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Derivation of the Nonlinear Schrodinger

Equation from Many Body Quantum Dynamics

Benjamin Schlein

Rutgers University, December 5, 2005

math-ph/0508010: Joint work with L. Erdos and H.-T. Yaumath-ph/0504051: Joint work with A. Elgartmath-ph/0410038: Joint work with L. Erdos and H.-T. Yaumath-ph/0410005: Joint work with A. Elgart, L. Erdos, and H.-T. Yau

Summary

1. Introduction2. The nonlinear Hartree equation3. Bose-Einstein Condensates4. The nonlinear Schrodinger equation5. The uniqueness problem6. The Gross-Pitaevskii Limit



1. Introduction

N-Boson System: Quantum mechanical N -boson systems are

described by a wave function

ψN ∈ L2s (R3N ), symmetric w.r.t. permutations.

The dynamics is governed by the Schrodinger equation

i∂ tψN,t = H N ψN,t .

H N is the Hamiltonian of the system,

H N = −N

j=1

∆x j +i<j

V (xi − x j) acts on L2s (R3N ) .

The expectation of H N

(ψN , HψN ) =

dxψN (x)(HψN )(x)

is the energy of the system.



Consider evolution of a condensate,

ψN,0(x) =N

j=1

ϕ(x j) (x = (x1, . . . , xN )).

If the factorization is preserved in time,

ψN,t(x) =N

j=1

ϕt(x j)

⇒ replace interaction by effective one-particle potential1

N

N i= j

V (xi − x j) 1

N

N i= j

dxi V (xi − x j)|ϕt(xi)|2 (V ∗ |ϕt|2)(x j)

Conjecture: if ψN,0(x) =N

j=1 ϕ(x j), then, as N → ∞,

ψN,t(x) N

j=1

ϕt(x j)

with ϕt being a solution of the nonlinear Hartree equation

i∂ tϕt = −∆ϕt + (V ∗ |ϕt|2)ϕt .



Previous Results:

• Hepp, 1974: Derivation of the Hartree equation for smoothpotentials.

• Spohn, 1980: Generalization to bounded potentials.

• Erdos and Yau, 2001: Derivation for the Coulomb potential

V (x) = ±1/|x|.

• Elgart and S., 2005: Derivation of the relativistic nonlinearHartree equation (application in cosmology: boson stars)

i∂ tϕt = (1 − ∆)1/2ϕt + (V ∗ |ϕt|2)ϕt

for Coulomb potential V (x) = λ/|x|, with λ > λcrit = −4/π.



2. General Strategy for Derivation of the Hartree equation

Marginal Densities:

• Density matrix

γ N,t = |ψN,tψN,t| ⇒ γ N,t(x;x) = ψN,t(x)ψN,t(x)

satisfies Heisenberg equation

i∂ tγ N,t = [H N , γ N,t], Tr γ N,t = 1.

• For k = 1, . . . , N , define the k-particle marginal density

γ (k)N,t (xk;x

k) =

dxN −k γ N,t(xk,xN −k;xk,xN −k) .

Here xk = (x1, . . . , xk), xN −k = (xk+1, . . . , xN ), Trγ (k)N,t = 1.

For every k-particle observable J (k):

ψN,t, (J (k)⊗1(N −k)) ψN,t = Tr(J (k)⊗1(N −k))γ N,t = Tr J (k)γ (k)N,t



The BBGKY Hierarchy: The family {γ (k)N,t }N

k=1 satisfies

i∂ tγ

(k)

N,t=

k

j=1

−

∆x j , γ

(k)

N,t +

1

N

k

i<j

V

(xi − x j

), γ

(k)

N,t

+

1 −

k

N

k j=1

Trk+1

V (x j − xk+1), γ

(k+1)N,t

.

Written in terms of the kernels,

i∂ tγ (k)N,t (xk;x

k) =k

j=1

−∆x j + ∆x

j

γ

(k)N,t (xk;x

k)

+1

N 1≤i<j≤k

(V (xi − x j) − V (xi − x j)) γ

(k)N,t (xk;x

k)

+

1 −

k

N

k j=1

dxk+1

V (x j − xk+1) − V (x

j − xk+1)

× γ (k+1)N,t (xk, xk+1;x

k, xk+1) .



The Hartree Hierarchy: As N → ∞, the BBGKY hierarchy

formally converges to the Hartree hierarchy

i∂ tγ (k)∞,t =

k j=1

−∆x j , γ (k)

∞,t

+

k j=1

Trk+1

V (x j − xk+1), γ (k+1)∞,t

⇒ Infinite system of coupled equations.

