Effective evolution equations from
many body quantum dynamics
Benjamin Schlein, University of Cambridge
Optical Lattices and Bose Gases, Warwick
March 1, 2010
Based on joint works with
L. Erdös, A. Michelangeli, I. Rodnianski, H.-T. Yau
1
I. Introduction.
N-Boson System: described by a wave function
ψN ∈ L2(R3N , dx1 . . . dxN )
symmetric w.r.t. permutations
Probabilistic interpretation:
|ψN (x1, . . . , xN )|2 = probability density
⇒
kψN k = 1
The dynamics is governed by the Schrödinger equation
i∂tψN,t = HN ψN,t
ψN,t = e−iHN t ψN
⇒
HN is the Hamiltonian of the system. For example,
HN =
N X
−∆xj + Vext(xj ) +
j=1
N
X
V (xi − xj )
i<j
2
Macroscopic Dynamics: in typical systems, N ' 103 − 1023 − ...
⇒ Impossible to solve the Schrödinger equation.
For practical purposes, it is enough to describe the macroscopic
evolution, resulting from averaging over the particles.
Mean-field systems: every particle interacts very weakly with
all other particles. Consider Hamiltonian
HN = −
N
X
∆xj + κ
N
X
V (xi − xj ) .
j=1
i<j
κ 1,
so that
in the regime
N 1,
κ0 = N κ ' 1
Kinetic and potential energy are O(N ); macroscopic evolution
described by an effective nonlinear one-particle equations.
3
Self-Consistent Evolution: consider evolution of a factorized
state,
N
Y
ψN,0(x) =
(x = (x1, . . . , xN )).
ϕ(xj )
j=1
If factorization is preserved in time,
ψN,t(x) '
N
Y
ϕt(xj )
j=1
we may replace interaction by an effective one-particle potential
κ
N
X
i6=j
V (xi −xj ) ' κ
N Z
X
dxi V (xi −xj )|ϕt(xi)|2 ' N κ (V ∗|ϕt|2)(xj )
i6=j
The orbital ϕt must solve the self-consistent Hartree equation
i∂tϕt = −∆ϕt + κ0 (V ∗ |ϕt|2)ϕt .
4
Reduced Densities: define the density matrix
γN,t = |ψN,tihψN,t|
kernel: γN,t(x; y) = ψN,t(x)ψ N,t(y)
as the orthogonal projection onto ψN,t.
For k = 1, .., N , the reduced k-particle density matrix is given by
(k)
γN,t = Trk+1,...,N γN,t
acting on L2(R3k )
(k)
γN,t is an operator on L2(R3k ) with kernel
(k)
γN,t (xk ; x0k ) =
Z
dxN −k γN,t(xk , xN −k ; x0k , xN −k ) ,
(k)
with xk = (x1, . . . , xk ), xN −k = (xk+1, . . . , xN ), Tr γN,t = 1.
For every k-particle observable J (k):
(k)
hψN,t, (J (k) ⊗ 1(N −k)) ψN,ti = Tr(J (k) ⊗ 1(N −k))γN,t = Tr J (k)γN,t
5
II. System of Gravitating Bosons (Boson Stars)
Consider system of N non-relativistic gravitating bosons.
In appropriate units, the Hamilton operator is given by
N
X
N
X
1
HN = −
∆xj − G
j=1
i<j |xi − xj |
where G is the gravitational constant.
In the regime characterized by
N 1,
and
λ := N G ' 1
it makes sense to consider macroscopic dynamics generated by
N
X
λ X
1
∆xj −
HN = −
N i<j |xi − xj |
j=1
in the limit N → ∞.
6
In 2000, Erdős-Yau proved that, if
ψN,t = e−iHN tϕ⊗N ,
(k)
γN,t = Trk+1,...,N |ψN,tihψN,t|
then, for any k ∈ N and t ∈ R fixed,
(k)
⊗k
Tr γN,t − |ϕtihϕt| → 0
as N → ∞
where ϕt solves the Hartree equation
!
i∂tϕt = −∆ϕt − λ
1
∗ |ϕt|2 ϕt
|.|
with initial data ϕt=0 = ϕ.
