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)

3

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.

5

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).

6

”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.

7

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.

8

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.

9

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

10

(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

11

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 .

12

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 )

14

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.

15

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.

16

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?

17

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!

18

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

20

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.

24

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?

26

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