Rate of convergence towards Hartree dynamics


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1 Rate of convergence towards Hartree dynamics Benjamin Schlein, LMU München and University of Cambridge Universitá di Milano Bicocca, October 22, 2007 Joint work with I. Rodnianski
2 1. Introduction boson system: quantum mechanical boson systems are described by a wave function ψ L 2 s (R3 ), symmetric w.r.t. permutations. ψ (x 1,..., x ) 2 = probability density ψ = 1 The dynamics is governed by the Schrödinger equation i t ψ,t = H ψ,t ψ,t = e ih t ψ. H is the Hamiltonian of the system, H = xj + i<j V (x i x j ) acts on L 2 s (R 3 ).
3 Mean field systems: boson system with Hamiltonian H = xj + 1 i<j V (x i x j ) on L 2 s (R 3 ) Evolution of a condensate: fix ϕ L 2 (R 3 ), consider the initial particle state ψ (x) = ϕ(x j ) and its timeevolution ψ,t = e ih t ψ If factorization is approximately preserved ψ,t (x) ϕ t (x j ) 1 V (x i x j ) 1 V (x i y) ϕ t (y) 2 dy = (V ϕ t 2 )(x i ) j i Hartree equation: i t ϕ t = ϕ t + (V ϕ t 2 )ϕ t
4 Marginal densities: let ψ,t = e ih t ψ and let γ,t = ψ,t ψ,t γ,t (x; x ) = ψ,t (x)ψ,t (x ) denote the density matrix associated with ψ,t. For k = 1,...,, define the kparticle marginal density γ (k),t (x k; x k ) = dx k γ,t (x k, x k ; x k, x k) with x k = (x 1,..., x k ), x k = (x 1,..., x k ), x k = (x k+1,..., x ) We use the convention Tr γ (k),t = 1 for all k,, t. For every kparticle observable J (k) on L 2 (R 3k ) ψ,t, ( J (k) 1 ( k)) ψ,t = Tr ( J (k) 1 ( k)) γ,t = Tr J (k) γ (k),t
5 Theorem (under appropriate assumptions on V ): Consider the factorized initial wave function ψ (x) = ϕ(x j ) for arbitrary ϕ H 1 (R 3 ) with ϕ = 1 and let γ (k),t be the kparticle marginal ass. with ψ,t = e ih t ψ. Then, for every fixed k 1 and t R, we have γ (k),t ϕ t ϕ t k as (in the trace class norm). Here ϕ t is the solution of the nonlinear Hartree equation with initial data ϕ t=0 = ϕ. i t ϕ t = ϕ t + (V ϕ t 2 )ϕ t In particular, for every fixed k, and every bounded J (k), as. ψ,t, ( J (k) 1 ( k)) ψ,t ϕ k t, J (k) ϕ k t
6 Proof of the theorem: Spohn, 1980: bounded potential. ErdösYau, 2000: derivation of Hartree for Coulomb potential (partial results also by BardosGolseMauser). ElgartS., 2005: derivation of relativistic Hartree i t ϕ t = ( ) 1 1 ϕ t + λ. ϕ t 2 ϕ t starting from H = 1 j + λ i<j 1 x i x j for all λ > 4/π (equation is critical) and ϕ H 1/2 (R 3 ).
