Erd¨os, Schlein and Yau [6, 7, 8, 9, 10, 11], based
X
on the notion of so-called Gross-Pitaevskii hierarchies. In their program an
r
a important step is to prove uniqueness to the Gross-Pitaevskii hierarchy via
Feynman graph (see [8, 14]).
Recently, T.Chen and N.Pavlovi´c [2] started to investigate the Cauchy
problem for the Gross-Pitaevskii hierarchy, using a Picard-type fixed point
argument. In the present paper, we continue this line of investigation. We
will prove local existence and uniqueness of solutions in certain Sobolev
type spaces Hα (for definition see Section 2 below) of sequences of marginal
ξ
density operators with α > n/2. Instead of using a fixed point principle
as in [2], here we use the fully expanded iterated Duhamel series, and a
2010 Mathematics Subject Classification: 35Q55, 81V70.
Key words: Gross-Pitaevskii hierarchy, nonlinear Schrodinger equation, Cauchy prob-
lem, Space-time typeestimate.
C.Liu is partially supported by NSFCgrants No.11071095.
1
2 Z.ChenandC.Liu
Cauchy convergence criterion, without additional conditions on any space-
time norms. The assumption of α > n/2 allows us to significantly simplify
the approaches and provide an improvement for the previous work [2] in all
cases α > n/2. Our proof involves the simple property that the interaction
operators B(k) are boundedmaps from the k+1-particle Hilbertspace Hα
k+1
to the k-particle Hilbert space Hα in the cubic case. A case of this type
k
has previously been presented by Chen-Pavlovi´c [4] in their derivation of
the quintic NLS for n =1. In the much more difficult situation α n/2, as
≤
done recently in [3], it is necessary to invoke the Strichartz estimates of the
type introduced in the pioneering work of Klainerman-Machedon [14].
The paper is organized as follows. In Section 2, some notations and
the main result are presented. Section 3 is devoted to present elementary
estimates which will be used later. In particular, we will prove the fact that
the interaction operators B(k) are bounded maps from the k + 1-particle
Hilbert space Hα to the k-particle Hilbert space Hα in the cubic case
k+1 k
for α > n/2. The proof is completely analogous to that of the classical
Sobolev inequality f L∞(Rn) C f Hα(Rn). In section 4, the main result
k k ≤ k k
is proved. Finally, in Section 5, we discuss the so-called quintic Gross-
Pitaevskii hierarchy and extend the result obtained in the previous sections
to that case.
2. Preliminaries and statement of the main result
2.1. Gross-Pitaevskii hierarchies. As follows, we denote by x a general
variable in Rn and by x= (x , ,x ) a point in RNn. We will also use the
1 N
···
notation x = (x ,...,x ) Rkn and x = (x ,...,x ) R(N−k)n.
k 1 k N−k k+1 N
For a function f on Rkn we∈let ∈
(Θ f)(x ,...,x )= f(x ,...,x )
σ 1 k σ(1) σ(k)
for any permutation σ Π (Π denotes the set of permutations on k
k k
∈
elements). Then, each Θ is a unitary operator on L2(Rkn). A bounded
σ
operator A on L2(Rkn) is called k-partite symmetric or simply symmetric if
(2.1) ΘσAΘσ−1 = A
foreveryσ Π .Evidently,adensityoperatorγ(k) onL2(Rkn)(i.e.,γ(k) 0
k
∈ ≥
and trγ(k) = 1) with the kernel function γ(k)(x ;x′) is k-partite symmetric
k k
if and only if
γ(k)(x ,...,x ;x′,...,x′)= γ(k)(x ,...,x ;x′ ,...,x′ )
1 k 1 k σ(1) σ(k) σ(1) σ(k)
for any σ Π .
