From many body quantum dynamics to nonlinear dispersive PDE, and back

Transcription

1 From many body quantum dynamics to nonlinear dispersive PDE, and back ataša Pavlović (based on joint works with T. Chen, and with T. Chen and. Tzirakis) University of Texas at Austin June 18, 2013 SF-CBMS Regional Research Conference in the Mathematical Sciences Kansas State University

2 Outline Introduction: from bosons to LS 1 Introduction: from bosons to LS 2 3 4

3 Introduction The mathematical analysis of interacting Bose gases is a hot topic in Math Phys. Major research directions are: 1 Proof of Bose-Einstein condensation Aizenman-Lieb-Seiringer-Solovej-Yngvason 2 Derivation of nonlinear Schrödinger or Hartree eq. Via Fock space: Hepp, Ginibre-Velo, Rodnianski-Schlein, Grillakis-Machedon-Margetis, Grillakis-Machedon, X. Chen Via BBGKY: Spohn, Elgart-Erdös-Schlein-Yau, Erdös-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: Fröhlich-Graffi-Schwarz, Fröhlich-Knowles-Pizzo, Anapolitanos-Sigal, Abou-Salem, Pickl

4 From bosons to LS following Erdös-Schlein-Yau [ ] Step 1: From -body Schrödinger to BBGKY hierarchy The starting point is a system of bosons whose dynamics is generated by the Hamiltonian (1.1) H := ( xj ) + 1 j=1 V (x i x j ), 1 i<j on the Hilbert space H = L 2 sym(r d ), whose elements Ψ(x 1,..., x ) are fully symmetric with respect to permutations of the arguments x j. Here with 0 < β 1. V (x) = dβ V ( β x),

5 The solutions of the Schrödinger equation i t Ψ = H Ψ with initial condition Ψ,0 H determine: the -particle density matrix γ (t, x ; x ) = Ψ (t, x )Ψ (t, x ), and its k-particle marginals γ (k) (t, x k; x k ) = dx k γ (t, x k, x k ; x k, x k), for k = 1,...,. Here x k = (x 1,..., x k ), x k = (x k+1,..., x ).

6 The BBGKY hierarchy is given by (1.2) (1.3) i t γ (k) + 1 = ( x k x k )γ (k) [V (x i x j ) V (x + k 1 i<j k k i=1 i x j )] γ(k) Tr k+1 [V (x i x k+1 ) V (x i x k+1 )] γ (k+1) In the limit, the sums weighted by combinatorial factors have the following size: In (1.2), k 2 0 for every fixed k and sufficiently small β. In (1.3), k 1 for every fixed k and V (x i x j ) b 0 δ(x i x j ), with b 0 = dx V (x).

7 Step 2: BBGKY hierarchy GP hierarchy As, one obtains the infinite GP hierarchy as a weak limit. (1.4) i t γ (k) = k ( xj x j ) γ (k) + b 0 j=1 k j=1 B j;k+1 γ (k+1) where the contraction operator is given via ( ) (1.5) B j;k+1 γ (k+1) (t, x 1,..., x k ; x 1,..., x k ) := dx k+1 dx k+1 [ δ(xj x k+1 )δ(x j x k+1 ) δ(x j x k+1 )δ(x j x k+1 ) ] γ (k+1) (t, x 1,..., x j,..., x k, x k+1 ; x 1,..., x k, x k+1 ).

8 Step 3: Factorized solutions of GP and LS It is easy to see that γ (k) = φ φ k := k j=1 φ(t, x j ) φ(t, x j ) is a solution of the GP if φ satisfies the cubic LS with φ 0 L 2 (R d ). i t φ + x φ b 0 φ 2 φ = 0

9 Step 4: Uniqueness of solutions of GP hierarchies While the existence of factorized solutions can be easily obtained, the proof of uniqueness of solutions of the GP hierarchy is the most difficult part in this analysis.

10 Why is it difficult to prove uniqueness? Fix a positive integer r. One can express the solution γ (r) to the GP hierarchy as: (1.6) γ (r) Z tr (tr, ) = 0 ei(tr t r+1 ) (r) ± Br+1 (γ (r+1) (t r+1 )) dt r+1 Z tr = 0 Z tr+1 0 e i(tr t r+1 ) (r) ± Br+1 e i(t r+1 t r+2 ) (r+1) ± Br+2 (γ (r+2) (t r+2 )) dt r+1 dt r+2 =... Z tr = 0... Z tr+n 1 J r (t r+n ) dt r+1...dt r+n, 0 where rx B r+1 = B j;r+1, j=1 t r+n = (tr, t r+1,..., t r+n ), J r (t r+n ) = e i(tr t r+1 ) (r) ± Br+1...e i(t r+(n 1) t r+n ) (r+(n 1)) ± B r+n (γ (r+n) (t r+n )). Since the interaction term involves the sum, the iterated Duhamel s formula has r(r + 1)...(r + n 1) terms.