7 Let H = j + 1 i<j 3β V ( β (x i x j )) ErdösS.Yau, 2006: for β < 1/2, derivation of nonlinear Schrödinger equation i t ϕ t = ϕ t + b 0 ϕ t 2 ϕ t with b 0 = dxv (x) Analogous results in 1dim by AdamiBardosGolseTeta. ErdösS.Yau, 2007: for β 1, derivation of i t ϕ t = ϕ t + σ ϕ t 2 ϕ t with σ = { b0 if β < 1 8πa 0 if β = 1
8 Spohn s strategy: study BBGKY hierarchy γ (k),t i t γ (k),t = k + [ j, γ (k),t ( 1 k ) k ] + 1 k i<j [ V (x i x j ), γ (k),t [ Tr k+1 V (x j x k+1 ), γ (k+1) is compact it has at least one limit point γ(k),t Limit points {γ (k),t } k 1 satisfy the infinite hierarchy i t γ (k),t = k Observe: γ (k) t [ j, γ (k),t ] + k ],t [ Tr k+1 V (x j x k+1 ), γ (k+1) = ϕ t ϕ t k, k 1 solves inf. hier. iff i t ϕ t = ϕ t + (V ϕ t 2 )ϕ t Proving the uniqueness of the infinite hierarchy, it follows that γ (k),t ϕ t ϕ t k for all k, t ],t ]
9 2. Rate of convergence towards Hartree dynamics. Question: What can be said about the error γ (1),t ϕ t ϕ t? Expanding the solution of the BBGKY hierarchy, one gets Tr γ(1),t ϕ t ϕ t C 1 for t < T 0 Iterating only leads to very bad bounds Tr γ(1),t ϕ t ϕ t C 1 2 t
10 Theorem (RodnianskiS., 2007): suppose that V 2 (x) const (1 ) Let ψ (x) = ϕ(x j ), for ϕ H 1 (R 3 ) and let ψ,t = e ih t ψ Moreover let ϕ t denote the solution of the Hartree equation i t ϕ t = ϕ t + (V ϕ t 2 )ϕ t Then there exists constants C, K > 0 such that, for all t R, Tr γ(1),t ϕ t ϕ t C ekt 1/4 Remarks: we can prove analogous bounds for kparticle marginal density. V (x) = λ/ x is allowed for arbitrary λ R. dependence is not optimal (1/ is expected).
11 Fock space representation (Hepp (1973), GinibreVelo (1979)): Bosonic Fock space: F = n 0 L 2 s (R 3n, dx 1... dx n ). Vectors in F are sequences ψ = {ψ (n) } n 1 with ψ (n) L 2 s (R 3n ). Special vectors: vectors like {0, 0,..., 0, ψ n, 0,... } F with ψ n L 2 (R 3n ) have a fixed number of particles (example Ω = {1, 0,... }). umber of particle operator: defined by ( ψ) (m) = mψ (m). Hamiltonian: we define (H ψ) (m) = H (m) ψ(m) with ote that H (m) = m xj + 1 m i<j V (x i x j ) e ith {0, 0,..., ψ, 0,... } = {0, 0,..., e ith ψ, 0,... }
12 Creation and annihilation operators: for f L 2 (R 3 ), define ( a (f)ψ ) (n) (x1,..., x n ) = 1 n n (a(f)ψ) (n) (x 1,..., x n ) = n + 1 f(x j )ψ (n 1) (x 1,..., ˆx j,..., x n ) dx f(x)ψ (n+1) (x, x 1,..., x n ) CCR: [a(f), a (g)] = (f, g) L 2 [a(f), a(g)] = [a (f), a (g)] = 0 Introduce also the operatorvalued distributions a x, a x s.t. a (f) = dx f(x)a x and a(f) = dx f(x) a x CCR: [a x, a y] = δ(x y) [a x, a y ] = [a x, a y] = 0
13 Then we have the representations = H = dx a xa x dx x a x x a x + 1 dxdyv (x y)a xa ya y a x Creation and annihilation operators are not bounded, but a(f)ψ f 1/2 ψ and a (f)ψ f ( +1) 1/2 ψ States in F can be obtained applying creation operators on the vacuum. For example, factorized states are given by ψ (x) = ϕ(x j ) {0,..., 0, ψ, 0,... } = a (ϕ)! Ω
14 Coherent states: for ϕ L 2 (R 3 ) define the Weyl operator W (ϕ) = exp(a (ϕ) a(ϕ)) The coherent state with wave function ϕ is then given by W (ϕ)ω = e ϕ 2 /2 j=0 a (ϕ) j Ω j! W (ϕ) = W (ϕ) 1 = W ( ϕ) W (ϕ)ω, W (ϕ)ω = ϕ 2 We have W (ϕ) a x W (ϕ) = a x + ϕ(x), W (ϕ) a x W (ϕ) = a x + ϕ(x) a x W (ϕ)ω = ϕ(x) W (ϕ)ω
15 Theorem (dynamics of coherent states): Suppose that V 2 (x) C(1 ) Choose ϕ H 1 (R 3 ), ϕ = 1. Consider the coherent initial state W ( { ϕ)ω = e /2 1, ϕ,..., j/2 } ϕ j,... j! and let Γ (1),t be the 1part. density ass. with ψ,t = e ih t W ( ϕ) Ω. where ϕ t solves Tr Γ(1),t ϕ t ϕ t C ekt i t ϕ t = ϕ t + (V ϕ t 2 )ϕ t with ϕ t=0 = ϕ.