k
∈
Also, we set
L2(Rkn) = f L2(Rkn) : Θ f = f, σ Π ,
s ∈ σ ∀ ∈ k
equipped with the inne(cid:8)r product of L2(Rkn). Clearly, L2((cid:9)Rkn) is a Hilbert
s
subspace of L2(Rkn). It is easy to check that any k-partite symmetric oper-
ator on L2(Rkn) preserves L2(Rkn).
s
Gross-Pitaevskiihierarchies 3
Definition 2.1. Given n 1, the n-dimensional Gross-Pitaevskii (GP)
≥
(k)
hierarchy refers to a sequence γ of k-partite symmetric density op-
{ t }k≥1
erators on L2(Rkn), where t 0, which satisfy the Gross-Pitaevskii infinite
≥
linear hierarchy of equations,
k
(2.2) i∂ γ(k) = ∆(k),γ(k) +µB(k)γ(k+1), ∆(k) = ∆ , µ = 1,
t t − t t xj ±
(cid:2) (cid:3) Xj=1
with initial conditions
(k) (k)
γ = γ , k = 1,2,....
t=0 0
Here, ∆ refers to the usual Laplace operator with respect to the variables
xj
x Rn and the operator B(k) is defined by
j
∈
k
B(k)γ(k+1) = tr δ(x x ),γ(k+1)
t k+1 j − k+1 t
Xj=1 (cid:2) (cid:3)
where the notation tr indicates that the trace is taken over the (k+1)-th
k+1
variable.
Asin[2], wereferto(2.2) as thecubicGP hierarchy. Forµ =1orµ = 1
−
werefertothecorrespondingGPhierarchiesasbeingdefocusingorfocusing,
respectively. We note that the cubic Gross–Pitaevskii hierarchy accounts
for two-body interactions between the Bose particles (e.g., see [5, 10] and
references therein for details).
Remark 2.1. In terms of the kernel functions γ(k)(x ;x′), we can rewrite
t k k
(2.2) as follows:
(2.3) i∂ + (k) γ(k)(x ;x′)= µ B(k)γ(k+1) (x ;x′),
t △± t k k t k k
(cid:0) (cid:1) (cid:2) (cid:3)
where △(±k) = kj=1(∆xj −∆x′j), with initial conditions
P
γ(k)(x ;x′) = γ(k)(x ;x′), k = 1,2,....
t=0 k k 0 k k
In particular, the action of B(k) on density operators with smooth kernel
functions, γ(k+1)(x ;x′ ) (R(k+1)n R(k+1)n), is given by
k+1 k+1 ∈ S ×
(2.4)
k
B(k)γ(k+1) (x ;x′)= dx dx′ γ(k+1)(x ,x ;x′,x′ )
t k k k+1 k+1 t k k+1 k k+1
Z
(cid:2) (cid:3) Xj=1
δ(x′ x ) δ(x x ) δ(x′ x ) .
× k+1− k+1 j − k+1 − j − k+1
(cid:2) (cid:3)
The action of B(k) can be extended to generic density operators. This will
be made precise in Lemma 3.2.
4 Z.ChenandC.Liu
Remark 2.2. Let ϕ H1(Rn), then one can easily verify that a particular
∈
solution to (2.3) with initial conditions
k
γ(k)(x ;x′) = ϕ(x )ϕ(x′), k = 1,2,...,
t=0 k k j j
jY=1
is given by
k
γ(k)(x ;x′) = ϕ (x )ϕ (x′) k = 1,2,...,
t k k t j t j
jY=1
where ϕ satisfies the cubic non-linear Schro¨dinger equation
t
(2.5) i∂ ϕ = ∆ϕ +µ ϕ 2ϕ , ϕ = ϕ,
t t t t t t=0
− | |
which is defocusing if µ = 1, and focusing if µ = 1.