In other words, for any k-particle observable J (k), we have
(k) ϕ⊗k i
hψN,t, (J (k) ⊗ 1(N −k))ψN,ti → hϕ⊗k
,
J
t
t
as N → ∞
7
In 2007, in a joint work with I. Rodnianski, we established that
for every k ∈ N there exist a constant Ck such that
(k)
Tr γN,t − |ϕtihϕt|⊗k eCk |t|
. √ .
N
Observe that explicit bounds are important; they justify the study
of the limit N → ∞.
Recently, Knowles-Pickl improved this result to more singular
potentials. Moreover, they showed that, if the solution of the
nonlinear equation scatters, the control is uniform in t ∈ R.
8
III. Semi-relativistic system of gravitating bosons.
To take into account relativity effects, consider Hamiltonian
HN =
N q
X
1 − ∆xj − G
j=1
N
X
1
i<j |xi − xj |
Again, we are interested in the regime where
N 1,
and
λ = NG ' 1
Hence we write
HN =
N q
X
j=1
N
λ X
1
1 − ∆xj −
N i<j |xi − xj |
and we consider the limit N → ∞.
We expect dynamics of factorized data to be approximated by
the semi-relativistic Hartree equation
i∂tϕt =
√
!
1 − ∆ ϕt − λ
1
∗ |ϕt|2 ϕt
|.|
9
Depending on value of λ > 0, we observe two different situations.
Subcritical case: for λ ≤ λcr = 2/π, potential energy is controlled by kinetic energy. Hence
HN =
N
1
λ X
≥0
1 − ∆xj −
N i<j |xi − xj |
N q
X
j=1
Moreover, the nonlinear Hartree equation
i∂tϕt =
√
!
1
1 − ∆ ϕt − λ
∗ |ϕt|2 ϕt
|.|
is globally well-posed (Lenzmann, 2005).
In 2005, in a joint work with A. Elgart, we proved that, if
ψN,t = e−iHN tϕ⊗N ,
⇒
and
(k)
γN,t = Trk+1,...,N |ψN,tihψN,t|
(k)
⊗k
Tr γN,t − |ϕtihϕt| → 0
as N → ∞
10
Supercritical case: if λ > λcr = 2/π, HN is not bounded below.

*
inf
kψk=1
N q
X
ψ, 
j=1

N
λ X
1
 ψ
1 − ∆xj −
N i<j |xi − xj |
+
= −∞
Moreover, there exist solutions of the Hartree equation
i∂tϕt =
√
!
1 − ∆ ϕt − λ
1
∗ |ϕt|2 ϕt
|.|
which blow-up in finite time, in the sense that
kϕtk2 1/2
H
(R3 )
=
Z
2 dx (1 − ∆)1/4ϕt → ∞
as t → T −
for some T < ∞ (Fröhlich-Lenzmann, 2006).
These solutions are supposed to describe the
phenomenon of stellar collapse (Chandrasekhar’s theory).
Question: is there a relation between the linear many body evolution and the nonlinear dynamics in the supercritical regime?
11
Mathematical problem: the operator
HN =
N
λ X
1
1 − ∆j −
N i<j |xi − xj |
N q
X
j=1
has no self-adjoint realization if λ > λcr = 2/π.
We introduce therefore regularized Hamiltonians
α =
HN
N
1
λ X
1 − ∆xj −
N i<j |xi − xj | + α
N q
X
j=1
where α = α(N ) > 0 is such that α(N ) → 0 as N → ∞.
Physically, the cutoff is justified because, on very short length
scales, the gravitational interaction is regularized by other forces.
α is bounded below,
For arbitrary α(N ) > 0, the Hamiltonian HN
and generates a one-parameter group of unitary evolutions.
12
Theorem 1 [Michelangeli-S., 2009]: Choose ϕ ∈ H 2(R3). Let
α t ⊗N
−iHN
ψN,t = e
ϕ
and
(k)
γN,t = Trk+1,...,N |ψN,tihψN,t|
Let ϕt be the solution of
√
!
