Derivation of the Nonlinear Schr odinger Equation from Many Body - - PowerPoint PPT Presentation

Derivation of the Nonlinear Schr¨

• dinger

Equation from Many Body Quantum Dynamics

Benjamin Schlein Rutgers University, December 5, 2005

math-ph/0508010: Joint work with L. Erd˝

• s and H.-T. Yau

math-ph/0504051: Joint work with A. Elgart math-ph/0410038: Joint work with L. Erd˝

• s and H.-T. Yau

math-ph/0410005: Joint work with A. Elgart, L. Erd˝

• s, and H.-T. Yau

Summary

• 1. Introduction
• 2. The nonlinear Hartree equation
• 3. Bose-Einstein Condensates
• 4. The nonlinear Schr¨
• dinger equation
• 5. The uniqueness problem
• 6. The Gross-Pitaevskii Limit

• 1. Introduction

N-Boson System: Quantum mechanical N-boson systems are described by a wave function ψN ∈ L2

s(R3N),

symmetric w.r.t. permutations. The dynamics is governed by the Schr¨

• dinger equation

i∂tψN,t = HNψN,t . HN is the Hamiltonian of the system, HN = −

N

• j=1

∆xj +

• i<j

V (xi − xj) acts on L2

s(R3N) .

The expectation of HN (ψN, HψN) =

• dxψN(x)(HψN)(x)

is the energy of the system.

Macroscopic Dynamics: In typical physical systems, N ≃ 1023. ⇒ Impossible to solve the Schr¨

• dinger equation.

For practical purposes ⇒ describe the macroscopic evolution. ⇒ Mean-field systems: very weak interaction. The Hamiltonian is given by HN = −

N

• j=1

∆xj + 1 N

N

• i<j

V (xi − xj) . ⇒ Kinetic and potential energy are = O(N); in mean field sys- tems the macroscopic evolution can be described by nonlinear

• ne-particle equations.

⇒ Application in atomic physics (dynamics of BEC), condensed matter physics (BCS superconductors), plasma physics, cosmol-

• gy (dynamics of bosonic stars),...

Consider evolution of a condensate, ψN,0(x) =

N

• j=1

ϕ(xj) (x = (x1, . . . , xN)). If the factorization is preserved in time, ψN,t(x) =

N

• j=1

ϕt(xj) ⇒ replace interaction by effective one-particle potential 1 N

N

• i=j

V (xi − xj) ≃ 1 N

N

• i=j
• dxi V (xi − xj)|ϕt(xi)|2 ≃ (V ∗ |ϕt|2)(xj)

Conjecture: if ψN,0(x) = N

j=1 ϕ(xj), then, as N → ∞,

ψN,t(x) ≃

N

• j=1

ϕt(xj) 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 smooth

potentials.

• Spohn, 1980: Generalization to bounded potentials.
• Erd˝
• s and Yau, 2001: Derivation for the Coulomb potential

V (x) = ±1/|x|.

• Elgart and S., 2005: Derivation of the relativistic nonlinear

Hartree 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

γN,t = |ψN,tψN,t| ⇒ γN,t(x; x′) = ψN,t(x)ψN,t(x′) satisfies Heisenberg equation i∂tγN,t = [HN, γN,t], Tr γN,t = 1.

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

γ(k)

N,t(xk; x′ k) =

• dxN−k γN,t(xk, xN−k; x′

k, 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))γN,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
• −∆xj, γ(k)

N,t

• + 1

N

k

• i<j
• V (xi − xj), γ(k)

N,t

• +
• 1 − k

N

• k
• j=1

Trk+1

• V (xj − xk+1), γ(k+1)

N,t

• .

Written in terms of the kernels, i∂tγ(k)

N,t(xk; x′ k) = k

• j=1
• −∆xj + ∆x′

j

• γ(k)

N,t(xk; x′ k)

+ 1 N

• 1≤i<j≤k

(V (xi − xj) − V (x′

i − xj)) γ(k) N,t(xk; x′ k)

+

• 1 − k

N

• k
• j=1
• dxk+1
• V (xj − 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
• −∆xj, γ(k)

∞,t

• +

k

• j=1

Trk+1

• V (xj − 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(xj)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=1 with 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 Hartree

hierarchy. ⇒ 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⊗k

t

, J(k)ϕ⊗k

t

• as N → ∞.

In this sense ψN,t(x) ≃ N

j=1 ϕt(xj)

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 λj|φjφj|, with 0 < λj ≤ 1, j λj = 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 EGP(ϕ) =

• dx
• |∇ϕ(x)|2 + Vext(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) + Vext(x)ϕt(x) + 8πa0|ϕt(x)|2ϕt(x) .

• 4. The Nonlinear Schr¨
• dinger Equation.