−
The Gross-Pitaevskii hierarchy (2.2) can be written in the integral form
t
(2.6) γ(k) = (k)(t)γ(k)+ ds (k)(t s)B˜(k)γ(k+1), k = 1,2,...,
t U0 0 Z U0 − s
0
where B˜(k) = iµB(k). Hereafter, the free evolution operator is defined by
−
(k)(t)A = exp it∆(k) Aexp it∆(k) , k = 1,2,...,
U0 −
(cid:0) (cid:1) (cid:0) (cid:1)
for every operator A on L2(Rkn). The action of (k)(t) on kernel functions
U0
γ(k) L2(Rkn Rkn) is given by
∈ ×
(2.7) (k)(t)γ(k)(x ,x′) = e−it△(±k)γ(k)(x ,x′).
U0 k k k k
(k)
Formally we can expand the solution γ of (2.6) for any m 1 as
t ≥
(2.8)
m−1 t s1 sj−1
γ(k) = (k)(t)γ(k) + ds ds ds (k)(t s )B˜(k)
t U0 0 Z 1Z 2···Z jU0 − 1 ···
Xj=1 0 0 0
(k+j−1)(s s )B˜(k+j−1) (k+j)(s )γ(k+j)
×U0 j−1− j U0 j 0
t s1 sm−1
+ ds ds ds (k)(t s )B˜(k)
Z 1Z 2···Z mU0 − 1 ···
0 0 0
(k+m−1)(s s )B˜(k+m−1)γ(k+m),
×U0 m−1− m sm
with the convention s = t. The terms in the summation contain only the
0
initial data. The last error term involves the density operator at an inter-
mediate time s .
m
Gross-Pitaevskiihierarchies 5
2.2. Statement of the main result. In order to state our main results,
we require some more notation. We will use γ(k),ρ(k) for denoting either
(density) operators or kernel functions. For k 1 and α > 0, we denote by
≥
Hα = Hα(Rkn Rkn) the space of measurable functions γ(k) = γ(k)(x ,x′)
k × k k
in L2(Rkn Rkn) such that
×
kγ(k)kHαk := kS(k,α)γ(k)kL2(Rkn×Rkn) < ∞,
where
k
S(k,α) := (1−∆xj)α2(1−∆x′j)α2 .
jY=1(cid:2) (cid:3)
Evidently, Hα is a Hilbert space with the inner product
k
γ(k),ρ(k) = S(k,α)γ(k), S(k,α)ρ(k) .
h i L2(Rkn×Rkn)
(cid:10) (cid:11)
(k)
Moreover, the norm k·kHαk is invariance under the action of U0 (t), that is,
kU0(k)(t)γ(k)kHαk = kγ(k)kHαk
because exp it∆(k) commutate with ∆ for any j.
{± } xj
Let 0 < ξ < 1 and α > 0, we define
∞ ∞
(2.9) Hξα = nΓ = {γ(k)}k≥1 ∈ Ok=1Hαk : kΓkHαξ := Xk=1ξkkγ(k)kHαk < ∞o.
Evidently, Hξα is a Banach space equipped with the norm k·kHαξ, which is
introduced in [2]. We remark that similar spaces are used in the isospectral
renormalization group analysis of spectral problems in quantum field theory
(see [1]).
Definition 2.2. For T > 0, Γ = γ(k) C([0,T], α) is said to be
t { t }k≥1 ∈ Hξ
a local (mild) solution to the Gross-Pitaevskii hierarchy (2.2) if for every
k = 1,2,...,
t
γ(k) = (k)(t)γ(k) + ds (k)(t s)B˜(k)γ(k+1), t [0,T],
t U0 0 Z U0 − s ∀ ∈
0
holds in Hα.
k
Our main result in this paper is the following theorem.
Theorem 2.1. Assume that n 1 and α> n/2. Suppose Γ = γ(k)
≥ 0 { 0 }k≥1 ∈
α for some 0 < ξ < 1. Then there exists a constant C = C depending
Hξ α,n
only on n and α such that, for a fixed 0 < T < ξ/C with η = ξ CT, the
−
following hold.