1
∗ |ϕt|2 ϕt .
i∂tϕt = 1 − ∆ ϕt − λ
|.|
Fix T > 0, and assume that
κ := sup kϕtkH 1/2(R3) < ∞ .
|t|≤T
Then for every k ∈ N, there exists a constant C, depending on
k, T and κ such that
(k)
C
⊗k
Tr γN,t − |ϕtihϕt| ≤ √
N
→0
as N → ∞
for all |t| ≤ T .
In words: as long as the Hartree equation does not blow-up, the
linear evolution is approximated by the Hartree dynamics.
13
Theorem 2 [Michelangeli-S., 2009]: As before, let ϕ ∈ H 2(R3),
αt
−iHN
ϕt be the solution of the Hartree equation, ψN,t = e
ϕ⊗N .
Assume that α(N ) ≥ N −` for some ` ∈ N.
Suppose that there exists T < ∞ such that kϕtkH 1/2 < ∞ for all
0 ≤ t < T , and
kϕtkH 1/2(R3) → ∞
as t → T − .
Then there exists N (t), defined for t ∈ [0, T ) such that N (t) → ∞
as t → T − and
(1)
lim Tr (1 − ∆)1/2 γN (t),t = ∞
t→T −
In words: if the solution of the nonlinear equation blows up at
time T < ∞, also the linear evolution collapses as N → ∞.
14
Remarks:
(1)
1) To prove Theorem 2, we need convergence of γN,t to |ϕtihϕt|
for all t < T , in the energy-norm. We show that
K
(1)
1/4
1/4
≤ √
γN,t − |ϕtihϕt| (1 − ∆)
Tr (1 − ∆)
N
for a constant K depending only on sup|τ |<t kϕτ kH 1/2 .
2) Proof based on approach introduced by Hepp, in 1973. Use
Fock space representation, and consider coherent states as initial
data.
Advantage: one can easily identify the Hartree component of
evolution. Need to control the fluctuations. Crucial observation:
fluctuation dynamics is “sub-critical”.
15
IV. Dynamics of Bose Einstein Condensates
In the last 15 years, BEC have become accessible to experiments.
In 2001, Cornell-Ketterle-Wieman received Nobel prize in physics
for experiments which first proved the existence of BEC for
trapped Bose gas.
Goal: give a mathematical description of these experiments.
16
Trapped Bose Gases: will be described by Hamiltonian
N N
X
X
trap
HN =
−∆xj + Vext(xj ) +
Va(xi − xj )
j=1
i<j
where Va has scattering length a > 0.
We will consider regime
N 1,
a 1,
a0 = N a ' 1.
Therefore, we write
N N
X
X
trap
VN (xi − xj )
HN =
−∆xj + Vext(xj ) +
j=1
i<j
where
VN (x) = N 2V (N x)
and V has fixed scattering length a0.
17
Scattering Length: Recall that the scattering length a0 of V
is defined by
1
−∆ + V (x) f (x) = 0
2
with f (x) → 1 for |x| → ∞ .
Then
a
f (x) ' 1 − 0 ,
|x|
for |x| large
⇒
8πa0 =
Z
dxV (x)f (x)
By scaling, VN (x) = N 2V (N x) has scattering length a = a0/N .
In fact, if
a
a0
=1−
fN (x) = f (N x) ' 1 −
N |x|
|x|
we find
1
−∆ + VN (x) fN (x) = 0,
2
and fN (x) → 1
as |x| → ∞.
18
Properties of the Ground State: in 2000, Lieb-Seiringertrap
Yngvason, proved that ground state energy of HN
is
EGS(N )
= inf EGP(ϕ)
lim
N →∞
N
ϕ:kϕk=1
with
EGP(ϕ) =
Z
dx |∇ϕ(x)|2 + Vext(x)|ϕ(x)|2 + 4πa0|ϕ(x)|4
a0 = scattering length of V (x)
In 2002, Lieb-Seiringer proved that the ground state exhibits
complete condensation in the minimizer of EGP, that is
(1)
γN
→ |φihφ|,
φ = minimizer of EGP
What happens if traps are switched off? The condensate evolves.
Our goal: show that the Gross-Pitaevskii theory also describe
the dynamics of the condensates, after traps are switched off.
