From many body quantum dynamics to nonlinear · PDF fileIntroduction: from bosons to NLS The...

Introduction: from bosons to NLSThe Cauchy problem for GP

Existence of solutions to GPDerivation of the cubic GP hierarchy in 3D

From many body quantum dynamics tononlinear dispersive PDE, and back

Natasa Pavlovic(based on joint works with T. Chen, and with T. Chen and N.

Tzirakis)

University of Texas at Austin

June 18, 2013

NSF-CBMS Regional Research Conference in the MathematicalSciences

Kansas State University

Natasa Pavlovic (UT Austin) From bosons to PDE, and back



Outline

1 Introduction: from bosons to NLS

2 The Cauchy problem for GP

3 Existence of solutions to GP

4 Derivation of the cubic GP hierarchy in 3D




Introduction

The mathematical analysis of interacting Bose gases is a hot topic in MathPhys. Major research directions are:

1 Proof of Bose-Einstein condensation

Aizenman-Lieb-Seiringer-Solovej-Yngvason

2 Derivation of nonlinear Schrodinger or Hartree eq.

Via Fock space: Hepp, Ginibre-Velo, Rodnianski-Schlein,Grillakis-Machedon-Margetis, Grillakis-Machedon, X. Chen

Via BBGKY: Spohn, Elgart-Erdos-Schlein-Yau, Erdos-Schlein-Yau,Adami-Bardos-Golse-Teta

Via BBGKY and PDE: Klainerman-Machedon,Kirkpatrick-Schlein-Staffilani, Chen-P., X. Chen,Gressman-Sohinger-Staffilani, X. Chen-Holmer

Other approaches: Frohlich-Graffi-Schwarz,Frohlich-Knowles-Pizzo, Anapolitanos-Sigal, Abou-Salem, Pickl




From bosons to NLS following Erdos-Schlein-Yau [2006-07]

Step 1: From N-body Schrodinger to BBGKY hierarchy

The starting point is a system of N bosons whose dynamics isgenerated by the Hamiltonian

HN :=N∑

j=1

(−∆xj ) +1N

∑1≤i<j≤N

VN(xi − xj ) ,(1.1)

on the Hilbert space HN = L2sym(RdN), whose elements Ψ(x1, . . . , xN)

are fully symmetric with respect to permutations of the arguments xj .

HereVN(x) = NdβV (Nβx),

with 0 < β ≤ 1.




The solutions of the Schrodinger equation

i∂t ΨN = HN ΨN

with initial condition ΨN,0 ∈ HN determine:the N-particle density matrix

γN(t , xN ; x ′N) = ΨN(t , xN)ΨN(t , x ′N),

and its k -particle marginals

γ(k)N (t , xk ; x ′k ) =

∫dxN−kγN(t , xk , xN−k ; x ′k , xN−k ) ,

for k = 1, . . . ,N.Here

xk = (x1, . . . , xk ),

xN−k = (xk+1, . . . , xN).




The BBGKY hierarchy is given by

i∂tγ(k)N = −(∆xk

−∆x′k )γ(k)N

+1N

∑1≤i<j≤k

[VN(xi − xj )− VN(x ′i − x ′j )] γ(k)N(1.2)

+N − k

N

k∑i=1

Trk+1[VN(xi − xk+1)− VN(x ′i − xk+1)] γ(k+1)N(1.3)

In the limit N →∞, the sums weighted by combinatorial factors havethe following size:

In (1.2), k2

N → 0 for every fixed k and sufficiently small β.

In (1.3), N−kN → 1 for every fixed k and VN(xi − xj )→ b0δ(xi − xj ),

with b0 =∫

dx V (x).