11 Summary of the method of ESY Roughly speaking, the method of Erdös, Schlein, and Yau for deriving the cubic LS justifies the heuristic explained above and it consists of the following two steps: (i) Deriving the Gross-Pitaevskii (GP) hierarchy as the limit as of the BBGKY hierarchy, for a scaling where the particle interaction potential tends to a delta distribution. (ii) Proving uniqueness of solutions for the GP hierarchy, which implies that for factorized initial data, the solutions of the GP hierarchy are determined by a cubic LS. The proof of uniqueness is accomplished by using highly sophisticated Feynman graphs.

12 A remark about ESY solutions of the GP hierarchy Solutions of the GP hierarchy are studied in L 1 -type trace Sobolev spaces of k-particle marginals with norms {γ (k) γ (k) h 1 < } (1.7) γ (k) h α := Tr( S (k,α) γ (k) ), where 1 S (k,α) := k xj α x j α. j=1 1 Here we use the standard notation: y := p 1 + y 2.

13 An alternative method for proving uniqueness of GP Klainerman and Machedon (CMP 2008) introduced an alternative method for proving uniqueness in a space of density matrices equipped with the Hilbert-Schmidt type Sobolev norm (1.8) γ (k) H α k := S (k,α) γ (k) L 2 (R dk R dk ). The method is based on: a reformulation of the relevant combinatorics via the board game argument and the use of certain space-time estimates of the type: B j;k+1 e it (k+1) ± γ (k+1) L 2 t Ḣ α (R R dk R dk ) γ(k+1) Ḣα (R d(k+1) R d(k+1) ).

14 The method of Klainerman and Machedon makes the assumption that the a priori space-time bound (1.9) B j;k+1 γ (k+1) L 1 t Ḣ 1 k < C k, holds, with C independent of k. Subsequently: Kirkpatrick, Schlein and Staffilani (AJM 2011) proved the bound for the cubic GP in d = 2. Chen and P. (JFA 2011) proved the relevant bound for the quintic GP in d = 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 and derivation of the 3D cubic GP via BBGKY and PDE techniques.

15 - joint work with T. Chen The work of Klainerman and Machedon inspired us to study well-posedness for the Cauchy problem for focusing and defocusing GP hierarchies.

16 Motivation for the word back in the title The theory of nonlinear dispersive PDE is nowadays spectacularly successful; fast progress and new powerful methods. Hence natural questions appear: Are properties of solutions of LS generically shared by solutions of the QFT it is derived from? Can methods of nonlinear dispersive PDE be lifted to QFT level?

17 Towards local well-posedness for the GP hierarchy Problem: The equations for γ (k) do not close & no fixed point argument. Solution: Endow the space of sequences with a suitable topology. Γ := ( γ (k) ) k.

18 Revisiting the GP hierarchy Recall, (k) ± = x k x k, with x k = kx xj. j=1 We introduce the notation: Γ = ( γ (k) ( t, x 1,..., x k ; x 1,..., x k) ) k, b ±Γ := ( (k) ± γ(k) ) k, bbγ := ( B k+1 γ (k+1) ) k. Then, the cubic GP hierarchy can be written as 2 (2.1) i tγ + b ±Γ = µ b BΓ. 2 Moreover, for µ = 1 we refer to the GP hierarchy as defocusing, and for µ = 1 as focusing.

19 p-gp hierarchy More generally, let p {2, 4}. We introduce the p-gp hierarchy as follows 3 (2.2) where (2.3) Here (2.4) where i tγ + b ±Γ = µ b BΓ, bbγ := ( B k+ p γ (k+ p 2 ) ) k. 2 B k+ p γ (k+ p 2 ) = B + 2 k+ p γ (k+ p 2 ) B 2 k+ p γ (k+ p 2 ), 2 B + k+ p 2 γ (k+ p 2 ) = B k+ p 2 γ (k+ p 2 ) = kx B + j;k+1,...,k+ p γ (k+ p 2 ), 2 j=1 kx B j;k+1,...,k+ p γ (k+ p 2 ), 2 3 We refer to (2.2) as the cubic GP hierarchy if p = 2, and as the quintic GP hierarchy if p = 4. Septic and so on considered in the recent work of Xie. j=1