16 Oneparticle density: in general, the kernel of the oneparticle density associated with ψ F is given by γ (1) 1 ψ (x; y) = ψ, ψ ψ, a ya x ψ Trγ (1) ψ = 1 Therefore, we are interested in Γ (1),t (x, y) = 1 e ih t W ( ϕ)ω, a y a x e ih t W ( ϕ)ω We expand = 1 Ω, W ( ϕ)e ih t a y a x e ih t W ( ϕ)ω e ih t a ye ih t = ϕ t (y) + e ih t (a y ϕ t (x))e ih t e ih t a x e ih t = ϕ t (x) + e ih t (a x ϕ t (x))e ih t around the solution of i t ϕ t = ϕ t + (V ϕ t 2 )ϕ t
17 Fluctuation dynamics: it was observed by Hepp that W ( ϕ)e ih t ( a y ϕ t (y) ) e ih t W ( ϕ) = U (t) a y U (t) where i t U (t) = L (t)u (t) with U (0) = 1 with timedependent generator L (t) = + dx ( x a x x a x + (V ϕ t 2 )(x)a xa x ) dxdy V (x y)ϕ t (x)ϕ t (y) a xa y + dxdy V (x y) ( ϕ t (x)ϕ t (y) a xa ) y + ϕ t (x)ϕ t (y)a x a y + 1 dxdy V (x y) a ( x ϕt (y)a y + ϕ t (y)a y) ax + 1 dxdy V (x y) a x a y a ya x
18 Therefore Γ (1),t (x, y) ϕ t(x)ϕ t (y) = 1 Ω, U (t)a y a x U (t)ω + ϕ t(x) Ω, U (t)a yu (t)ω + ϕ t(y) Ω, U (t)a xu (t)ω Main problem: show that Ω, U (t) U (t)ω Ce Kt
19 First attempt: d dt Ω,U (t) U (t)ω = Ω, U (t)[il (t), ]U (t)ω = 2 dxdy V (x y) ( ϕ t (x)ϕ t (y) Ω, U (t)a xa yu (t)ω + c.c. ) + 1/2 dxdyv (x y) ( ϕ t (y) Ω, U (t)a xa ya x U (t)ω + c.c. ) (since L (t) = + dx ( x a x x a x + (V ϕ t 2 )(x)a xa x ) dxdy V (x y)ϕ t (x)ϕ t (y) a xa y + dxdy V (x y) ( ϕ t (x)ϕ t (y) a x a y + ϕ ) t(x)ϕ t (y)a x a y + 1 dxdy V (x y) a ( x ϕt (y)a y + ϕ t (y)a y) ax + 1 dxdy V (x y) a xa ya y a x ) d dt Ω, U (t) U (t)ω C Ω, U (t) U (t)ω
20 Instead: define new Hamiltonian with cutoff in cubic term L (t) = + dx ( x a x xa x + (V ϕ t 2 )(x)a x a x dxdy V (x y)ϕ t (x)ϕ t (y) a xa y + dxdy V (x y) ( ϕ t (x)ϕ t (y) a xa ) y + ϕ t (x)ϕ t (y)a x a y + 1 dxdy V (x y) ( ϕ t (y) a xa y χ( ) a x + h.c. ) + 1 dxdy V (x y) a xa ya y a x and the unitary evolution i Ũ(t) = L (t)ũ(t) with Ũ(0) = 1 Conclude proof by showing that Ω, Ũ (t) Ũ(t)Ω Ce Kt ) Ω, U (t) U (t)ω C
