Propagation of Monokinetic Measures with Rough Momentum Profiles I

Transcription

1 with Rough Momentum Profiles I Ecole Polytechnique Centre de Mathématiques Laurent Schwartz Quantum Systems: A Mathematical Journey from Few to Many Particles May 16th 2013 CSCAMM, University of Maryland. Work in collaboration with Claude Bardos, Peter Markowich and Thierry Paul

2 Statement of the problem Def: Monokinetic probability measure in the 1-particle phase space R N x R N ξ, with density ρin and momentum profile U in U in (x) R N µ in (x, dξ):=ρ in (x)δ U in (x)(ξ) where ρ in 0 a.e., and ρ in (x)dx =1 R N Hamiltonian flow Φ t : R N x R N ξ (x, ξ) (X, Ξ)(t; x, ξ) RN x R N ξ generated by Hamiltonian system (H) Ẋ = D ξ H(X, Ξ), Ξ = D x H(X, Ξ)

3 Statement of problem II I.e. t Φ t (x, ξ) = solution of (H) s.t. (x, ξ) at Φ 0 (x, ξ) = (x, ξ) If T : X Y is measurable and µ is a probability measure on X, define a probability measure ν on Y by ν(b) := µ(t 1 (B)), denoted ν = T #µ Propagated measure: with Hamiltonian flow Φ t, we define µ(t) := Φ t #µ in Space marginal of µ(t): with the notation Π : (x, ξ) x, we set ρ(t) := Π#µ(t) = F t #(ρ in L N ) i.e. ρ(t, ) = µ(t,, dξ) R N

4 Statement of problem III To study the structure of the propagated phase space probability measure µ(t) and of its space marginal ρ(t) for all t R For instance Is µ(t) still a monokinetic measure? if not Is µ(t) representable in terms of monokinetic measures? Is ρ(t) a probability density for all t R? if not What can be said of the singular component of ρ(t)? Moreover, we are interested in answering these questions under the most general regularity assumptions possible on ρ in and U in. Earlier research in this direction: Gasser-Markowich ( 94), Sparber- Markowich-Mauser ( 03)...

5 Motivation: classical limit of Schrödinger s equation Classical limit of Schrödinger s equation for x R N : iɛ t ψ ɛ ɛ2 x ψ ɛ = V (x)ψ ɛ, ψ ɛ (0, x) = a in (x)e isin (x)/ɛ Wigner function at scale ɛ: for each wave function Ψ L 2 (R N ) W ɛ [Ψ](x, ξ) := 1 (2π) N R N Ψ(x ɛy)ψ(x 1 2 ɛy)e iξ y dy Case of WKB ansatz: for a in L 2 (R N ) and S in W 1,1 loc (RN ) W ɛ [a in e Sin /ɛ ](x, ξ) a in (x) 2 δ x S in (x)(ξ)

6 From Schrödinger to Liouville Thm (Lions-Paul 93): Assume a in L 2 (R N ) and S in W 1,1 loc (RN ) Let V C 2 (R N ) satisfy, for some α > N/2, the condition V (x)=o( x ) and V (x)=o( x α ) as x Set ψ ɛ (t, ) := e i t ɛ ( ɛ2 2 x +V (x) ) a in e isin /ɛ Then (a) W ɛ [ψ ɛ ] µ 0 in S (R N x R N ξ ) as ɛ 0+ and (b) { t µ + ξ x µ x V (x) ξ µ = 0 µ t=0 = a in (x) 2 δ x S in (x)(ξ)

7 Hamiltonian propagation of Wigner measure Hence µ(t) = (X t, Ξ t )#µ in with µ in (x, ξ) := a in (x) 2 δ x S in (x)(ξ) with (X t, Ξ t )=flow generated by Hamiltonian of classical mechanics H(x, ξ) := 1 2 ξ 2 + V (x) Propagation of Wigner measure for WKB ansatz requires much less regularity of V, and of S in and a in than the WKB method Propagation of Wigner measure for WKB ansatz is global on R R N not limited by caustic onset See also Gérard-Markowich-Mauser-Poupaud CPAM 97 for other PDEs.

