Dispersive Estimates for Charge transfer models

Transcription

1 Dispersive Estimates for Charge transfer models I. Rodnianski PU, W. S. CIT, A. Soffer Rutgers Motivation for this talk about the linear Schrödinger equation: NLS with initial data = sum of several solitons (plus some small perturbation) which are far apart and do not approach each other at later times. We address the asymptotic stability problem: ( ) i t ψ ψ + β( ψ 2 )ψ =0 in R d, d 3. Initial data ψ 0 (x) = Here w j are solitons k j=1 w j (0,x)+R 0 (x). w j (t, x) = w(t, x; σ j (0)) = e iθ j(t,x) φ(x xj (t),α j (0)) θ j (t, x) = v j (0) x 1 2 ( v j(0) 2 α 2 j (0))t + γ j(0) x j (t) = v j (0)t + D j (0). 1

2 It is well-known that if φ = φ(,α)satisfies ( ) 1 2 φ α2 2 φ + β( φ 2 )φ =0 then w j satisfies ( ) witharbitraryconstantparameters σ j (0) = (v j (0),D j (0),γ j (0),α j (0)). Solutions of ( ) areknowntoexistundersuitableconditions on β. We need the nonlinear stability condition α φ, φ > 0. Theorem 1. Assume separation, nonlinear stability, and some spectral conditions. Then ɛ>0 such that for R 0 satisfying s k=0 k R 0 L 1 L 2 <ɛ for some integer s> 2 d,thereexistsapathσ(t) in R 2d+2 such that ψ(t) k j=1 w j (t, x; σ j (t)) (1 + t) d 2 as t. The parameters σ j (t) converge as t. 2

3 The proof is perturbative. Linearizing leads to the system for R = ( R R) (R = R(t) =perturbation) ( ) i t R + ( R t=0 = R 0, where ) R + k j=1 V j (t, v j t) R = F v j are distinct vectors in R d (not the velocities of the initial solitons) V j are matrix potentials V j (t, x) = ( U j (x) e iθj(t,x) W j (x) e iθj(t,x) W j (x) U j (x) ), θ j (t, x) =( v j 2 + α 2 j )t +2x v j + γ j U j,w j are exponentially decaying F contains all the rest: interaction terms between different solitons O(w i w j )fori j, fullynonlinear term R p (at least for nonlinearity ψ p 1 ψ), mixed terms, terms due to difference between straight path and time-dependent path (better considered as a small potential that perturbs the left-hand side). 3

4 Each moving matrix potential in ( ) hasastationary counterpart H j = α2 j + U j W j W j α2 j U j via a modulation and Galilei transform. Scalar Galilean transformation: (1) g v,d (t) =e i v 2 2 t e ix v e i(d+tv)p where p = i ; analogueofx x D tv, p p v. Vector-valued version G vj (t) ( f ) 1 (g vj,0(t)f 1 = f 2 g vj,0(t) f 2 Suppose S j (t) solvess j (0) = Id, ( ) 12 0 i t S j (t) V j (t, t v j )S j (t) =0. Then S j (t) = G vj (t) 1 M j (t) 1 e ith jm j (0)G vj (0), ). where for some linear function ω j (t) M j (t) = ( e iω j (t) 0 0 e iω j(t) ). 4

5 Our goal: Prove L 1 L estimates for the system ( ). Obstacle: Traveling bound states. If H l ψ = E ψ, then e ith l ψ = e ite ψ and R(t) := G vl (t) 1 M l (t) 1 e ite ψ solves i t R + ( ) R + k j=1 V j (t, v j t) R = o(1). So need to impose the condition that G vl (t)m l (t)ψ(t) has no component (relative to a suitable projection) on the eigenfunctions of H j. Note that H j is not selfadjoint. It turns out (because of symmetries of the nonlinear equation) that H j has a nontrivial root-space at zero: H j ψ = φ 0,H j φ =0. Thene ith jψ = ψ + itφ. So we do not even have L 2 -boundedness here. spectral issues related to systems will be discussed later in more detail. 5

6 Start with the scalar homogeneous case in R d, d 3: ( ) i t ψ ψ + [ V 1 + V 2 ( v t ) ] ψ =0. Yajima, Graf proved asymptotic completeness of these charge transfer models (more on this later). Here V j are nice enough so that (JSS) e it(1 2 +V j) Pj f C t d 2 f 1, 1 P j =orthogonal projection onto the bound states of 1 2 +V j. Estimate (JSS)wasobtainedbyJourné, Soffer, Sogge. For ( ) with evolution U(t) define ψ 0 L 2 (R d )tobeascattering state if (1 P 1 )U(t)ψ (1 P 2 ) g v (t)u(t)ψ Theorem 2. For such ψ 0 one has the decay U(t)ψ 0 L 2 +L t d/2 ψ 0 L 1 L 2. Here t =(1+t 2 ) 1 2 and L 2 + L =(L 1 L 2 ). 6