Step 2: BBGKY hierarchy→ GP hierarchy

As N →∞, one obtains the infinite GP hierarchy as a weak limit.

i∂tγ(k)∞ = −

k∑j=1

(∆xj −∆x′j ) γ(k)∞ + b0

k∑j=1

Bj;k+1γ(k+1)∞(1.4)

where the “contraction operator” is given via(Bj;k+1γ

(k+1)∞

)(t , x1, . . . , xk ; x ′1, . . . , x

′k )(1.5)

:=

∫dxk+1dx ′k+1[

δ(xj − xk+1)δ(xj − x ′k+1)− δ(x ′j − xk+1)δ(x ′j − x ′k+1)]

γ(k+1)∞ (t , x1, . . . , xj , . . . , xk , xk+1; x ′1, . . . , x

′k , x′k+1) .




Step 3: Factorized solutions of GP and NLS

It is easy to see that

γ(k)∞ =

∣∣φ ⟩⟨φ ∣∣⊗k:=

k∏j=1

φ(t , xj )φ(t , x ′j )

is a solution of the GP if φ satisfies the cubic NLS

i∂tφ + ∆xφ − b0 |φ|2 φ = 0

with φ0 ∈ L2(Rd ).




Step 4: Uniqueness of solutions of GP hierarchies

While the existence of factorized solutions can be easily obtained, theproof of uniqueness of solutions of the GP hierarchy is the mostdifficult part in this analysis.




Why is it difficult to prove uniqueness?

Fix a positive integer r . One can express the solution γ(r) to the GP hierarchyas:

γ(r)(tr , ·) =

Z tr

0e

i(tr−tr+1)∆(r)± Br+1(γ(r+1)(tr+1)) dtr+1

=

Z tr

0

Z tr+10

ei(tr−tr+1)∆

(r)± Br+1e

i(tr+1−tr+2)∆(r+1)± Br+2(γ(r+2)(tr+2)) dtr+1 dtr+2

= ...

=

Z tr

0...

Z tr+n−10

Jr (tr+n) dtr+1...dtr+n,(1.6)

where

Br+1 =rX

j=1Bj;r+1,

tr+n = (tr , tr+1, ..., tr+n),

Jr (tr+n) = ei(tr−tr+1)∆

(r)± Br+1...e

i(tr+(n−1)−tr+n)∆(r+(n−1))± Br+n(γ(r+n)(tr+n)).

Since the interaction term involves the sum, the iterated Duhamel’s formulahas r(r + 1)...(r + n − 1) terms.




Summary of the method of ESY

Roughly speaking, the method of Erdos, Schlein, and Yau for deriving thecubic NLS justifies the heuristic explained above and it consists of thefollowing two steps:

(i) Deriving the Gross-Pitaevskii (GP) hierarchy as the limit as N→∞of the BBGKY hierarchy, for a scaling where the particle interactionpotential tends to a delta distribution.

(ii) Proving uniqueness of solutions for the GP hierarchy, which impliesthat for factorized initial data, the solutions of the GP hierarchy aredetermined by a cubic NLS. The proof of uniqueness is accomplished byusing highly sophisticated Feynman graphs.




A remark about ESY solutions of the GP hierarchy

Solutions of the GP hierarchy are studied in “L1-type traceSobolev” spaces of k -particle marginals

{γ(k) | ‖γ(k)‖h1 < ∞}

with norms

‖γ(k)‖hα := Tr(|S(k,α)γ(k)|) ,(1.7)

where1

S(k,α) :=k∏

j=1

〈∇xj 〉α〈∇x′j 〉α .

1Here we use the standard notation: 〈y〉 :=p

1 + y2.Natasa Pavlovic (UT Austin) From bosons to PDE, and back



An alternative method for proving uniqueness of GP

Klainerman and Machedon (CMP 2008) introduced an alternativemethod for proving uniqueness in a space of density matricesequipped with the Hilbert-Schmidt type Sobolev norm

‖γ(k)‖Hαk := ‖S(k,α)γ(k)‖L2(Rdk×Rdk ).(1.8)

The method is based on:a reformulation of the relevant combinatorics via the “boardgame argument” andthe use of certain space-time estimates of the type:

‖Bj;k+1 eit∆(k+1)± γ(k+1)‖L2

t Hα(R×Rdk×Rdk ) . ‖γ(k+1)‖Hα(Rd(k+1)×Rd(k+1)).




