Pair exitations and the mean eld approximation of ...matei/Bosonstalk.pdf · (˚satis es the...
Transcript of Pair exitations and the mean eld approximation of ...matei/Bosonstalk.pdf · (˚satis es the...
Pair exitations and the mean fieldapproximation of interacting Bosons.
Based on current work with M. Grillakis
and earlier work with Grillakis and D. Margetis
(2010, 2011)
1
The problem
Describe
ψN(t, ·) = eitHNψN(0, ·)ψN(0, x1, · · · , xN) = φ0(x1)φ0(x2)φ0(xN)
‖ψN(t, ·)‖L2(R3N) = 1
The Hamiltonian is
HN =N∑j=1
∆xj −1
N
∑i<j
vN(xi − xj)
N is large, but fixed.
The potential v ≥ 0, v ∈ C∞0 , spherically sym-
metric and decreasing, and 0 < β ≤ 1.
vN = N3βv(Nβx)→ (∫v)δ(x)
or else β = 0, vN = v, singular.
2
Motivation: In the presence of a trap the ground
state on HN looks like
ΨN(x1, x2, · · · , xN) w φ0(x1)φ0(x2)φ0(xN)
Results suggesting this: Lieb, Seiringer:
γN1 (x, x′)→ φ0(x)φ0(x′)
where
γN1 (x, x′) =∫
ΨN(x, x2, · · · , xN)ΨN(x′, x2, · · · , xN)
dx2 · · · dxNHere ‖φ0‖L2 = 1 and φ0 minimizes the Gross-
Pitaevskii functional.
The trap is removed and the evolution is ob-
served. (More about this interesting point later)
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GOAL: if
ψN(0, x1, · · · , xN) = φ0(x1)φ0(x2)φ0(xN)
(or a finer refinement) then
ψN(t, x1, · · · , xN) w eiNχ(t)φ(t, x1)φ(t, x2)φ(t, xN)
where
i∂tφ+ ∆φ− cφ|φ|2 = 0
c = 8πa, a = the scattering length of v if β = 1
(φ satisfies the Gross-Pitevskii equation),
c =∫v if 0 < β < 1 (NLS)
while if β = 0, φ satisfies the Hartree equation.
The meaning of w is to be made precise, and
' is not expected to be true in L2(R3N).
One way around this : average out most vari-
ables and use marginal densities, as originally
used by Erdos, Schlein and Yau.
(Another way: introduce a unitary operator eB
(pair excitation) to refine the approximation
and then estimate in L2(R3N).
4
Background
Original approach of Erdos, Schlein and Yau:
construct the orthogonal projection kernels
γN(ψN)(t,xN ,x′N) = ψN(t,xN)ψN(t,x′N)
and the k-particle marginal density:
γ(k)N (ψN)(t,xk,x
′k)
=∫γN(t,xk,xN−k,x
′k,xN−1) dxN−k
Here xN−k represents the last N − k variables.
Their result:
γ(1)N (t, x1, x
′1)→ φ(t, x1)φ(t, x′1)
in the trace norm, as N →∞. φ satisfies NLS
or GP.
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Strategy: The limits(as N → ∞) of γ(k)N (ψN)
satisfies the Gross-Pitaevskii hierarchy, and so
do solutions made out of φ(t, x1)φ(t, x2)φ(t, xN),
and solutions to the Gross-Pitaevskii hierarchy
are unique.
”Uniqueness part” substantially simplified by
Klainerman and M., in different space-time norms
(using ideas from harmonic analysis and a com-
binatorial ”boardgame” argument inspired by
the original approach of Erdos, Schlein and
Yau).
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”Existence” part (showing that the relevant
solutions to the Gross-Pitaevskii hierarchy sat-
isfy the needed space-time estimates) started
by K. Kirkpatrick, B. Schlein and G. Staffilani
in 2d, and T. Chen, N. Pavlovic in 3d.
Current result using space-time norms: β <
3/5 due to X. Chen and J. Holmer.
The original result of Erdos, Schlein and Yau
works in the range β ≤ 1.
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The space-time norms method is more flexible,
and opened up the problem in a bounded do-
main ( Kirkpatrick, Schlein and Staffilani) or a
R3 with trap, or even a switchable trap
HN =N∑j=1
∆xj −1
N
∑i<j
vN(xi − xj) + χ(t)N∑j=1
|xj|2
(Xuwen Chen, not entirely completed). This
involves the lens transform.
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Improvement due to Rodnianski and Schlein:
rate of convergence in the case β = 0, v ∼Coulomb. This brings us to Fock space.
The problem discused is a ”classical PDE”
problem, the number of particles stays fixed.
For technical reasons we move over in Fock
space where all possible numbers of particles
are considered at the same time, and only at
the end look at just the Nth component. The
algebra is much easier in Fock space.