20 with B + j;k+1,...,k+ p 2 γ (k+ p 2 ) (t, x 1,..., x k ; x 1,..., x k) = γ (k+ p 2 ) (t, x 1,..., x j,..., x k, x j, x j ; x 1,..., x k, x j,, x j ), {z } {z } p p 2 2 B j;k+1,...,k+ p 2 γ (k+ p 2 ) (t, x 1,..., x k ; x 1,..., x k) = γ (k+ p 2 ) (t, x 1,..., x k, x j, x j ; x 1,..., x j,... x k, x j,, x j ). {z } {z } p p 2 2

21 Spaces Let G := M L 2 (R dk R dk ) k=1 be the space of sequences of density matrices Γ := ( γ (k) ) k. As a crucial ingredient of our arguments, we introduce Banach spaces H α ξ = { Γ G Γ H α ξ < } where Γ H α ξ := X k ξ k γ (k) H α (R dk R dk ). Properties: Finiteness: Γ H α ξ < C implies that γ (k) H α (R dk R dk ) < Cξ k. Interpretation: ξ 1 upper bound on typical H α -energy per particle.

22 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 T 3. Gressman-Sohinger-Staffilani (arxiv 2012)

23 - 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.

24 Local in time existence and uniqueness - Chen, P. (DCDS 2010) We prove local in time existence and uniqueness of solutions to the cubic and quintic GP hierarchy with focusing or defocusing interactions, in a subspace of Hξ α, for α A(d, p), which satisfy a spacetime bound (3.1) BΓ L 1 t I H α ξ <, for some ξ > 0.

25 Flavor of the proof: ote that the GP hierarchy can be formally written as a system of integral equations (3.2) (3.3) Z t Γ(t) = e it b ± Γ 0 iµ bbγ(t) = b B e it b ± Γ 0 iµ 0 Z t ds e i(t s)b ± b BΓ(s) 0 ds b B e i(t s)b ± b BΓ(s), where (3.3) is obtained by applying the operator b B on the linear non-homogeneous equation (3.2). We prove the local well-posedness result by applying the fixed point argument in the following space: (3.4) W α ξ (I) := { Γ L t IH α ξ b BΓ L 1 t IH α ξ }, where I = [0, T ].

26 A new proof of existence of solutions to GP - Chen, P. (PAMS 2013) From our work on local well-posedness for GP, the question remained whether the condition (3.5) BΓ L 1 t H α ξ is necessary for both existence and uniqueness. In this work we prove that for the existence part, this a priori assumption is not required. However, as an a posteriori result, we show that the solution that we obtain satisfies the property (3.5).

27 Flavor of the proof: Fix K. We consider solutions Γ K (t) of the GP hierarchy, (3.6) i tγ K = b ±Γ K + µ b BΓ K, for the truncated initial data Γ K (0) = (γ (1) 0,..., γ(k ) 0, 0, 0,... ), where m-th component of Γ K (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 = Step 3 Existence of the strong limit: Γ = lim bbγ K L 1 t IH α ξ K. lim K ΓK L t IHξ α, that satisfies the GP hierarchy, given the initial data Γ 0 H α ξ. Step 4 Via obtaining equations that Θ and Γ satisfy we prove that bbγ = Θ.

28 - Chen, P. (Ann. H. Poincare 2013) Above mentioned results on the GP: from LS to GP Long term goal: from GP to BBGKY... First result in that direction is the following derivation of the cubic GP.

29 Theorem Let d = 2, 3 and let δ > 0 be an arbitrary small, fixed number. Let 0 < β < 1 d+1+2δ. Let Φ denote a solution of the -body Schrödinger equation, for which Γ Φ (0) = (γ (1) (0),..., γ () (0), 0, 0,... ) Φ Φ has a strong limit Γ 0 = lim Γ Φ (0) H 1+δ ξ. Denote by (4.1) Γ Φ (t) := (γ (1) Φ (t),..., γ () Φ (t), 0, 0,..., 0,... ) 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 () = b 0 log, for a sufficiently large constant b 0 > 0, we have (4.2) lim P K () ΓΦ = Γ strongly in L t I H1 ξ, and (4.3) lim bb P K () Γ Φ = b BΓ strongly in L 2 t [0,T ] H1 ξ, for ξ > 0 sufficiently small. In particular, Γ solves the cubic, defocusing GP hierarchy with Γ(0) = Γ 0, and it is an element of the space W 1 ξ ([0, T ]) used in our previous works.