8 Proper Hamiltonian flows Hamiltonian H(x, ξ) satisfies, for some κ>0 and h(r)=o(r) at (H) { ξ H(x, ξ) κ(1 + ξ ), D 2 H(x, ξ) κ x H(x, ξ) h( x ) + κ ξ Prop: Under assumptions (H), Hamiltonian H generates a global flow Φ t = (X t, Ξ t ) that is C 1 in all its variables. Besides a) for each T, η > 0 there exists C T,η > 0 s.t. sup X t (x, ξ) x C T,η (1 + ξ ) + η x t T b) for each t > 0, one has DΦ t (x, ξ) I e κ t 1

9 The dynamics in configuration space Assume initial momentum profile U in C(R N ; R N ) satisfies (SL) U in (x) = o( x ) as x With the Hamiltonian flow Φ t = (X t, Ξ t ), define the map F t : R N y X t (y, U in (y)) R N Lemma: The map F t satisfies the following properties F t (y) = y +o( y ) as y for all t R F t is proper deg(f t, B(0,R), x) = 1 for x R N and R 1 F t is onto

10 Rough (non C 1 ) momentum profiles U in Assume initial momentum profile U in C(R N ; R N ) satisfies { U in (x) = o( x ) as x (SL) DU in L N,1 loc (RN ) (DU) (Variant of) Rademacher s thm: L N (E) = 0 where E := {y R N U in not differentiable} Jacobian: J t (y) := det(df t (y)), P t := J 1 t ((0, )), Z t := J 1 t ({0}) Caustic fiber (for rough momentum profiles): C t := {x R N F 1 t ({x}) (Z t E) }

11 Structure of µ(t) and ρ(t) outside caustic fiber Thm A: Assume Hamiltonian H satisfies condition (H) and that momentum profile U in satisfies (SL+DU). Then (a) for a.e. x R N and all t R, the set Ft 1 ({x}) is finite (b) the following conditions are equivalent ρ(t)(c t ) = 0 ρ(t)(r N \ C t ) = 1 ρ in = 0 a.e. on Z t (c) under the equivalent conditions in (b), ρ(t) L N and ρ(t, x) := dρ(t) dl N (x) = F t(y)=x ρ in (y) J t (y) for a.e. x R N (d) under the equivalent conditions in (b) µ(t, x, ) = F t(y)=x ρ in (y) J t (y) δ Ξ t(y,u in (y)) for a.e. x R N

12 Structure of µ(t) outside caustic fiber Outside caustic fiber, µ(t) = a.e. finite sum of monokinetic measures Analogy with ψ ɛ caustic locally finite sum of WKB ansatz away from More than an analogy: if a k L 2 (R N ) and S k W 1,1 loc (RN ), then [ n ] n W ɛ a k e is k/ɛ (x, ) a k (x) 2 δ Sj (x) k=1 k=1 in S (R N R N ) as ɛ 0 provided that S 1 (x)..., S n (x) linearly independent for a.e. x R N

13 Solving for y the equation F t (y) = x Counting function: we define N (t, x) := #Ft 1 ({x}) The counting function measures the complexity of the structure of the propagated measure µ(t) If momentum profile U in satisfies (SL) then M T (R) := sup F t (y) y / y 0 as R y R t T Let R T > 0 satisfy M T (R T ) < 1 2 ; then, for all R > 2R T one has x R and F t (y) = x y 2R (Indeed y > 2R > R T F t(y) y 1 2 y F t(y) 1 2 y >R.)

14 Estimates on the set Ft 1 ({x}): rough (non C 1 ) case Thm B: Assume Hamiltonian H satisfies (H) and momentum profile U in satisfies (SL+DU). Let T > 0 and R T > 0 s.t. M T (R T ) < 1 2. (a) The caustic fibers satisfy L N (C t ) = 0 for all t R. (b) For each t [ T, T ] and each R > 2R T L N ({x s.t. x R & N (t, x) n}) eκn t 1+ Du in N n L N (B(0,2R)) (c) For a.e. x R N H 1 ({(t, y) s.t. t T & F t (y) = x}) <

15 Solving for y the equation F t (y) = x: smooth (C 1 ) case Caustic: define C := {(t, x) R R N s.t. x C t } Thm B : Assume Hamiltonian H satisfies (H) and momentum profile U in C 1 b (RN, R N ) satisfies (SL) (a) The caustic C is closed in R R N. (b) The integer N (t, x) is odd for each (t, x) R R N \ C. (c) There exists a<0<b s.t. a< t <b C t = and N (t, x)=1. (d) The integer-valued counting function N is constant on each connected component of R R N \ C (e) Set F 1 t ({x}) := {y j (t, x) j = 1,..., N (t, x)} for all x / C t ; then each map y j C 1 (O j ), where O j := {(t, x) s.t. N (t, x) j}.