Fock space is a Hilbert space with a creation
and annihilation operator satisfying the canon-
ical commutation relations. There are several
models of Fock space, including one of holo-
morphic functions on Cn.
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Background on Fock space
Symmetric Fock space F over L2(R3): The
elements of F are vectors of the form
ψ = (ψ0, ψ1(x1), ψ2(x1, x2), · · · )
where ψ0 ∈ C and ψn ∈ L2s are symmetric in
x1, · · · , xn. The Hilbert space structure of F is
given by (φ,ψ) =∑n∫φnψndx.
We only care about the Nth component, but
the algebra is much simpler if we carry all com-
ponents. Extracting information about the rel-
evant component may or may not be easy.
For f ∈ L2(R3) the (unbounded, closed, densely
defined) creation operator a∗(f) : F → F and
annihilation a(f) : F → F are defined by
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(a∗(f)ψn−1
)(x1, x2, · · · , xn) =
1√n
n∑j=1
f(xj)ψn−1(x1, · · · , xj−1, xj+1, · · ·xn)
and (a(f)ψn+1
)(x1, x2, · · · , xn) =√
n+ 1∫ψ(n+1)(x, x1, · · · , xn)f(x)dx
as well as the operator valued distributions a∗xand ax defined by
a∗(f) =∫f(x)a∗xdx
a(f) =∫f(x)axdx
These satisfy the canonical relations
[ax, a∗y] = δ(x− y)
[ax, ay] = [a∗x, a∗y] = 0
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Fix N , define the Fock space Hamiltonian H :
F → F defined by
HN =∫a∗x∆axdx−
1
2N
∫v(x− y)a∗xa
∗yaxaydxdy
HN is a diagonal operator on F which acts on
each component ψn as a PDE Hamiltonian
HN,n =n∑
j=1
∆xj −1
2N
∑1≤i 6=j≤n
v(xi − xj)
Most interesting component: n = N .
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We need two more ingredients, unitary opera-
tors eA(φ), (A linear in creation and annihila-
tion operators) and eB(k) , (B(k) quadratic in
a, a∗). These will be introduced a bit later,
and are generalizations of the Stone-von Neu-
mann representation of the Heisenberg group
and the Segal-Shale-Weil or metaplectic repre-
sentation of the symplectic group.
13
The Fock space approach and eA were intro-
duced, (for this problem, in Math) by Hepp
and also Ginibre and Velo, to study the Fock
space evolution
eitHNψ0
where the initial data is a coherent state
ψ0 = (c0, c1φ0(x1), c2φ0(x1)φ0(x2), · · · )
= e−√NA(φ0)Ω
(A(φ),Ω defined in the following slides )
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Fix φ(t, x) (to be determined later). Define the
skew-Hermitian closed unbounded Weyl oper-
ator
A(φ) = a(φ)− a∗(φ)
Let Ω = (1,0,0, · · · ) ∈ F and
ψ0 = e−√NA(φ0)Ω
= e−N/2
1, · · · ,(Nn
n!
)1/2
φ0(x1) · · ·φ0(xn), · · ·
is a coherent state (initial data). Proof: Baker-
Campbell-Hausdorff.
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and
ψfirst app(t) = e−√NA(φ(t,·))Ω
= e−N/2
1, · · · ,(Nn
n!
)1/2
φ(t, x1) · · ·φ(t, xn), · · ·
is a first candidate for an approximation for
eitHNψ0 which works from the point of view of
marginal densities, but not in the Fock space
norm. Our goal: to refine this to a second or-
der approximation which works in Fock space.
This brings us to the pair excitation k.
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Introduce
B =1
2
∫ (k(t, x, y)axay − k(t, x, y)a∗xa
∗y
)dxdy
Notice B is skew-Hermitian and eB is unitary.
Look at a more refined approximation of
Ψexact = eitHNe−√NA(φ0)Ω
of the form
Ψapprox = e−√NA(φ(t))e−B(k(t))Ω
The idea is that the Nth component of Ψapprox
looks, roughly, like
φ(t, x1) · · ·φ(t, xN)∏i,j
f(t, xi, xj)
What is this good for?
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Could it be that, in the presence of a trap and
suitable kN , φN ,
‖Ψground state − e−√NA(φN)e−B(kN)Ω‖F → 0
as N → ∞ ?? This is believed to be true (at
least for the Nth component), but I don’t know
of any such rigorous results. This would be a
satisfying result!