The method of Klainerman and Machedon makes the assumption that the apriori space-time bound

‖Bj;k+1γ(k+1)‖L1

t H1k< Ck ,(1.9)

holds, with C independent of k .

Subsequently:

Kirkpatrick, Schlein and Staffilani (AJM 2011) proved the bound for thecubic GP in d = 2.

Chen and P. (JFA 2011) proved the relevant bound for the quintic GP ind = 1, 2.

Xie (arXiv 2013) proved the bound for the general power GP in d = 1, 2.

Key problem: proof of (1.9) for solutions to the cubic GP in d = 3 andderivation of the 3D cubic GP via BBGKY and PDE techniques.




The Cauchy problem for GP - joint work with T. Chen

The work of Klainerman and Machedon inspired us to studywell-posedness for the Cauchy problem for focusing and defocusingGP hierarchies.




Motivation for the word “back” in the title

The theory of nonlinear dispersive PDE is nowadays spectacularlysuccessful; fast progress and new powerful methods.

Hence natural questions appear:

Are properties of solutions of NLS generically shared bysolutions of the QFT it is derived from?Can methods of nonlinear dispersive PDE be “lifted” to QFTlevel?




Towards local well-posedness for the GP hierarchy

Problem: The equations for γ(k) do not close & no fixed pointargument.

Solution: Endow the space of sequences

Γ := ( γ(k) )k∈N.

with a suitable topology.




Revisiting the GP hierarchy

Recall,

∆(k)± = ∆xk −∆x′k

, with ∆xk =kX

j=1

∆xj .

We introduce the notation:

Γ = ( γ(k)( t , x1, . . . , xk ; x ′1, . . . , x′k ) )k∈N ,

b∆±Γ := ( ∆(k)± γ

(k) )k∈N ,

bBΓ := ( Bk+1γ(k+1) )k∈N .

Then, the cubic GP hierarchy can be written as2

i∂t Γ + b∆±Γ = µbBΓ .(2.1)

2Moreover, for µ = 1 we refer to the GP hierarchy as defocusing, and for µ = −1 asfocusing.




p-GP hierarchy

More generally, let p ∈ {2, 4}. We introduce the p-GP hierarchy as follows3

i∂t Γ + b∆±Γ = µbBΓ ,(2.2)

where bBΓ := ( Bk+ p2γ(k+ p

2 ) )k∈N .(2.3)

Here

Bk+ p2γ(k+ p

2 ) = B+

k+ p2γ(k+ p

2 ) − B−k+ p

2γ(k+ p

2 ) ,(2.4)

where

B+

k+ p2γ(k+ p

2 ) =kX

j=1

B+

j;k+1,...,k+ p2γ(k+ p

2 ),

B−k+ p

2γ(k+ p

2 ) =kX

j=1

B−j;k+1,...,k+ p

2γ(k+ p

2 ),

3We refer to (2.2) as the cubic GP hierarchy if p = 2, and as the quintic GPhierarchy if p = 4. Septic and so on considered in the recent work of Xie.




with

“B+

j;k+1,...,k+ p2γ(k+ p

2 )”

(t , x1, . . . , xk ; x ′1, . . . , x′k )

= γ(k+ p2 )(t , x1, . . . , xj , . . . , xk , xj , · · · xj| {z }

p2

; x ′1, . . . , x′k , xj , · · · , xj| {z }

p2

),

“B−

j;k+1,...,k+ p2γ(k+ p

2 )”

(t , x1, . . . , xk ; x ′1, . . . , x′k )

= γ(k+ p2 )(t , x1, . . . , xk , x ′j , · · · x ′j| {z }

p2

; x ′1, . . . , x′j , . . . x

′k , x′j , · · · , x ′j| {z }

p2

).




Spaces

Let

G :=∞M

k=1

L2(Rdk × Rdk )

be the space of sequences of density matrices

Γ := ( γ(k) )k∈N.