(i) There exists a solution Γ = γ(k) C([0,T], α) to the Gross-
t { t }k≥1 ∈ Hη
Pitaevskii hierarchy (2.2) with the initial data Γ satisfying
0
η
(2.10) Γt C([0,T],Hα) Γ0 Hα.
k k η ≤ ξk k ξ
6 Z.ChenandC.Liu
(ii) For T = ξ/(5C), if Γ and Γ′ in C([0,T], α) are two solutions to (2.2)
t t Hη
with initial conditions Γ = Γ and Γ′ = Γ′ in α respectively,
t=0 0 t=0 0 Hξ
then
4
(2.11) kΓt−Γ′tkC([0,T],Hαη) ≤ 5kΓ0−Γ′0kHαξ.
Consequently, the solutionΓ tothe initialproblem (2.2) withthe initial
t
data in α is unique in C([0,T], α) for any 0 < T < ξ/C.
Hξ Hη
Remark 2.3. We will prove this theorem by the method of infinitely iterat-
ing the Duhamel series, and proving Cauchy convergence without additional
conditions on spacestime bounds, and however, our argument won’t work if
α n/2. The assumption of α > n/2 allows us to significantly simplify
≤
the approaches in the previous work [2] and improve the corresponding one.
In the much more difficult situation α n, as done recently in [3], it is
≤ 2
necessary to invoke the Strichartz estimates of the type introduced in the
pioneering work of Klainerman-Machedon [14].
3. Preliminary estimates
In the sequel, we will mostly work in Fourier (momentum) space. Fol-
lowing [8], we use the convention that variables p,q,r,p′,q′,r′ always refer
to n dimensional Fourier variables, while x,x′,y,y′,z,z′ denote the position
space variables. With this convention, the usual hat indicating the Fourier
transform will be omitted. For example, for k 1 the kernel of a bounded
operatorAonL2(Rkn)inpositionspaceisK(x ≥;x′),theninthemomentum
k k
space it is given by the Fourier transform
K(qk;q′k)= K,e−ih·,qkieih·,q′ki = dxkdx′kK(xk;x′k)e−ihxk,qkieihx′k,q′ki,
Z
(cid:10) (cid:11)
withthe slight abuseof notation of omitting the haton left handside. Here,
k
x ,q = x q , x = (x ,...,x ),q = (q ,...,q ) Rkn.
k k j j k 1 k k 1 k
h i · ∀ ∈
Xj=1
Gross-Pitaevskiihierarchies 7
Thus, on kernels in the momentum space B(k) in (2.4) acts according to
B(k)γ(k+1) (p ;p′)
k k
(cid:2) (cid:3)
k
= dq dq′
k+1 k+1
Z
Xj=1
γ(k+1)(p ,...,p q +q′ ,...,p ,q ;p′,q′ )
× 1 j − k+1 k+1 k k+1 k k+1
n
(3.1) −γ(k+1)(pk,qk+1;p′1,...,p′j +qk+1−qk′+1,...,p′k,qk′+1)
o
k k
= dq dq′ δ(p q )δ(p′ q′)
Xj=1Z k+1 k+1hYl6=j l − l l − l i
γ(k+1)(q ;q′ ) δ(p′ q′)δ p [q +q q′ ]
× k+1 k+1 j − j j − j k+1− k+1
n
(cid:0) (cid:1)
δ(p q )δ p′ [q′ +q′ q ] .
− j − j j − j k+1− k+1
o
(cid:0) (cid:1)
We begin with the following simple lemma.