19
Theorem [Erdös - S. - Yau, 2006-2008]: suppose V ≥ 0,
|V (x)| ≤ Chxi−σ , for some σ > 5, and let
HN =
N
X
−∆xj +
j=1
N
X
N 2V (N (xi − xj )) .
i<j
Assume that ψN has finite energy per particle hψN , HN ψN i ≤ CN
and that it exhibits complete BEC
(1)
γN
→ |ϕihϕ|
for some ϕ ∈ L2(R3)
Let ψN,t = e−iHN tψN . Then, for every fixed t ∈ R, k ∈ N,
(k)
⊗k
Tr γN,t − |ϕtihϕt| → 0
as N → ∞ ,
where ϕt is the solution of the Gross-Pitaevskii equation
i∂tϕt = −∆ϕt + 8πa0|ϕt|2ϕt
with ϕt=0 = ϕ.
20
Remarks:
• The result concerns the stability of condensation with respect
to the time-evolution.
• Examples of initial data include (by the results of Lieb-Seiringertrap
Yngvason) the ground state of HN
, the Hamiltonian with traps.
• Since
N
1 X
N 3V (N (xi − xj ))
HN =
−∆xj +
N i<j
j=1
N
X
HN looks like a mean-field Hamiltonian with potential
vN (x) = N 3V (N x).
Physically, however, we are not at all in a mean-field situation,
because here interactions are very rare and very strong.
21
Evolution of Marginal Densities: recall the definition of the
(k)
k-particle marginal density γN,t associated with ψN,t:
(k)
γN,t = Trk+1,...,N |ψN,tihψN,t|
Its kernel is given by
(k)
γN,t (xk ; x0k ) =
Z
dxN −k ψN,t(xk , xN −k )ψ N,t(x0k , xN −k )
(k)
The family {γN,t }N
k=1 satisfies the BBGKY Hierarchy
k X
X
(k)
(k)
(k)
−∆xj , γN,t +
N 2V (N (xi − xj )), γN,t
i∂tγN,t =
j=1
1≤i<j≤k
k
X
(k+1)
.
Trk+1 N 2V (N (xj − xk+1)), γN,t
+ (N − k)
j=1
22
For k = 1, we find
(1)
(1)
(2)
i∂tγN,t = −∆, γN,t + (N − 1)Tr2 N 2V (N (x1 − x2)), γN,t .
(1)
(1)
(2)
(2)
Naively, if γN,t → γ∞,t and γN,t → γ∞,t, we obtain
(1)
(1)
(2)
i∂tγ∞,t = −∆, γ∞,t + b0Tr2 δ(x1 − x2), γ∞,t
because
(N − 1)N 2V (N x) ' N 3V (N x) → b0 δ(x)
with
b0 =
Z
dxV (x).
For condensates,
(1)
(2)
γ∞,t = |ϕtihϕt|
γ∞,t = |ϕtihϕt|⊗2
(1)
γ∞,t(x1; x01) = ϕt(x1)ϕt(x01)
(2)
γ∞,t(x1, x2; x01, x02) = ϕt(x1)ϕt(x2)ϕt(x01)ϕt(x02)
⇒ i∂tϕt = −∆ϕt + b0|ϕt|2ϕt
Right equation but wrong coupling constant.
23
Emergence of Scattering Length: correlations effect.
(1)
γN,t (x1; x01) ' ϕt(x1)ϕt(x01)
(2)
γN,t (x1, x2; x01, x02) ' fN (x1 − x2) fN (x01 − x02)ϕt(x1)ϕt(x2)ϕt(x01)ϕt(x02)
where
−∆ +
N2
2
!
V (N x) fN (x) = 0
⇒
a0
fN (x) ' 1 −
N |x|
From
(1)
(1)
(2)
i∂tγN,t = −∆, γN,t + (N − 1)N 2Tr V (N (x1 − x2)), γN,t .
we obtain
i∂tϕt(x) = − ∆ϕt(x) +
lim
N →∞
Z
dx N 3V (N x)f (N x) |ϕt(x)|2ϕt(x)
= − ∆ϕt(x) + 8πa0|ϕt(x)|2ϕt(x)
⇒ Gross-Pitaevskii equation with correct coupling constant.