16 The manifold Λ t For each t R, we define Λ t := Φ t ({(x, ξ) s.t. ξ = U in (x)}) Λ := {(t, x, ξ) (x, ξ) Λ t } Therefore Λ t ({x} R N ) = {(x, Ξ t (y, U in (y))) F t (y) = x} #(Λ t ({x} R N )) N (t, x) Smooth case: let O n R N \ C be a connected component, then N (t, x) = n for all (t, x) O n n Λ (O n R N ) = {(t, x, Ξ t (y j (t, x), U in (y j (t, x)) (t, x) O n } j=1

17 Example of Λ t : free dynamics of cubic lagrangian Free flow H(x, ξ) = 1 2 ξ2 in space dimension N = 1 initial profile U in inverse of y 8y 3y 3, time t = 0, 8, 16

18 Example of caustic Caustic (simple cusp) in case of cubic lagrangian

19 Proof of Thm B Proof of (b): apply area formula (Maly-Swanson-Ziemer 02) J t (y)dy = #(Ft 1 ({x}) (B(0, 2R)))dx B(0,2R) R N #(Ft 1 ({x}) (B(0, 2R)))dx B(0,R) = N (t, x)dx By Bienaymé-Chebyshev s inequality L N ({x B(0, R) s.t. N (t, x) n}) 1 n B(0,R) B(0,2R) J t (y)dy

20 By the estimate on the gradient DΦ t of the Hamiltonian flow D x X t (y, U in (y)) e κ t, D ξ X t (y, U in (y)) (e κ t 1) so that, by Hadamard s inequality J t (y) = det(d x X t (y, U in (y)) + D ξ X t (y, U in (y))du in (y)) Therefore (e κ t + (e κ t 1) DU in (y) ) N L N ({x B(0, R) s.t. N (t, x) n}) eκn t 1+ DU in N n L N (B(0,R))

21 Proof of (c): consider the map F : [ T, T ] R N (t, y) F (t, y) R N Jacobian DF (t, y) is the column-wise partitioned matrix DF (t, y) = [V (t, y), M(t, y)], t T, y R N \ E with { V (t, y) = ξ H(Φ t (y, U in (y))) M(t, y) = D x X t (y, U in (y)) + D ξ X t (y, U in (y))du in (y). so that DF (t, y)df (t, y) T = V (t, y)v (t, y) T + M(t, y)m(t, y) T

22 For each m > 0, both sets K m := F 1 (B(0, m)) K m := {y R N there exists t [ T, T ] s.t. (t, y) K m } are compact since F t is proper uniformly in t T. Therefore VV T + MM T N/2 L N/2 (K m) 2N/2 1 V N L (K m) L N+1 (K m ) +2 N/2 T M N L N (K m ) <

23 By the co-area formula (Maly-Swanson-Ziemer 02), for each m > 0 H 1 (F 1 ({x}) K m )dx R N = det(vv T + MM T )(t, y)dtdy < K m In particular, for each m > 0 H 1 (F 1 ({x})) = H 1 (F 1 ({x}) K m ) < a.e. in x m so that H 1 (F 1 ({x})) < for a.e. x R N

24 Proof of Thm A Proof of (a): using the bound on the counting function in Thm B L N ({x s.t. x R and N (t, x) = }) = 0 Proofs of (c+d): by definition of µ(t) = Φ t #µ in µ(t), χ = µ in, χ Φ t = χ(f t (y), Ξ t (y, U in (y)))ρ in (y)dy Since ρ in = 0 a.e. on Z t := Jt 1 ({0}), define a positive measurable function b by the formula b(y) = ρ in (y)/j t (y) if y P t, and b(y) = 0 if y / P t

25 Therefore µ(t), χ = χ(f t (y), Ξ t (y, U in (y)))b(y)j t (y)dy = b(y)ψ(x, Ξ t (y, U in (y))) dx = y Ft 1 ({x}) y Ft 1 ({x}) by the area formula, so that µ(t, x, ) = y Ft 1 ({x}) b(y) δ Ξt(y,U in (y)), ψ(x, ) dx b(y)δ Ξt(y,U in (y))

26 Conclusion We have obtained a detailed description of the propagation of monokinetic measures by proper Hamiltonian flows under the assumption that the initial momentum profile is sublinear at infinity with L N,1 loc gradient In the complement of the caustic fiber, a Lebesgue-negligeable set, the propagated measure is an a.e. finite sum of monokinetic measures We have obtained an estimate on the distribution of values of the number of terms in this sum The proof of these results is based on the area formula from GMT