Lemma 3.1. If β > n, then
p β
(3.2) sup dqdq′ h i < .
p∈RnZRn p+q′ q β q β q′ β ∞
h − i h i h i
Proof. Since
1
p+q′ q β q β q′ β
h − i h i h i
2β 1 1
+
≤ p+q′ β q′ β p+q′ q β q β
(cid:16) (cid:17)
h i h i h − i h i
22β 1 1 1 1
+ + ,
≤ p β p+q′ β q′ β p+q′ q β q β
(cid:16) (cid:17)(cid:16) (cid:17)
h i h i h i h − i h i
the inequality (3.2) is concluded from the assumption β > n. (cid:3)
As in [14], we introduce for γ(k+1)(x ,x′ ) (R(k+1)n R(k+1)n),
k+1 k+1 ∈ S ×
[B1 γ(k+1)](x ,x′)
j,k k k
= dx dx′ δ(x x′ )δ(x x )γ(k+1)(x ,x′ ),
Z k+1 k+1 k+1− k+1 j − k+1 k+1 k+1
and
[B2 γ(k+1)](x ,x′)
j,k k k
= dx dx′ δ(x x′ )δ(x′ x )γ(k+1)(x ,x′ ),
Z k+1 k+1 k+1− k+1 j − k+1 k+1 k+1
8 Z.ChenandC.Liu
where j = 1,...,k. Then, by (2.4) we have
k
B(k) = B1 B2
j,k − j,k
Xj=1 (cid:0) (cid:1)
acting on smooth kernel functions γ(k+1) (R(k+1)n R(k+1)n).
∈ S ×
The following estimate is crucial for the proof of Theorem 2.1. In fact,
the proof of the lemma is completely analogous to that of f L∞(Rn)
k k ≤
C f Hα(Rn) based on Fourier analysis.
k k
n
Lemma 3.2. Suppose that α > and n 1. Then, there exists a constant
2 ≥
C > 0 depending only on α and n such that, for any γ(k+1) (R(k+1)n
α,n
∈S ×
R(k+1)n),
kBjl,kγ(k+1)kHαk ≤ Cα,nkγ(k+1)kHαk+1, l = 1,2,
for all k 1, where j = 1, ,k. Consequently,
≥ ···
(3.3) B(k)γ(k+1) Hα Cα,nk γ(k+1) Hα
k k k ≤ k k k+1
for any γ(k+1) (R(k+1)n R(k+1)n) and all k 1.
∈ S × ≥
Remark 3.1. The estimate (3.3) indicates that the operator B(k), originally
defined on Schwarz functions, can be extended to a bounded operator from
Hα to Hα. In this case, we still denote it by B(k).
k+1 k
Proof. We first consider B1 . For γ(k+1) (R(k+1)n R(k+1)n), from the
1,k ∈ S ×
Plancherel’s theorem it is concluded that
k
B1 γ(k+1) 2 = p 2α p′ 2αdp dp′ dqdq′
k 1,k kHαk Z jY=1h ji h ji k k(cid:12)(cid:12)Z
(cid:12)
2
γ(k+1)(p +q′ q,p , ,p ,q;p′, ,p′,q′) .
1 − 2 ··· k 1 ··· k (cid:12)
(cid:12)
(cid:12)
Gross-Pitaevskiihierarchies 9
Then, we obtain
B1 γ(k+1) 2
k 1,k kHαk
k
p 2α p′ 2αdp dp′
≤Z h ji h ji k k
jY=1
1
dqdq′
×(cid:16)Z p1+q′ q 2α q 2α q′ 2α(cid:17)
h − i h i h i
dqdq′ p +q′ q 2α q 2α q′ 2α
1
×(cid:16)Z h − i h i h i
γ(k+1)(p +q′ q,p , ,p ,q;p′, ,p′,q′)2
×| 1 − 2 ··· k 1 ··· k |
(cid:17)
p 2α
sup dqdq′ h 1i
≤p1∈RnZ hp1+q′−qi2αhqi2αhq′i2α
k
p 2α p′ 2αdp dp′dqdq′
×Z h ji h ji k k
jY=2
p +q′ q 2α p′ 2α q 2α q′ 2α
× h 1 − i h 1i h i h i
n
γ(k+1)(p +q′ q,p , ,p ,q;p′, ,p′,q′)2
×| 1 − 2 ··· k 1 ··· k |
o
p 2α
= sup dqdq′ h 1i γ(k+1) 2
p1∈RnZ hp1+q′−qi2αhqi2αhq′i2αk kHαk+1
C γ(k+1) 2 ,
≤ α,nk kHαk+1
wherewe have used Lemma3.1 in the last inequality. For the operator B2 ,
1,k
we have the same estimate
kB12,kγ(k+1)kHαk ≤ Cα,nkγ(k+1)kHαk+1.