24
Strategy to prove theorem:
(k)
(1) Show compactness of γN,t w.r.t. appropriate weak topology.
(k)
(2) Show existence of short scale correlation structure in γN,t .
(k)
(k)
(1)+(2) ⇒ γN,t has limit points γ∞,t with
(k)
(k)
γN,t ' fN (xi − xj )γ∞,t
when |xi − xj | ' N −1.
(k)
(3) Prove that every limit point γ∞,t satisfies the infinite hierarchy
(k)
i∂tγ∞,t =
k X
(k)
−∆xj , γ∞,t +8πa0
k
X
(k+1)
Trk+1 δ(xj − xk+1), γ∞,t
j=1
j=1
(k)
and observe γ∞,t = |ϕtihϕt|⊗k is a solution iff ϕt solves GP eq.
(4) Theorem follows if we show uniqueness for infinite hierarchy.
25
Existence of correlation structure: follows from a-priori bounds
of the form
Z
ψN,t(x) 2
≤C
dx ∇xi ∇xj
fN (xi − xj ) uniformly in N ∈ N and in t ∈ R.
Remarks:
• It is crucial that we divide by fN . In fact
Z
dx
2
∇x1 ∇x2 ψN,t(x) ' N
• A-priori estimates follow from energy conservation. More precisely, we show that
2 ψ i ≥ CN 2
hψN , HN
N
Z
dx
2
ψN (x) ∇xi ∇xj
fN (xi − xj ) 26
Remark: theorem also applies to factorized initial data.
This suggests that correlations are formed dynamically within
very short times (Erdős-Michelangeli-S., 2008).
The formation of correlations lowers the local energy; the excess
energy is scattered away through incoherent waves, which do not
influence the macroscopic dynamics.
In particular, this implies that, in general, we do not have convergence in the energy-norm.
Open Problems:
Control on rate of convergence?
What happens if a0 < 0?
27
Uniqueness for infinite hierarchy: we prove uniqueness in the
class of densities with
(k)
Tr (1 − ∆x1 ) . . . (1 − ∆xk ) γt
≤ Ck
with a constant C independent of k ≥ 1 and t.
(k)
Need to prove that any limit point {γ∞,t}k≥1 of the marginals
(k)
{γN,t }N
k=1 satisfies these a-priori bounds.
Problem: the estimates
(k)
Tr (1 − ∆x1 ) . . . (1 − ∆xk ) γN,t ≤ C k
cannot be true uniformly in N , because of short scale structure.
28
Choose a length scale ` with N `2 1 and N `3 1. For j =
1, . . . , N define
(
θj (x) '
1 if |xi − xj | `
0 otherwise
∀i 6= j
Proposition (higher order energy estimates):
hψN , (HN +N )k ψN i ≥ C k N k
⇒
Z
Z
dx θ1(x) . . . θk−1(x) |∇x1 . . . ∇xk ψN (x)|2
dx θ1(x) . . . θk−1(x) |∇x1 . . . ∇xk ψN,t(x)|2 ≤ C k
The cutoff θj (x) is effective only when xj falls into a volume of
order N `3 in R3.
Since N `3 → 0 as N → ∞, the cutoff can be removed in the limit
N → ∞, and we obtain the a-priori bounds
(k)
Tr (1 − ∆x1 ) . . . (1 − ∆xk )γ∞,t ≤ C k .
29
Theorem: given a family {γ (k)}k≥1 with
Tr (1 − ∆x1 ) . . . (1 − ∆xk )γ (k) ≤ C k
(k)
there exists at most one solution {γt
(k)
i∂tγt
k X
(k)
−∆xj , γt
=
+ 8πa0
j=1
k
X
}k≥1 of
(k+1)
Trk+1 δ(xj − xk+1), γt
j=1
such that
(k)
Tr(1 − ∆1) . . . (1 − ∆k )γt
≤ Ck
for all
t ∈ R.
Main difficulty: in 3 dimensions,
δ(x) 6≤ C(1 − ∆)
(δ(x) ≤ C(1 − ∆)α
only if α > 3/2)
A-priori bounds are not enough to show uniqueness; instead we
need to make use of the smoothing effects of free evolution.