7 Norm in L 2 + L is f 2+ =inf{ f f 2 : f 1 + f 2 = f}. Two easy estimates: (1) Decay for a single small time-dependent potential: H 0 = 2 1, U(t) =e ith 0 + i t 0 ei(t s)h 0 V (s)u(s) ds Clearly, e ith 0f 2+ C 0 t d 2 f 1 2. Assume that U(t)f 2+ C 1 t d 2 f 1 2. Then from Duhamel (for times t 1) with sup s V (s) 1 <ε t d 2 U(t)f 2+ C 0 f t d 2 + t d 2 t 1 1 t C 0 t s d 2 V (s) 1 2 C 1 s d 2 ds f 1 2 t 1 V (s) 2 C 1 s d 2 ds f t 2 d C 0 t s d 2 V (s) 2 ds f 2 0 [ C 0 +(1+C 0 )C 1 ε ] f 1 2. Thus if ε small, then C 1 C 0. This argument needs d 3alsofornon-technicalreasons: Indimensions d =1, 2smallpotentialscanhaveboundstates. 7

8 (2) From local decay to global decay (Ginibre): Before J-S-S both Rauch (exponentially decaying potentials) and Jensen, Kato (power like decay) proved estimates of the form (LD) χe ith P c χf 2 t d 2 f 2 for some decaying weight χ. Let H 0 = 2 1,H = H 0 + V : (3) e ith P c = e ith 0P c + i = e ith 0P c + i (4) P c e ish 0 = P c e ish i = P c e ish i Inserting (4) into (3) yields e ith P c = e ith 0 P c + i t s 0 t t 0 ei(t s)h 0 Ve ish P c ds t s 0 ei(t s)h 0 VP c e ish ds 0 P c e i(s τ)h Ve iτh 0 dτ s 0 ei(s τ)h P c Ve iτh 0 dτ. 0 ei(t s)h 0 VP c e ish 0 ds 0 ei(t s)h 0 Ve i(s τ)h P c Ve iτh 0 dτ. 8

9 Apply the local decay estimate to the middle piece of the last term: Ve i(s τ)h P c V with V playing the role of χ. Conclusion: Local L 2 -decay (LD) impliesthat e ith P c f 2+ t d 2 f 1 2. J-S-S removed L 2 under the additional assumption that ˆV 1 < since then sup 1 p e it1 2 Ve it1 2 p p ˆV 1. In the dispersive estimates for charge transfer models as in Theorem 2 one can remove L 2 from the left-hand side. This passage from local to global proved to be very useful for systems, since one can adapt an old technique of Rauch for exponentially decaying potentials to the case of systems. 9

10 Proof of Theorem 2: for scattering states projections onto (moving) bound states decay exponentially due to the exponential decay of eigenfunctions of V with positive eigenvalues (Agmon s estimate) and the exponential decay of the potentials. reduce to the dispersive bound for individual potentials by means of a channel decomposition, i.e., split evolution U(t) =χ 1 (t, )U(t)+χ 2 (t, )U(t)+χ 3 (t, )U(t) with cut-offs χ 1 (t, x) =χ ( x/(δt) ), χ 2 (t, x) =χ ( (x t v)/(δt) ). Use Duhamel to compare χ 1 (t, )U(t) to χ 1 (t, )e ith 1 etc., where H 1 = 1 2 +V 1. One can factor in the projections P 1 for free from the previous comment: if ψ 0 is scattering, then χ 1 (t, )(1 P 1 )U(t)ψ 0 e αt ψ

11 χ 1,χ 2 at two times (first with one stationary potential, and then with two moving ones): 11

12 As in the easy argument involving small potentials we will use a bootstrap argument. The required smallness for that comes from two sources: interaction between channels is small for projections onto small momenta (propagation estimates or low velocity estimates). Easy: For fixed T,M sup t T as y. χe it F ( p M)χ( y) Harder (also for H): If F ( p M) is a smooth cut-off that vanishes when p >M and α>1then χ( x >α(r + Mt)) e it 2 F ( p M)χ( x <R) 2 2 C N (R + Mt) N for all N>0. for high momenta one obtains a gain via Kato s smoothing estimate sup B R A 0 ( ) 1 4 e ith f 2 L 2 (B R ) dt C R A f 2 2 A on RHS needed: bound states of H. 12