Remark: the factorized family of densities {γ (k)t }k≥1 with

γ (k)t (xk;x

k) =k

j=1

ϕt(x j)ϕt(x j)

γ

(k)t = |ϕtϕt|⊗k

is a solution of the Hartree hierarchy if ϕt satisfies

i∂ tϕt = −∆ϕt + (V ∗ |ϕt|2)ϕt .



Strategy for Rigorous Derivation:

• Prove the compactness of {γ (k)

N,t}N

k=1with respect to some

weak topology

⇒ there exists at least one limit point {γ (k)∞,t}k≥1 of {γ

(k)N,t }N

k=1.

• Prove that the limit point {γ (k)∞,t}k≥1 is a solution of the infi-

nite Hartree equation.

• Prove the uniqueness of the solution of the infinite Hartreehierarchy.

⇒ for every k ≥ 1, t ∈ R, γ (k)

N,t

→ γ (k)

t

= |ϕtϕt|⊗k .

⇒ ψN,t, (J (k) ⊗ 1(N −k))ψN,t → ϕ⊗kt , J (k)ϕ⊗k

t as N → ∞.

In this sense ψN,t(x) N

j=1 ϕt(x j) for large N .



3. Bose-Einstein Condensation.

Definition: BEC exists if

max σ(γ (1)N ) = O(1) as N → ∞

( In general γ (1)N =

j λ j|φ jφ j|, with 0 < λ j ≤ 1,

j λ j = 1 )

Interpretation: a macroscopic number of particles occupies the

same one-particle state.

Recently, Lieb-Seiringer proved

γ (1)N (x; x) → φ(x) φ(x) as N → ∞

for the ground state of a trapped Bose gas.

⇒ Complete condensation into φ, the minimizer of the Gross-

Pitaevskii energy functional

E GP(ϕ) =

dx

|ϕ(x)|2 + V ext(x)|ϕ(x)|2 + 4πa0|ϕ(x)|4

,

where a0 = scattering length of pair potential.



Experiments on BEC: in 2001, Cornell-Ketterle-Wieman re-

ceived Nobel prize in physics for experiments which first proved

the existence of BEC for trapped Bose gas.

In the experiments gases are trapped in small volumes by strong

magnetic fields, and cooled down at very low temperatures.

Then one observes the dynamical evolution of the condensate

when the trap is removed.



To interpret these experiments one need an accurate descrip-

tion of the dynamics of the trapped condensate. To this end,

physicists use the time dependent Gross-Pitaevskii equation

i∂ tϕt(x) = −∆ϕt(x) + V ext(x)ϕt(x) + 8πa0|ϕt(x)|2ϕt(x) .



4. The Nonlinear Schrodinger Equation.

Delta-Potential: we choose smooth V (x) ≥ 0 and define

V N (x) = N 3β V (N β x) , with β > 0 .

We consider the system described by the Hamiltonian

H N = −N

j=1

∆x j +1

N

N

i<j

V N (xi − x j) .

As N → ∞,

V N (x) → b0δ(x) with b0 =

dx V (x) .

We expect that the macroscopic dynamics is described by theone-particle nonlinear Schrodinger equation

i∂ tϕt = −∆ϕt + b0|ϕt|2ϕt

= −∆ϕt +

b0δ ∗ |ϕt|2

ϕt

.

Problem much more difficult because, in 3 dim, δ ≤ −const · ∆.



Main Result: Consider the initial data

ψN,0(x) =N

j=1

ϕ(x j), with ϕ ∈ H 1(R3).

Let

V N (x) = N 3β V (N β x), with 0 < β < 1/2, and V ≥ 0.

Then, for all t ∈ R, k ≥ 1,

γ (k)N,t → γ (k)

t = |ϕtϕt|⊗k as N → ∞.

Here ϕt is the solution of

i∂ tϕt = −∆ϕt + b0|ϕt|2ϕt, ϕt=0 = ϕ, b0 =

dx V (x)

The convergence is in the weak* topology of L1(L2(R3k)); forevery compact operator J (k) on L2(R3k), we have

ψN,t,

J (k) ⊗ 1(N −k)

ψN,t = Tr J (k)γ (k)N,t

→ Tr J (k)γ (k)t = ϕ⊗k

t , J (k)ϕ⊗kt .