To this end, we developed diagrammatic expansion in terms of
Feynman graphs.
30
Hierarchy in Integral Form: rewrite infinite hierarchy
(k)
i∂tγt
=
k X
(k)
−∆xj , γt
+ 8πa0
j=1
k
X
(k+1)
Trk+1 δ(xj − xk+1), γt
j=1
as
(k)
γt
(k)
= U (k)(t)γ0
+
Z t
0
(k+1)
ds U (k)(t − s)B (k)γs
,
k≥1
with

U (k)(t)γ (k) = exp it
k
X

∆xj

 γ (k) exp −it
j=1
B (k)γ (k+1) = −i8πa0
k
X
k
X

∆xj 
j=1
Trk+1 δ(xj − xk+1), γ (k+1)
j=1
31
(k)
Duhamel Series: expand arbitrary solution γt
as
n−1
X (k)
(k)
(k)
(k)
(k)
γt = U (t)γ0 +
ξm,t + ηn,t
m=1
with
(k)
ξm,t =
(k)
ηn,t =
Z t
0
Z t
0
ds1 . . .
Z s
m−1
dsm U (k)(t − s1) B (k) U (k+1)(s1 − s2) B (k+1) . . .
0
. . . U (k+m−1)(s
ds1 . . .
Z s
n−1
(k+m)
(k+m−1) U (k+m) (s )γ
m 0
m−1 − sm )B
dsn U (k)(t − s1)B (k)U (k+1)(s1 − s2)B (k+1) . . .
0
. . . U (k+n−1)(s
(k+n−1) γ (k+n)
sn
n−1 − sn )B
with
B (k)γ (k+1) = −i8πa0
k
X
Trk+1 δ(xj − xk+1), γ (k+1)
j=1
32
For example:
(k)
ξm,t = (−8πia0)m
k
X
k+1
X
···
Z t
k+m−1
X
ds1 . . .
0
j1 =1 j2 =1
jm =1
h
(k)
× U (t − s1) Trk+1 δ(xj1 − xk+1),
h
(k+1)
×U
(s1 − s2)Trk+2 δ(xj2 − xk+2), . . .
Z s
m−1
0
dsm
...
(k+m)
× U (k+m−1)(sm−1 − sm) Trk+m δ(xjm − xk+m), U (k+m)(sm)γ0
...
Classical Graphs: the graphs should describe the collision history of the different terms. For example, for k = 2, m = 4,
1
3
5
δ ( x1− x3) δ ( x2− x4 ) δ ( x3− x5 ) δ ( x2− x6)
2
4
6
33
(k)
More generally, contributions to ξm,t can be represented by ordered forests of k disjoint trees with m vertices
1
k+1
2
k+2
k roots
k+m leaves
k
Number of ordered graphs =
(k + m)!
k!
34
Doubled Graphs: because of the commutators, for every collision we have a binary choice. To represent all contributions we
double the classical graphs. For example (k = 2, m = 4)
1
3
5
δ ( x1− x3) δ ( x2− x6) γ(6) δ ( x3− x5 ) δ ( x2− x4 )
2
4
6
The vertices are still completely ordered, and
number of doubled graphs = 2m
(k + m)!
k!
35
Removing the Order: next we combine the contributions of
topologically equivalent ordered graphs.
1
1
2
2
3
4
3
I
I) =
Z t
0
ds1
4
II
Z s
1
0
ds2
Z s
2
III
ds3 U (1)(t − s1)Tr2,3,4 δ(x1 − x2)U (2)(s1 − s2)
0
(4)
× δ(x1 − x3)U (3)(s2 − s3)δ(x2 − x4)U (4)(s3)γ0
II) =
Z t
0
ds1
Z s
1
0
ds2
Z s
2
ds3 U (1)(t − s1)Tr2,3,4 δ(x1 − x2)U (2)(s1 − s2)
0
(4)
× δ(x2 − x3)U (3)(s2 − s3)δ(x1 − x4)U (4)(s3)γ0
Z t
Z s
Z s
1
3
=
ds1
ds3
ds2 U (1)(t − s1)Tr2,3,4 δ(x1 − x2)U (2)(s1 − s2)
0
0
0
(4)
× δ(x1 − x3)U (3)(s2 − s3)δ(x2 − x4)U (4)(s3)γ0
36
Removing the Order: next we combine the contributions of
topologically equivalent ordered graphs.