30 Remarks The result implies that the -BBGKY hierarchy has a limit in the space introduced by [CP], which is based on the space considered by [KM]. For factorized solutions, this provides the derivation of the cubic defocusing LS in those spaces. We assume that the i.d. has a limit, Γ φ (0) Γ 0 H 1+δ ξ as, which does not need to be factorized. We note that in [ESY], i.d. is assumed to be asymptotically factorized. In [ESY], the limit γ (k) Φ γ (k) of solutions to the BBGKY to solutions to the GP holds in the weak, subsequential sense, for an arbirary but fixed k. In our approach, we prove strong convergence for a sequence of suitably truncated solutions to the BBGKY, in an entirely different space. For brevity, we will below write (Γ, Θ) W α ξ (I) := Γ L t I H α ξ + Θ L 2 t I Hα ξ

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

32 Main steps of the proof 1. Prove existence of solution Γ K to -BBGKY hierarchy in W 1+δ ξ ([0, T ]) with truncated initial data P K Γ (0). 2. Compare to solution Γ K of GP with truncated initial data P K Γ 0, (Γ K (), B b Γ K () ) (Γ K (), BΓ b K () ) W 1 ξ ([0,T ]) 0 ( ) 3. Also compare to the truncated solution P K () Γ Φ of -BBGKY, (Γ K (), B b Γ K () ) (P K () Γ Φ, B b P K () Γ Φ ) W 1 ξ ([0,T ]) 0 4. Prove that (Γ K (), b BΓ K () ) (Γ, b BΓ) in W 1 ξ([0, T ]) where Γ solves the cubic defocusing GP hierarchy. Already done in PAMS paper.

33 Step 1: Solve -BBGKY hierarchy with K -truncated initial data. Let K < be fixed. Goal: Find solution Γ K (t) of the BBGKY hierarchy, with K -truncated initial data i tγ K = b ±Γ K + b B Γ K Γ K (0) = P K Γ (0) = (γ (1) ) (0),..., γ(k (0), 0, 0,... ). Strategy: a fixed point argument (using Duhamel, Strichartz,...) Γ K L t I H 1+δ ξ, b B Γ K L 2 t I H 1+δ ξ C(T, ξ, ξ ) Γ K (0) H 1+δ ξ bounded, uniformly in K, for ξ < ξ.

34 Step 2: Compare Γ K to solution Γ K of GP hierarchy with K -truncated initial data P K Γ 0. Let b 0 > 0 be a sufficiently large constant, and We prove the following convergence: (a) In the limit, Γ K () K () = b 0 log satisfies (4.4) () lim ΓK Γ K () L t H 1 ξ = 0. (b) In the limit, b B Γ K () satisfies (4.5) lim b B Γ K () BΓ b K () L 2 t H 1 ξ = 0. The proof of these limits makes use of the δ amount of extra regularity of the initial data (beyond H 1 ξ ).

35 Step 3: Compare Γ K (-BBGKY hierarchy with K -truncated data) to P K () Γ Φ (K -truncation of -BBGKY). Equal at t = 0! Letting K () = b 0 log for a sufficiently large constant b 0 > 0 (the same as in the previous step), we prove that (4.6) and (4.7) () lim ΓK P K () Γ Φ L t I H 1 ξ lim B Γ K () B P K () Γ Φ L 2 t I H 1 ξ = 0. = 0. The proof of this limit involves the a priori energy bounds for the -body Schrödinger system established by Erdös-Schlein-Yau.

36 Step 4: Finally, use the strong limit proven in [CP (PAMS 2013)] (4.8) lim b BΓ K () b BΓ L 2 t H 1 ξ = 0. Here, Γ K solves GP with K -truncated initial data P K Γ 0 Γ solves full GP hierarchy with initial data Γ 0.

37 End of proof Combining Steps 1-4, we find that for K () = b 0 log, (4.9) lim b B P K () Γ Φ b BΓ L 2 t H 1 ξ = 0. This straightforwardly implies that lim P K ()Γ Φ Γ L t H 1 ξ = 0. Thus, we have proven the strong convergence in W 1 ξ([0, T ]), (P K () Γ Φ, b B P K () Γ Φ ) (Γ, b BΓ), of -body Schrödinger via BBGKY to a solution of the GP hierarchy. From (4.9): Limit satisfies Klainerman-Machedon condition!

38 Summary of the proof Start with solution of -body Schrödinger equation. Artificially close BBGKY and GP hierarchies by truncation at K. Eliminates [KM] a priori assumption on spacetime norms. For suitable K (), strong limit in W 1 ξ (I) as. Obtain cubic defocusing GP hierarchy in 3D. The limit satisfies the Klainerman-Machedon condition. This implies that the Klainerman-Machedon type spaces for solutions of the GP hierarchy are natural.

39 Related works Chen-Taliaferro: (arxiv 2013): Derivation of cubic GP in R 3 for H 1 ξ data. X.Chen-Holmer (arxiv 2013): Derivation of cubic GP in R 3 for β (0, 2/3). Via weak-* limit of γ (k) uniformly in. Proof uses X 1,b spaces. and proof that the KM condition is satisfied,