As a crucial ingredient of our arguments, we introduce Banach spacesHαξ = { Γ ∈ G | ‖ Γ ‖Hα

ξ<∞} where

‖ Γ ‖Hαξ

:=Xk∈N

ξk ‖ γ(k) ‖Hα(Rdk×Rdk ) .

Properties:

Finiteness: ‖ Γ ‖Hαξ< C implies that ‖ γ(k) ‖Hα(Rdk×Rdk ) < Cξ−k .

Interpretation: ξ−1 upper bound on typical Hα-energy per particle.




Recent results for the GP hierarchy

1 Local in time existence and uniqueness of solutions to GP.Chen, P. (DCDS 2010)

2 Blow-up of solutions to the focusing GP hierarchies in certain cases.Chen, P., Tzirakis (Ann. Inst. H. Poincare 2010)

3 Morawetz identities for the GP hierarchy.Chen, P., Tzirakis (Contemp. Math. 2012)

4 Global existence of solutions to the GP hierarchy in certain cases.Chen, P. (CPDE to appear)Chen, Taliaferro (arXiv 2013 )

5 A new proof of existence of solutions to the GP hierarchy.Chen, P. (PAMS 2013)

6 Derivation of the cubic GP hierarchy in [KM] spaces.Chen, P. (Ann. H. Poincare 2013)X. Chen (ARMA to appear)X. Chen, Holmer (arXiv 2013)

7 Uniqueness of the cubic GP hierarchy on T3.Gressman-Sohinger-Staffilani (arXiv 2012)




Existence of solutions to GP - joint works with T. Chen

Here we present two results:1 Local in time existence and uniqueness via a fixed point.2 Local in time existence via a truncation based argument.




Local in time existence and uniqueness - Chen, P. (DCDS 2010)

We prove local in time existence and uniqueness of solutions tothe cubic and quintic GP hierarchy with focusing or defocusinginteractions, in a subspace of Hαξ , for α ∈ A(d ,p), which satisfy aspacetime bound

‖BΓ‖L1t∈IH

αξ<∞,(3.1)

for some ξ > 0.




Flavor of the proof:

Note that the GP hierarchy can be formally written as a system of integralequations

Γ(t) = eit b∆±Γ0 − iµZ t

0ds ei(t−s)b∆± bBΓ(s)(3.2)

bBΓ(t) = bB eit b∆±Γ0 − iµZ t

0ds bB ei(t−s)b∆± bBΓ(s) ,(3.3)

where (3.3) is obtained by applying the operator bB on the linearnon-homogeneous equation (3.2).

We prove the local well-posedness result by applying the fixed pointargument in the following space:

Wαξ (I) := { Γ ∈ L∞t∈IHαξ | bBΓ ∈ L1

t∈IHαξ },(3.4)

where I = [0,T ].




A new proof of existence of solutions to GP - Chen, P. (PAMS 2013)

From our work on local well-posedness for GP, the question remainedwhether the condition

BΓ ∈ L1tHαξ(3.5)

is necessary for both existence and uniqueness.

In this work we prove that for the existence part, this a prioriassumption is not required.However, as an a posteriori result, we show that the solution thatwe obtain satisfies the property (3.5).




Flavor of the proof:

Fix K ∈ N. We consider solutions ΓK (t) of the GP hierarchy,

i∂t ΓK = b∆±ΓK + µbBΓK ,(3.6)

for the truncated initial data ΓK (0) = (γ(1)0 , . . . , γ

(K )0 , 0, 0, . . . ), where m-th

component of ΓK (t) = 0 for all m > K and establish:

Step 1 Existence of solutions to the truncated GP (3.6)

Step 2 Existence of the strong limit:

Θ = limK→∞

bBΓK ∈ L1t∈IHαξ′′ .