Delta-Potential: we choose smooth V (x) ≥ 0 and define VN(x) = N3βV (Nβx) , with β > 0 . We consider the system described by the Hamiltonian HN = −

N

• j=1

∆xj + 1 N

N

• i<j

VN(xi − xj) . As N → ∞, VN(x) → b0δ(x) with b0 =

• dx V (x) .

We expect that the macroscopic dynamics is described by the

• ne-particle nonlinear Schr¨
• dinger 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(xj), with ϕ ∈ H1(R3). Let VN(x) = N3βV (Nβx), 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)); for every 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⊗k

t

.

The BBGKY Hierarchy: the densities {γ(k)

N,t}N k=1 satisfy the

BBGKY hierarchy i∂tγ(k)

N,t = k

• j=1
• −∆xj, γ(k)

N,t

• + 1

N

k

• i<j
• VN(xi − xj), γ(k)

N,t

• +
• 1 − k

N

N

• j=1

Trk+1

• VN(xj − xk+1), γ(k+1)

N,t

• .

As N → ∞ ⇒ infinite hierarchy i∂tγ(k)

∞,t = k

• j=1
• −∆xj, γ(k)

∞,t

• + b0

N

• j=1

Trk+1

• δ(xj − xk+1), γ(k+1)

∞,t

• .

Strategy to derive nonlinear Schr¨

• dinger 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

≤ Ck . ⇒ Tr (1 − ∆x1) . . . (1 − ∆xk) γ(k)

N,t ≤ Ck

A-priori estimates follow from energy estimates, (HN + N)k ≥ CkNk(1 − ∆x1) . . . (1 − ∆xk) and from conservation of the energy: ψN,t, (1 − ∆x1) . . . (1−∆xk) ψN,t ≤ CkN−kψN,t , (HN + N)kψN,t = CkN−kψN,0, (HN + N)kψN,0 ≤ Ck.

Compactness of γ(k)

N,t: Let Sj = (1 − ∆xj)1/2.

A-priori estimate ⇒ γ(k)

N,tHk = Tr

• S1 . . . Sk γ(k)

N,t Sk . . . S1

• = Tr (1 − ∆x1) . . . (1 − ∆xk) γ(k)

N,t ≤ Ck

⇒ 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

• S1 . . . Sk γ(k)

∞,t Sk . . . S1

• = Tr (1 − ∆x1) . . . (1 − ∆xk) γ(k)

∞,t ≤ Ck

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

• VN(xj−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

−∆xj+ 1 N

k

• i<j

VN(xi−xj) Claim: the limiting family of densities {γ(k)

∞,t}k≥1 satisfies the

infinite hierarchy γ(k)

∞,t = U(k)

(t)γ(k)

∞,0−ib0 k

• j=1

t

0 ds U(k)

(t−s)Trk+1

• δ(xj−xk+1), γ(k+1)

∞,s

• with γ(k)

∞,0 = |ϕϕ|⊗k and

U(k) (t)γ(k) = e−itH(k)

0 γ(k)eitH(k) 0 ,

H(k) =

k

• j=1

−∆xj

Proof of the claim: compare term by term. Use following lemma to control the VN(x) → δ(x) convergence. Lemma: suppose J(k) is a compact operator on L2(R3k) with smooth kernel, decaying sufficiently fast at infinity. Then

• Tr J(k)(VN(xj−xk+1) − δ(xj − xk+1))γ(k+1)
• ≤ const N−β/2 Tr (1 − ∆xj)(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) (t)γ(k) +

t

0 ds U(k)

(t − s)B(k)γ(k+1)

s

, k ≥ 1 with U(k) (t)γ(k) = exp

 it

k

• j=1

∆xj

  γ(k) exp  −it

k

• j=1

∆xj

 

B(k)γ(k+1) = −ib0

k

• j=1

Trk+1

• δ(xj − 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) (t)γ(k) +

n−1

• m=1

ξ(k)

m,t + η(k) n,t

with ξ(k)

m,t =

t

0 ds1 . . .

sm−1

dsm U(k) (t − s1) B(k) U(k+1) (s1 − s2) B(k+1) . . . . . . U(k+m−1) (sm−1 − sm)B(k+m−1)U(k+m) (sm)γ(k+m) η(k)

n,t =

t

0 ds1 . . .

sn−1

dsn U(k) (t − s1)B(k)U(k+1) (s1 − s2)B(k+1) . . . . . . U(k+n−1) (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∈Fm,k

Tr J(k)KΓ,tγ(k+m)

:

) = 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) leaves 2k roots

Γ

E (

Tr J(k) KΓ,t γ(k+m) = =

• e∈E(Γ)

dαedpe αe − p2

e + iτeηe

• v∈V (Γ)