21 Remark: Hepp (1973) and GinibreVelo (1979) proved that where s lim U (t) = U(t) with generator L(t) = + + i t U(t) = L(t)U(t) with U(0) = 1 dx ( x a x x a x + (V ϕ t 2 )(x)a xa x ) dxdy V (x y)ϕ t (x)ϕ t (y) a x a y dxdy V (x y) ( ϕ t (x)ϕ t (y) a xa ) y + ϕ t (x)ϕ t (y)a x a y They only need to control the growth of the number of particles w.r.t. the limiting dynamics L(t), in the sense Ω, U (t) U(t)Ω Ce Kt
22 From coherent states to factorized states: note that {0,..., 0, ϕ, 0,... } = c 2π 0 dθ 2π eiθ W ( e iθ ϕ)ω with c =! e /2 1/4 /2 In fact, using that we get 2π 0 W ( e iθ ϕ)ω = e /2 j 0 dθ 2π eiθ W ( e iθ ϕ)ω = e /2 j/2 j 0 j! = e /2 /2! e ijθ (a (ϕ)) j Ω j/2 j! a (ϕ) j Ω a (ϕ) Ω! 2π = 1 c {0, 0,..., 0, ϕ, 0,... } 0 dθ 2π ei( j)θ
23 Let γ (1),t be the 1part. dens. of {0,.., 0, e ih t ϕ, 0,..}, then γ (1),t (x, y) = 1 {0,.., 0, ϕ, 0,..}, e ih t a ya x e ih t {0,.., 0, ϕ, 0,..} = c2 2π 2π dθ 1 dθ 2 0 2π 0 2π e iθ 1e iθ 2 Ω, W ( e iθ 1ϕ)e iht a y a x e iht W ( e iθ 2ϕ)Ω Expanding e ih t a y e ih t = e iθ 1 ϕ t (y) + e ih t ( a y e iθ 1ϕ t (y) ) e ih t e ih t a x e ih t = e iθ 2 ϕ t (x) + e ih t ( a x e iθ 2ϕ t (x) ) e ih t and using that c 2π 0 dθ 2π ei( 1)θ W ( e iθ ϕ)ω = ϕ ( 1)
24 we obtain γ (1),t (x, y) ϕ t(x)ϕ t (y) = c2 dθ1 dθ 2 (2π) 2 e iθ 1e iθ 2 Ω, W ( e iθ 1ϕ)e ih t ( a y e iθ 1ϕ t (y) ) + c ϕ t (y) ( a x e iθ 2ϕ t (x) ) W ( e iθ 2ϕ)Ω dθ2 (2π) eiθ 2 ϕ ( 1), e ih t (a x e iθ 2ϕ t (x))e ih t W ( e iθ 2ϕ)Ω + c ϕ t (x) dθ1 (2π) e iθ 1 Ω, W ( e iθ 1ϕ)e ih t (a y e iθ 1 ϕ t (y))e ih t ϕ ( 1) This implies the theorem for the evolution of factorized states.
25 Why V 2 (1 )? In the commutator [il (t), ] we have to deal with terms like because dxdy V (x y)ϕ t (x)ϕ t (y) Ω, U (t) a x a y U (t)ω = dxϕ t (x) a x U (t)ω, a (V (x.)ϕ t )U (t)ω dx ϕ t (x) a x U (t)ω a (V (x.)ϕ t )U (t)ω sup x V (x.)ϕ t L 2 ( + 1) 1/2 U (t)ω 2 a (f)ψ f L 2 ( + 1) 1/2 ψ With the assumption V 2 (1 ), we can bound V (x.)ϕ t 2 = uniformly in time. dyv 2 (x y) ϕ t (y) 2 ϕ t H 1 const
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