Step 3 Existence of the strong limit:

Γ = limK→∞

ΓK ∈ L∞t∈IHαξ ,

that satisfies the GP hierarchy, given the initial data Γ0 ∈ Hαξ′ .Step 4 Via obtaining equations that Θ and Γ satisfy we prove thatbBΓ = Θ.




Derivation of the cubic GP hierarchy in 3D - Chen, P. (Ann. H. Poincare 2013)

Above mentioned results on the GP: “from NLS to GP”

Long term goal: “from GP to BBGKY”...First result in that direction is the following derivation of the cubicGP.




TheoremLet d = 2, 3 and let δ > 0 be an arbitrary small, fixed number. Let 0 < β < 1

d+1+2δ . Let ΦN denote a solution of the N-body

Schrodinger equation, for which

ΓΦN (0) = (γ(1)ΦN

(0), . . . , γ(N)ΦN

(0), 0, 0, . . . )

has a strong limit Γ0 = limN→∞ ΓΦN (0) ∈ H1+δξ′

. Denote by

ΓΦN (t) := (γ(1)ΦN

(t), . . . , γ(N)ΦN

(t), 0, 0, . . . , 0, . . . )(4.1)

the solution to the associated BBGKY hierarchy. Define the truncation operator P≤K by P≤K Γ = (γ(1), . . . , γ(K ), 0, 0, . . . ) .

Then, letting K = K (N) = b0 log N , for a sufficiently large constant b0 > 0, we have

limN→∞

P≤K (N)ΓΦN = Γ(4.2)

strongly in L∞t∈IH1ξ , and

limN→∞

bBN P≤K (N)ΓΦN = bBΓ(4.3)

strongly in L2t∈[0,T ]

H1ξ , for ξ > 0 sufficiently small.

In particular, Γ solves the cubic, defocusing GP hierarchy with Γ(0) = Γ0 , and it is an element of the space W1ξ([0, T ]) used in our

previous works.




Remarks

The result implies that the N-BBGKY hierarchy has a limit in the spaceintroduced by [CP], which is based on the space considered by [KM].For factorized solutions, this provides the derivation of the cubicdefocusing NLS in those spaces.

We assume that the i.d. has a limit, ΓφN (0)→ Γ0 ∈ H1+δξ′ as N →∞,

which does not need to be factorized. We note that in [ESY], i.d. isassumed to be asymptotically factorized.

In [ESY], the limit γ(k)ΦN

⇀ γ(k) of solutions to the BBGKY to solutions tothe GP holds in the weak, subsequential sense, for an arbirary but fixedk . In our approach, we prove strong convergence for a sequence ofsuitably truncated solutions to the BBGKY, in an entirely different space.

For brevity, we will below write

‖(Γ , Θ)‖Wαξ

(I) := ‖Γ‖L∞t∈IHαξ

+ ‖Θ‖L2t∈IH

αξ




The key idea of the proof - the use of auxiliary truncations fromour recent proof of existence of solutions to the GP.




Main steps of the proof

1. Prove existence of solution ΓKN to N-BBGKY hierarchy in W1+δ

ξ ([0,T ])with truncated initial data P≤K ΓN(0).

2. Compare to solution ΓK of GP with truncated initial data P≤K Γ0,

‖(ΓK (N)N , bBNΓ

K (N)N )− (ΓK (N) , bBΓK (N) )‖W1

ξ([0,T ]) → 0 (N →∞)

3. Also compare to the truncated solution P≤K (N)ΓΦN of N-BBGKY,

‖(ΓK (N)N , bBNΓ

K (N)N )− (P≤K (N)Γ

ΦN , bBNP≤K (N)ΓΦN )‖W1

ξ([0,T ]) → 0

4. Prove that (ΓK (N), bBΓK (N))→ (Γ, bBΓ) in W1ξ([0,T ]) where Γ solves the

cubic defocusing GP hierarchy. Already done in PAMS paper.




Step 1: Solve N-BBGKY hierarchy with K -truncated initial data.

Let K < N be fixed.