The BBGKY Hierarchy: the densities {γ (k)N,t }N

k=1 satisfy the

BBGKY hierarchy

i∂ tγ (k)N,t =

k j=1

−∆x j , γ (k)

N,t

+ 1

N

ki<j

V N (xi − x j), γ (k)

N,t

+

1 −

k

N

N j=1

Trk+1

V N (x j − xk+1), γ

(k+1)N,t

.

As N → ∞ ⇒ infinite hierarchy

i∂ tγ (k)∞,t =

k j=1

−∆x j , γ

(k)∞,t

+ b0

N j=1

Trk+1

δ(x j − xk+1), γ

(k+1)∞,t

.

Strategy to derive nonlinear Schrodinger equation as before: first

prove compactness, then convergence, and finally uniqueness.



A-priori Estimates: Let 0 < β < 3/5. Then ∃C :

ψN,t, (1−∆x1) . . . (1 − ∆xk) ψN,t

=

dx(1 − ∆x1)1/2 . . . (1 − ∆xk)1/2ψN,t(x)

2

≤ C k .

⇒ Tr (1 − ∆x1) . . . (1 − ∆xk) γ (k)N,t ≤ C k

A-priori estimates follow from energy estimates,

(H N + N )k ≥ C kN k(1 − ∆x1) . . . (1 − ∆xk)

and from conservation of the energy:

ψN,t, (1 − ∆x1) . . . (1−∆xk) ψN,t

≤ C kN −kψN,t , (H N + N )kψN,t

= C kN −kψN,0, (H N + N )kψN,0 ≤ C k.



Compactness of γ (k)N,t : Let S j = (1 − ∆x j )1/2.

A-priori estimate ⇒

γ (k)N,t Hk

= Tr

S 1 . . . S k γ (k)N,t S k . . . S 1

= Tr (1 − ∆x1) . . . (1 − ∆xk) γ

(k)N,t ≤ C k

⇒ γ (k)N,t is a uniformly bounded sequence in Hk

⇒ Banach-Alaoglu Theorem implies that γ (k)N,t is a compact se-

quence in Hk w.r.t. weak* topology.

Every weak* limit point γ (

k)

∞,t of γ (

k)

N,t satisfies

γ (k)∞,tHk

= Tr

S 1 . . . S k γ (k)∞,t S k . . . S 1

= Tr (1 − ∆x1) . . . (1 − ∆xk) γ

(k)∞,t ≤ C k



Convergence to the infinite hierarchy: Rewrite BBGKY hier-archy in integral form

γ (k)N,t = U (k)(t)γ (k)N,0−i

k j=1

t

0 ds U (k)(t−s)Trk+1

V N (x j−xk+1), γ (k+1)N,s

with

U (k)(t)γ (k) = e−itH (k)γ (k)eitH (k)

, H (k) =k

j=1

−∆x j +1

N

ki<j

V N (xi−x j)

Claim: the limiting family of densities {γ (k)∞,t}k≥1 satisfies the

infinite hierarchy

γ (k)∞,t = U

(k)0 (t)γ

(k)∞,0−ib0

k

j=1

t

0

ds U (k)0 (t−s)Trk+1δ(x j−xk+1), γ

(k+1)∞,s

with γ (k)∞,0 = |ϕϕ|⊗k and

U (k)0 (t)γ (k) = e−itH

(k)0 γ (k)eitH

(k)0 , H

(k)0 =

k j=1

−∆x j



Proof of the claim: compare term by term. Use following lemma

to control the V N (x) → δ(x) convergence.

Lemma: suppose J (k) is a compact operator on L2(R3k) with

smooth kernel, decaying sufficiently fast at infinity. ThenTr J (k)(V N (x j−xk+1) − δ(x j − xk+1))γ (k+1)

≤ const N −β/2 Tr (1 − ∆x j

)(1 − ∆xk+1

)γ (k+1)

Morally, if δα(x) = α−3f (α−1x),

± (δα(x − y) − δ(x − y)) “ ≤ ”const · α1/2(1 − ∆x)(1 − ∆y)



5. Uniqueness of the infinite hierarchy

Write infinite hierarchy as

γ (k)t = U

(k)0 (t)γ

(k)0 +

t

0ds U

(k)0 (t − s)B(k)γ

(k+1)s , k ≥ 1

with

U (k)0 (t)γ (k) = exp

it

k

j=1

∆x j

γ (k) exp

−it

k

j=1

∆x j

B(k)γ (k+1) = −ib0

k j=1

Trk+1

δ(x j − xk+1), γ (k+1)

B(k)γ (k+1)Hk≤ constγ (k+1)Hk+1

, because δ(x) ≤ const(1−∆).