Similarly, wecan prove thesame boundfor B1 andB2 whenj = 2, ,k.
j,k j,k ···
Consequently, we conclude the estimate (3.3). (cid:3)
4. Proof of Theorem 2.1
Now we are ready to prove Theorem 2.1. The proof is divided into two
parts as follows.
Proof. (i) Let α > n/2 and 0 < ξ < 1. Given Γ = γ(k) α. For
0 { 0 }k≥1 ∈ Hξ
m 1, set
≥
t
(4.1) γ(k) = (k)(t)γ(k) + ds (k)(t s)B˜(k)γ(k+1), t > 0,k 1,
m,t U0 0 Z U0 − m−1,s ≥
0
10 Z.ChenandC.Liu
with the convention γ(k) γ(k), where B˜(k) = iµB(k) (e.g., (2.6)). By
0,t ≡ 0 −
expansion, for every m 1 one has
≥
m−1 t t1 tj−1
γ(k) = (k)(t)γ(k)+ dt dt dt (k)(t t )B˜(k)
m,t U0 0 Z 1Z 2···Z jU0 − 1 ···
Xj=1 0 0 0
(k+j−1)(t t )B˜(k+j−1) (k+j)(t )γ(k+j)
×U0 j−1− j U0 j 0
t t1 tm−1
+ dt dt dt (k)(t t )B˜(k)
Z 1Z 2···Z mU0 − 1 ···
0 0 0
(k+m−1)(t t )B˜(k+m−1)γ(k+m)
×U0 m−1− m 0
m
, Ξ(k),
j,t
Xj=0
with the convention t = t. Then, for j = 1, ,m 1, by Lemma 3.2 we
0
··· −
have
t t1 tj−1
kΞ(jk,t)kHαk ≤Z0 dt1Z0 dt2···Z0 dtj(cid:13)U0(k)(t−t1)B˜(k)···
(cid:13)
(k+j−1)(t t )B˜(k+j(cid:13)−1) (k+j)(t )γ(k+j)
×U0 j−1− j U0 j 0 (cid:13)Hα
k
(cid:13)
t t1 tj−1 (cid:13)
dt dt dt k
1 2 j
≤Z Z ···Z ···
(4.2) 0 0 0
×(k+j −1)(Cα,n)jkU0(k+j)(tj)γ0(k+j)kHαk+j
tj
=j!k···(k+j −1)(Cα,n)jkγ0(k+j)kHαk+j
k+j 1
=(cid:18) j − (cid:19)(Cα,nt)jkγ0(k+j)kHαk+j,
and
t t1 tm−1
kΞ(mk,)tkHαk ≤Z0 dt1Z0 dt2···Z0 dtm(cid:13)U0(k)(t−t1)B˜(k)···
(cid:13)
(k+m−1)(t t )B˜(k+(cid:13)m−1)γ(k+m)
×U0 m−1− m 0 (cid:13)Hα
k
(cid:13)
t t1 tm−1 (cid:13)
dt dt dt k
1 2 m
≤Z Z ···Z ···
0 0 0
×(k+m−1)(Cα,n)mkγ0(k+m)kHαk+m
tm
≤m!k···(k+m−1)(Cα,n)mkγ0(k+m)kHαk+m
k+m 1
=(cid:18) m− (cid:19)(Cα,nt)mkγ0(k+m)kHαk+m.