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Back to what I know:
Let
S(s) :=1
ist + gT s+ s g
W(p) :=1
ipt + [gT , p]
g(t, x, y) := ∆xδ(x− y) + (vN ∗ |φ|2)(t, x)δ(x− y)
+ vN(x− y)φ(t, x)φ(t, y)
mN(x, y) := −vN(x− y)φ(x)φ(y)
sh(k) := k +1
3!k k k + . . . ,
ch(k) := δ(x− y) +1
2!k k + . . . ,
19
Most recent Grillakis-M result:
If 0 ≤ β < 13, let φ and k satisfies the system
1
i∂tφ−∆φ+
(vN ∗ |φ|2
)φ = 0
S (sh(2k)) = mN ch(2k) + ch(2k) mN
W(ch(2k)
)= mN sh(2k)− sh(2k) mN .
(k(0) = 0), then∥∥∥∣∣∣ψexact(t)− e−iχ(t)∣∣∣ψapprox(t)
∥∥∥F
≤C(1 + t) log4(1 + t)
N(1−3β)/2.
with
Ψapprox = e−√NA(φ(t))e−B(k(t))Ω
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The history of such a construction (as far as I
know it). In the static case, a different, non-
unitary eB (using only creation operators) used
by Lee, Huang and Yang in physics (1959),
and then, rigorously, by Erdos, Schlein and Yau
(2009), and also Yau and Yin, to find an upper
bound for ground state energy of H in a box.
In the time-dependent case, another non-unitary
eB used by Wu (1961), Margetis, together with
a (different) equation for k to study the evo-
lution problem.
In ”pure” math, this is the metaplectic, or
Segal-Shale-Weil representation.
This particular construction of a unitary eB in-
troduced by Grillakis, Margetis and M.M.
21
Very recently, eB was used by Benedikter, de
Oliveira and Schlein ( with a suitable choice of
k), to explain the transition from the coupling
constant c =∫v to c = 8πa in the equation for
φ
i∂tφ+ ∆φ− cφ|φ|2 = 0
as β becomes 1, and also provide estimates
on the rate of convergence for the marginal
densities.
22
Ideas in our proof: Notice ‖Ψapp −Ψexact‖F =
‖e−√NA(t)e−B(t)e−iχ(t)Ω− eitHNe−
√NA(0)Ω‖F
= ‖Ω− eB(t)e√NA(t)eiχ(t)eitHNe−
√NA(0)Ω‖F
Call
Ψred = eB(t)e√NA(t)eitHNe−
√NA(0)Ω
where B(t) = B(k)(t), A(t) = A(φ(t))
Compute reduced Hamiltonain Hred so(1
i
∂
∂t−Hred
)(Ψred)
= 0
and decide on PDEs for φ, k such that
‖(
1
i
∂
∂t−Hred
)Ω‖F is small.
Prove this smallness estimate.
23
Algebra
Computing Hred involves
∂
∂t
(eB(t)e
√NA(t)eitHNe−
√NA(0)Ω
)which involves calculations such as(
∂
∂te√NA(t)
)(e−√NA(t)
)=√NA+
N
2![A, A]
conjugated by eB, as well as(∂
∂teB(t)
) (e−B(t)
)= B +
1
2![B, B] + . . .
The series is infinite, unlike the corresponding
one for A. First difficulty: writing this in closed
form.
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The Segal-Shale-Weil representation of the real
symplectic group is just the right tool. Con-
sider d, k = kT , l = lT d, k, l ∈ L2(dxdy)) .(d k
l −dT
)→ I
(d k
l −dT
)
= −∫d(x, y)
axa∗y + a∗yax2
dx dy
+1
2
∫k(x, y)axay dx dy
−1
2
∫l(x, y)a∗xa
∗y dx dy
LHS∈ sp(C), RHS operator, not necessarily
skew-adjoint.Lie algebra isomorphism. Restrict
this to a Lie subalgebra spc(R) to make RHS
skew-adjoint. (espc(R)=Bogoliubov transforma-
tions.)
25
Under this isomorphism, if
B =1
2
∫ (k(t, x, y)axay − k(t, x, y)a∗xa
∗y
)dxdy
then B = I(K) for
K =
(0 kk 0
)∈ spc(R)
and, for instance,(∂
∂teB(t)
) (e−B(t)
)= I
( ∂∂teK(t)
) (e−K(t)
)which is elementary to compute, since
eK =
(ch(k) sh(k)sh(k) ch(k)
)This is how sh(k), ch(k) come in. How do
sh(2k), ch(2k) come in?
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Now that we can compute Hred,
we want:(
1i∂∂t −Hred
)Ω small.(
1
i
∂
∂t−Hred
)Ω
= N(function)Ω (this becomes a phase function)
+N12(linear terms)Ω +N0(quadratic terms)Ω
+N−12(cubic terms) +N−1(quartic terms)
The term N(function)Ω is not small, but con-
tributes to the phase function χ.