⇒ Need to use smoothing effect of free evolution!



Duhamel series: expand arbitrary solution γ (k)t as

γ (k)t = U

(k)0 (t)γ

(k)0 +

n−1

m=1

ξ(k)m,t + η

(k)n,t

with

ξ(k)m,t =

t

0ds1 . . .

sm−1

0dsm U

(k)0 (t − s1) B(k) U

(k+1)0 (s1 − s2) B(k+1) . . .

. . . U (k+m−1)0 (sm−1 − sm)B(k+m−1) U

(k+m)0 (sm)γ

(k+m)0

η(k)n,t =

t

0ds1 . . .

sn−1

0dsn U

(k)0 (t − s1)B(k) U

(k+1)0 (s1 − s2)B(k+1) . . .

. . . U (k+n−1)0 (sn−1 − sn)B(k+n−1)γ

(k+n)sn

Diagrammatic expansion of ξ(k)m,t:

We expand

TrJ (k)ξ(k)m,t =

Γ∈F m,k

Tr J (k)K Γ,tγ (k+m)0



:

) = set of edges of Γ Γ

V ( )= set of vertices of Γ Γ

L ( )= set of leaves of Γ Γ

R ( )= set of roots of Γ Γ 2(k+m) leaves2k roots

Γ

E (

Tr J (k) K Γ,t γ (k+m)0 =

=

e∈E (Γ)

dαed pe

αe − p2e + iτ eηe

v∈V (Γ)

δ

e∈v

±αe

δ

e∈v

± pe

× J (k)

{( pe, pe)}e∈R(Γ)

γ

(k+m)0

{( pe, p

e)}e∈L(Γ)

× exp(−it

e∈R(Γ)

τ e(αe + iτ eηe)), τ e = ±1



Remark: e−itp2= +∞

−∞ dα e−it(α+iη)

α− p2+iη

Control of the integral: use x = (1 + x2)1/2.Tr J (k) K Γ,t γ (k+m)0

≤ C mtm

×

e∈E (Γ)

dαed pe

αe − p2e

v∈V (Γ)

δ

e∈v

±αe

δ

e∈v

± pe

×

J (k)

{( pe, pe)}e∈R(Γ)

γ

(k+m)0

{( pe, p

e)}e∈L(Γ)

Singularity of potential 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.



Typical example:

pr rα pu u

α pv v

α pw w

α

Integrate first the α-variables of the son-edges

dαudαvdαw

δ(αr = αu + αv − αw)

αu − p2uαv − p2

v αw − p2w ≤

const

αr − p2u − p2

v + p2w1−ε

Then integrate over the momenta of the son-edges

d pud pvd pw

| pu|2+λ

| pv|2+λ

| pw|2+λ

δ( pr = pu + pv − pw)

αr − p2

u − p2

v + p2

w1−ε

≤const

| pr|2+λ

After integrating out all vertices

⇒

Tr J (k)K Γ,tγ

(k+m)0

≤ C mtm ∀Γ ∈ F m,k



Convergence of the expansion: Since |F m,k| ≤ C m, we find

Tr J (k)ξ(k)m,t

≤

Γ∈F m,k

Tr J (k)K Γ,tγ (k+m)0

≤ C mtm.

Analogously, we prove that Tr J (k)η(k)n,t ≤ C ntn.

⇒ if γ (k)1,t , γ

(k)2,t are two solutions with same initial data

Tr J (k)

γ

(k)1,t − γ

(k)2,t

≤ C ntn

Since n ∈ N is arbitrary ⇒ uniqueness for short time.

A-priori estimates are uniform in time ⇒ uniqueness for all times.



6. The Gross-Pitaevskii Limit

Our result holds for

V N (x) = N 3β V (N β x) β < 1/2

Same result is expected to hold for all β < 1.

What happens at β = 1?

The Hamiltonian of the system is given by

H N = −N

j=1

∆x j +N

i<j

N 2V (N (xi − x j))

In this case the BBGKY Hierarchy is given by:

i∂ tγ (k)N,t =

k j=1

−∆x j , γ

(k)N,t

+

ki<j

N 2V (N (xi − x j)), γ

(k)N,t

+

1 −

k

N

k j=1

Trk+1

N 3V (N (x j − xk+1)), γ

(k+1)N,t

.