Goal: Find solution ΓKN (t) of the BBGKY hierarchy,

i∂t ΓKN = b∆±ΓK

N + bBNΓKN

with K -truncated initial data

ΓKN (0) = P≤K ΓN(0) = (γ

(1)N (0), . . . , γ

(K )N (0), 0, 0, . . . ).

Strategy: a fixed point argument (using Duhamel, Strichartz, ...)

‖ ΓKN ‖L∞t∈IH

1+δξ

, ‖ bBNΓKN ‖L2

t∈IH1+δξ≤ C(T , ξ, ξ′) ‖ΓK

N (0)‖H1+δξ′

bounded, uniformly in K , N for ξ < ξ′.




Step 2: Compare ΓKN to solution ΓK of GP hierarchy with K -truncated initial

data P≤K Γ0.

Let b0 > 0 be a sufficiently large constant, and

K (N) = b0 log N

We prove the following convergence:

(a) In the limit N →∞, ΓK (N)N satisfies

limN→∞

‖ΓK (N)N − ΓK (N)‖L∞t H

1ξ

= 0 .(4.4)

(b) In the limit N →∞, bBNΓK (N)N satisfies

limN→∞

‖bBNΓK (N)N − bBΓK (N)‖L2

tH1ξ

= 0 .(4.5)

The proof of these limits makes use of the δ amount of extra regularity of theinitial data (beyond H1

ξ′ ).




Step 3: Compare ΓKN (N-BBGKY hierarchy with K -truncated data) to

P≤K (N)ΓΦN (K -truncation of N-BBGKY). Equal at t = 0 !

Letting K (N) = b0 log N for a sufficiently large constant b0 > 0 (the same asin the previous step), we prove that

limN→∞

‖ΓK (N)N − P≤K (N)Γ

ΦN‖L∞t∈IH1ξ

= 0 .(4.6)

and

limN→∞

‖BNΓK (N)N − BNP≤K (N)Γ

ΦN‖L2t∈IH

1ξ

= 0 .(4.7)

The proof of this limit involves the a priori energy bounds for the N-bodySchrodinger system established by Erdos-Schlein-Yau.




Step 4: Finally, use the strong limit proven in [CP (PAMS 2013)]

limN→∞

‖ bBΓK (N) − bBΓ ‖L2tH

1ξ

= 0 .(4.8)

Here,

ΓK solves GP with K -truncated initial data P≤K Γ0

Γ solves full GP hierarchy with initial data Γ0.




End of proof

Combining Steps 1 - 4 , we find that for K (N) = b0 log N,

limN→∞

‖ bBNP≤K (N)ΓΦN − bBΓ ‖L2

tH1ξ

= 0 .(4.9)

This straightforwardly implies that

limN→∞

‖P≤K (N)ΓΦN − Γ ‖L∞t H

1ξ

= 0 .

Thus, we have proven the strong convergence in W1ξ([0,T ]),

(P≤K (N)ΓΦN , bBNP≤K (N)Γ

ΦN ) −→ (Γ , bBΓ) ,

of N-body Schrodinger via BBGKY to a solution of the GP hierarchy.

From (4.9): Limit satisfies Klainerman-Machedon condition !




Summary of the proof

Start with solution of N-body Schrodinger equation.Artificially close BBGKY and GP hierarchies by truncation at K .Eliminates [KM] a priori assumption on spacetime norms.For suitable K (N)→∞, strong limit in W1

ξ(I) as N →∞.Obtain cubic defocusing GP hierarchy in 3D.The limit satisfies the Klainerman-Machedon condition.

This implies that the Klainerman-Machedon type spaces for solutionsof the GP hierarchy are natural.




Related works

Chen-Taliaferro: (arXiv 2013): Derivation of cubic GP in R3 forH1ξ data.

X.Chen-Holmer (arXiv 2013): Derivation of cubic GP in R3 forβ ∈ (0,2/3).

Via weak-* limit of γ(k)N and proof that the KM condition is satisfied,

uniformly in N.Proof uses X 1,b spaces.


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