The vanishing of the coefficient of the a∗ coef-
ficient of N12(linear terms) is the Hartree equa-
tion for φ, since the linear terms are∫dxh(t, x)a∗x + h(t, x)ax
where h := −(1/i)∂tφ+ ∆φ−
(vN ∗ |φ|2
)φ.
27
The vanishing of the a∗a∗ coefficient of
N0(quadratic terms) is the new equation for k,
but this has to be written down in a useable
form.
The last two are error terms, and have to be
estimated.
In sh(k), ch(k) coordinates the coefficient of
a∗a∗ is complicated. This is where sh(2k), ch(2k)
come in.
28
New trick: For any M , the of the a∗a∗ com-
ponent of I(M) is the lower-left component
of M . The vanishing of that is equivalent to
[M,N ] = 0, where N is the ”number operator”
matrix
N =
(I 00 −I
)The reduced Hamiltonian comes from some
M = eK(
1
i
∂
∂t+H1
)e−K
H1 =
(g m
−m −gT
)where
g(t, x, y) := ∆xδ(x− y) + (vN ∗ |φ|2)(t, x)δ(x− y)
+ vN(x− y)φ(t, x)φ(t, y)
mN(x, y) := −vN(x− y)φ(x)φ(y)
29
Writing 0 = [M,N ] = eK[1i∂∂t+H1, e
−KNeK]e−K
e−KNeK =
(ch(2k) sh(2k)−sh(2k) −ch(2k)
)
leads to the equation for the pair excitation k:
[1
i
∂
∂t+H1,
(ch(2k) sh(2k)−sh(2k) −ch(2k)
) ]= 0
or, explicitly,
S (sh(2k)) = mN ch(2k) + ch(2k) mN
W(ch(2k)
)= mN sh(2k)− sh(2k) mN .
30
Analysis: In order to control the N−1/2, N−1
error terms, need Lp and Hs estimates for the
solutions to
1
i∂tφ−∆φ+
(vN ∗ |φ|2
)φ = 0
S (sh(2k)) = mN ch(2k) + ch(2k) mN
W(ch(2k)
)= mN sh(2k)− sh(2k) mN .
mN(t, x, y) = −N3βv(Nβ|x−y|)φ(t, x)φ(t, y) with
N3βv(Nβ|x−y|)φ(t, x)φ(t, y)→ δ(x−y)φ(t, x)φ(t, y).
31
The estimates for the Hartree equation
1
i∂tφ−∆φ+
(vN ∗ |φ|2
)φ = 0
are standard (but not trivial) in the case of
NLS (vN = δ), and the proofs can be adapted
to Hartree. The basic estimates needed are
‖φ‖L4([0,∞)×R3) ≤ C
( Colliander, Keel, Staffilani, Takaoka and
Tao, 2002),
‖φ(t, ·)‖Hs ≤ Cs ∀t .
(Bourgain, 1998) and, finally,
‖φ(t, ·)‖L∞ ≤C
t32
and also
‖∂tφ(t, ·)‖L∞ ≤C
t32
(Lin and Strauss, 1978). All easier to prove
with modern tools, for either NLS or Hartree.
32
For the linear equation
1
i∂tsh(2k)−∆x,ysh(2k) = mN + · · ·
sh(2k)(0, ·, ·) = 0
recalling
mN → −δ(x− y)φ(t, x)φ(t, y)
the estimate proved is
‖sh(2k)(t, ·)‖L2(R6) + ‖(ch(2k)− δ)(t, ·)‖L2(R6)
≤ C log(1 + t)
33
This is a new estimate which follows from a
new (but elementary) fixed time elliptic esti-
mate: if
∆R6u(x, y) = δ(x− y)φ(x)φ(y)
then
‖u‖L2 ≤ C‖φ‖2L3
Using Duhamel’s formula,
‖sh(2k)(t, ·)‖L2(R6)
≤ C(‖φ(0, ·)‖2
L3 + ‖φ(t, ·)‖2L3
+∫ t
0‖φ(s, ·)‖L3‖∂sφ(s, ·)‖L3ds
)≤ C log(1 + t)
34
Recall the strategy:(1
i
∂
∂t−Hred
)(Ω−Ψreduced)
= −HredΩ should be small.
Estimate for typical term in ‖HredΩ‖F :
N−1‖vN(y1 − y2)sh(k)(y3, y1)sh(k)(y2, y4)‖L2
≤ N−1‖vN‖L∞‖sh(k)‖2L2 ≤ CN3β−1 log2(1 + t)
35
Near future state of the project: Coupling the
equations
1
i∂tφ−∆φ+
(vN ∗ |φ|2
)φ = 0
S (sh(2k)) = mN ch(2k) + ch(2k) mN
W(ch(2k)
)= mN sh(2k)− sh(2k) mN .
in order to handle the case β = 1. (Coming
this fall).
36
Thank you!
37