In particular, for k = 1, in terms of kernels

i∂ tγ (1)N,t (x1; x

1) = (−∆x1 + ∆x1

)γ (1)N,t (x1; x

1)

+

dx2

N 3V (N (x1 − x2)) − N 3V (N (x1 − x2))

γ (2)N,t (x1, x2; x1, x2)

Naively, as N → ∞,

dx2 N 3V (N (x1 − x2))γ (2)

N,t

(x1, x2; x1, x2)

→ b0

dx2δ(x1 − xk+1)γ

(2)∞,t(x1, x2; x

1, x2)

with b0 =

dxV (x), and where γ (k)∞,t is a weak limit of γ

(2)N,t .

But this is wrong!! We are neglecting the correlations amongthe particles.



Ground State Wave Function: good approximation

W N (x) =N

i<j

f (N (xi − x j))

where f (x) is solution of one-body problem−∆ +

1

2V (x)

f (x) = 0, f (x) → 1 as |x| → ∞

f (x) = 1 − a0|x|

, for large |x|, a0 = 18π

dx V (x)f (x)

Condensate wave function: we consider initial states

ψN (x) = W N (x)φN (x), with φN (x) ∼N

j=1

ϕ(x j).

Then, assuming the structure is preserved by the time evolution

γ (2)N,t (x1, x2; x

1, x2) ∼ f (N (x1 − x2))f (N (x

1 − x2))γ

(2)∞,t(x1, x2; x

1, x2)

γ (2)∞,t(x1, x2; x

1, x2) = ϕt(x1)ϕt(x

1)ϕt(x2)ϕt(x2)



Equation for γ (1)N,t :

i∂ tγ (1)N,t (x1; x

1) = (−∆x1 + ∆x1

)γ (1)N,t (x1; x

1)

+

dx2

N 3V (N (x1 − x2)) − N 3V (N (x

1 − x2))

γ (2)N,t (x1, x2; x

1, x2)

As N → ∞, last term converges towards

dx2 N

3

V (N (x1 − x2))γ

(2)

N,t (x1, x2; x

1, x2)

dx2 N 3V (N (x1 − x2))f (N (x1 − x2))γ (2)∞,t(x1, x2; x

1, x2)

→ 8πa0

dx2 δ(x1 − x2)γ

(2)∞,t(x1, x2; x

1, x2)

because dxN 3V (N x)f (N x) =

dxV (x)f (x) = 8πa0

Correlations interplay with potential ⇒ b0 changes to 8πa0



Conjecture: choose initial state

ψN,0(x) = W N (x)N

j=1

ϕ(x j)

where W N approximate ground state wave function. Then

γ (k)N,t → γ

(k)t = |ϕtϕt|⊗k

where ϕt is the solution of Gross-Pitaevskii equation

i∂ tϕt = −∆ϕt + 8πa0|ϕt|2ϕt

The Gross-Pitaevskii equation is widely used in the physics liter-

ature to describe dynamics of Bose Einstein Condensates!



Our result: assume

ψN,0(x) = W N (x)N

j=1

ϕ(x j)

⇒ marginal densities {γ (k)N,t }N

k=1 is a compact sequence.

Moreover, any limit point γ (k)∞,t satisfies the infinite GP-hierarchy

i∂ tγ (k)∞,t =

k j=1

−∆x j , γ (k)

∞,t

+ 8πa0

k j=1

Trk+1

δ(x j − xk+1), γ

(k+1)∞,t

But: we need small modification of the Hamiltonian to avoid

triple collisions.

Still missing: prove of strong a-priori bounds for γ (k)∞,t (needed to

apply our uniqueness result).



Main difficulty: a N -particle wave functions ψN only satisfies

ψN , H 2N ψN ≤ CN 2

if it has the right short scale structure!!

If ψN,t(x) =N

j=1 ϕt(x j) ⇒ ψN,t, H 2N ψN,t ≤ CN 2.

But, if ψN has short scale structure on scale 1/N , it cannot

satisfy a-priori estimates.

To prove a-priori estimates, need first to isolate the singular part

of ψN (x). Write ψN,t(x) = W N (x)φN,t(x). Then we can prove

dx W N (x)2|i jφN,t(x)|2 ≤ C i = j

As N → ∞, this implies thatTr (1 − ∆xi)(1 − ∆x j )γ

(k)∞,t ≤ C i = j

uniformly in t.

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