1
1
2
2
3
4
4
3
I
II
III
III) := I) + II)
=
Z t
0
ds1
Z s
1
0
ds2
Z s
1
ds3 U (1)(t − s1)Tr2,3,4 δ(x1 − x2)U (2)(s1 − s2)
0
(4)
× δ(x1 − x3)U (3)(s2 − s3)δ(x2 − x4)U (4)(s3)γ0
37
(k)
Feynman Graphs: different contributions to ξm,t will be represented by graphs in
Fm,k = set of forests with 2k disjoint paired trees
with m partially ordered vertices
2 k roots
2 (k+m) leaves
Number of graphs in Fm,k ≤ C m+k .
(k)
Diagrammatic Expansion of ξm,t: we expand
(k)
TrJ (k)ξm,t =
X
(k+m)
Tr J (k)KΓ,tγ0
Γ∈Fm,k
38
Γ:
E ( Γ ) = set of edges of Γ
V ( Γ )= set of vertices of Γ
R ( Γ )= set of roots of Γ
2(k+m) leaves
2k roots
L ( Γ )= set of leaves of Γ
(k+m)
Tr J (k) KΓ,t γ0
=

=
Z
 

Y
X
X
dαedpe




±α
±p
δ
δ
e
e
α − p2
e + iτe ηe v∈V (Γ)
e∈v
e∈v
e∈E(Γ) e
Y
(k+m)
(k)
0
0
×J
{(pe, pe)}e∈R(Γ) γ0
{(pe, pe)}e∈L(Γ)
× exp(−it
X
τe(αe + iτeηe)),
τe = ±1
e∈R(Γ)
39
Control of the Integral: use hxi = (1 + x2)1/2.
(k+m) m m/4
Tr J (k) K
Γ,t γ0
≤ C t

×
Z
 

Y
X
X
dαedpe



δ
±αe δ
±pe
2
hαe − pe i v∈V (Γ)
e∈v
e∈v
e∈E(Γ)
Y
(k) × J
{(pe, p0e)}e∈R(Γ) (k+m) γ
{(pe, p0e)}e∈L(Γ) 0
Singularity at x = 0 ⇒ large momentum problem!!
From a-priori estimates ⇒ decay in the momenta of leaves.
Perform integration over all α and p, starting from the leaves
and moving towards the roots. At each vertex, we propagate
the decay from the son-edges to the father-edge.
40
Typical Example:
αr pr
αu pu
αv pv
αw pw
Integrate first the α-variables of the son-edges
Z
dαudαv dαw
δ(αr = αu + αv − αw )
const
≤
2
2
2
2 1−ε
hαu − p2
hαr − p2
u ihαv − pv ihαw − pw i
u − pv + pw i
Then integrate over the momenta of the son-edges
Z
δ(pr = pu + pv − pw )
const
dpudpv dpw
≤
2
2 1−ε
|pu|2+λ|pv |2+λ|pw |2+λ hαr − p2
|pr |2+λ
u − pv + pw i
After integrating out all vertices
(k+m) (k)
m tm/4
⇒ Tr J KΓ,tγ0
≤
C
∀Γ ∈ Fm,k
41
Convergence of the Expansion: since |Fm,k | ≤ C m, we find
Tr J (k)ξ (k) ≤
m,t X
Γ∈Fm,k
Tr J (k)K γ (k+m) ≤ C mtm/4.
Γ,t 0
(k) (k)
Analogously, we prove that Tr J ηn,t ≤ C ntn/4.
(k)
(k)
⇒ if γ1,t , γ2,t are two solutions with same initial data
Tr J (k) γ (k) − γ (k) ≤ C ntn/4
1,t
2,t Since n ∈ N is arbitrary ⇒ uniqueness for short time.
A-priori estimates are uniform in time ⇒ uniqueness for all times.
42