δ

 

e∈v

±αe

  δ  

e∈v

±pe

 

× J(k) {(pe, p′

e)}e∈R(Γ)

• γ(k+m)
• {(pe, p′

e)}e∈L(Γ)

• × exp(−it
• e∈R(Γ)

τe(αe + iτeηe)), τe = ±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) KΓ,t γ(k+m)
• ≤ Cmtm

×

• e∈E(Γ)

dαedpe αe − p2

e

• v∈V (Γ)

δ

 

e∈v

±αe

  δ  

e∈v

±pe

 

×

• J(k)

{(pe, p′

e)}e∈R(Γ)

• γ(k+m)
• {(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:

p

r r

α p

u u

α p

v v

α p

w w

α

Integrate first the α-variables of the son-edges

• dαudαvdαw

δ(αr = αu + αv − αw) αu − p2

uαv − p2 vαw − p2 w ≤

const αr − p2

u − p2 v + p2 w1−ε

Then integrate over the momenta of the son-edges

• dpudpvdpw

|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)KΓ,tγ(k+m)
• ≤ Cmtm

∀Γ ∈ Fm,k

Convergence of the expansion: Since |Fm,k| ≤ Cm, we find

• Tr J(k)ξ(k)

m,t

• ≤
• Γ∈Fm,k
• Tr J(k)KΓ,tγ(k+m)
• ≤ Cmtm.

Analogously, we prove that

• Tr J(k)η(k)

n,t

• ≤ Cntn.

⇒ 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

• ≤ Cntn

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 VN(x) = N3βV (Nβx) 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 HN = −

N

• j=1

∆xj +

N

• i<j

N2V (N(xi − xj)) In this case the BBGKY Hierarchy is given by: i∂tγ(k)

N,t = k

• j=1
• −∆xj, γ(k)

N,t

• +

k

• i<j
• N2V (N(xi − xj)), γ(k)

N,t

• +
• 1 − k

N

• k
• j=1

Trk+1

• N3V (N(xj − 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 + ∆x′

1)γ(1)

N,t(x1; x′ 1)

+

• dx2
• N3V (N(x1 − x2)) − N3V (N(x′

1 − x2))

• γ(2)

N,t(x1, x2; x′ 1, x2)

Naively, as N → ∞,

• dx2 N3V (N(x1 − x2))γ(2)

N,t(x1, x2; x′ 1, 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 among the particles.

Ground State Wave Function: good approximation WN(x) =

N

• i<j

f(N(xi − xj)) 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 = 1 8π

• dx V (x)f(x)

Condensate wave function: we consider initial states ψN(x) = WN(x)φN(x), with φN(x) ∼

N

• j=1

ϕ(xj). Then, assuming the structure is preserved by the time evolution γ(2)

N,t(x1, x2; x′ 1, x′ 2) ∼ f(N(x1 − x2))f(N(x′ 1 − x′ 2))γ(2) ∞,t(x1, x2; x′ 1, x′ 2)

γ(2)

∞,t(x1, x2; x′ 1, x′ 2) = ϕt(x1)ϕt(x′ 1)ϕt(x2)ϕt(x′ 2)

Equation for γ(1)

N,t:

i∂tγ(1)

N,t(x1; x′ 1) = (−∆x1 + ∆x′

1)γ(1)

N,t(x1; x′ 1)

+

• dx2
• N3V (N(x1 − x2)) − N3V (N(x′

1 − x2))

• γ(2)

N,t(x1, x2; x′ 1, x2)

As N → ∞, last term converges towards

• dx2 N3V (N(x1 − x2))γ(2)

N,t(x1, x2; x′ 1, x2)

≃

• dx2 N3V (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

• dxN3V (Nx)f(Nx) =
• dxV (x)f(x) = 8πa0

Correlations interplay with potential ⇒ b0 changes to 8πa0

Conjecture: choose initial state ψN,0(x) = WN(x)

N

• j=1

ϕ(xj) where WN 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) = WN(x)

N

• j=1

ϕ(xj) ⇒ 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
• −∆xj, γ(k)

∞,t

• + 8πa0

k

• j=1

Trk+1

• δ(xj − 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, H2

NψN ≤ CN2

if it has the right short scale structure!! If ψN,t(x) = N

j=1 ϕt(xj)

⇒ ψN,t, H2

NψN,t ≤ CN2.

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

• f ψN(x). Write ψN,t(x) = WN(x)φN,t(x). Then we can prove
• dx WN(x)2|∇i∇jφN,t(x)|2 ≤ C

i = j As N → ∞, this implies that Tr (1 − ∆xi)(1 − ∆xj)γ(k)

∞,t ≤ C

i = j uniformly in t.