13 More details: Decay of the bound states. Split U(t)ψ 0 = ψ 2 + ψ 1 = m i=1 a i (t)u i + ψ 1 (t, ) where H 1 u j = λ j u j, {u j } orthonormal. Project i t ψ ψ + [ V 1 + V 2 ( v t ) ] ψ =0 onto u j : iȧ j + λ j a j + ψ, V 2 ( t v)u j =0. Since u j are exponentially decaying, and a j ( ) =0, we get a j (t) Ce αt. Channel decomposition. With χ 1 = χ 1 (t, ), χ 1 P 1 U(t)ψ 0 = χ 1 e ith 1P 1 ψ 0 +iχ 1 t A 0 e i(t s)h 1P 1 V 2 ( s v)u(s)ψ 0 ds t +iχ 1 t A ei(t s)h 1P 1 V 2 ( s v)u(s)ψ 0 ds. Treat first integral as in previous bootstrap argument, get small constant if A large. Second integral more difficult. Expand again, but first project: 13

14 χ 1 t t A ei(t s)h 1P 1 V 2 ( s v)u(s)ψ 0 ds = χ 1 t t A ei(t s)h 1P 1 V 2 ( s v)(1 P 2 (s))u(s)ψ 0 ds +χ 1 t t A ei(t s)h 1P 1 V 2 ( s v)p 2 (s)u(s)ψ 0 ds. Here P 2 (s) =g v (s)p 2 g v (s). Integral with 1 P 2 (s) is exponentially small. The second one is expanded again by Duhamel one easy term that does not involve U(t) (weskipit)andanother χ 1 t t A s 0 ei(t s)h 1P 1 V 2 ( s v)g v (s)e i(s τ)h 2P 2 V 1 ( + τ v)g v (τ)u(τ)ψ 0 dτds. Split integration into 0 τ s B, s B τ s. The former gives desired small constant if B large. For the latter use low velocity estimate (δ >0small) χ 1 (t, )e i(t s)h 1P 1 F ( p M)V 2 ( s v) L 2 L 2 AM δt, t s A ( small factor) andkatosmoothing. 14

15 The remaining Kato term t s χ 1 t A s B ei(t s)h 1P 1 F ( p >M)V 2 ( s v) g v (s)e i(s τ)h 2P 2 V 1 ( + τ v)g v (τ)u(τ)ψ 0 dτds is estimated as follows: Estimate entire expression in L 2, get rid of χ 1 e i(t s)h 1P 1. Move F ( p >M)throughV 2 ( s v)g v (s), which changes F ( p > M) to F ( p v > M), and also introduces the commutator [V 2,F( p >M)] which has small norm Remains to deal with t t A s s B V 2F ( p v M)e i(s τ)h 2P 2 g v (τ)v 1 U(τ)ψ 0 L 2 dτds. Apply Kato s smoothing estimate to evolution e i(s τ)h 2 with cut-off V 2. Gain small factor M

16 Asymptotic completeness of charge-transfer models ( ) i t ψ + 1 k 2 ψ + V j ( t v j )ψ =0 j=1 goes back to Yajima (1980) and Graf (1990). Review of basic scattering for V : Wave operators Ω := s lim t e ith e it 2 exist by Cook s method if V (x) < (1 + x ) 1 ε. Are isometries with Ran(Ω) H a.c. (H). If Ω :=s lim t e it 2 e ith P a.c. exists, then Ran(Ω) = H a.c. (H). Asymptotic completeness means this and L 2 = H a.c. (H)+H p.p. (H), where p.p.-spectrum = finite number of positive bound states. Dynamically: f L 2 g L 2 s.t. e ith f = e it 2 g + m j=1 where Hg j = E j g j, E j > 0. e ite jg j + o(1) L 2 16

17 Suppose V is nice enough that Ω exists and that one has the dispersive bound e ith P a.c. f 2+ t d 2 f 1 2. Then asymptotic completeness follows: 0 e it 2 Ve ith f 2 dt V 2 e ith f 2+ dt 0 0 t d 2 f 1 2 dt < since d 3 (Cook s method). Thus Ω is unitary onto Ran(P a.c. ). Similar logic applies to the charge transfer case. First, meaning of asymptotic completeness for charge transfer models (in Graf s terminology): Ω := s lim t U(t) 1 e it 2 and Ω j := s lim t U(t) 1 g v j (t)p b (H j )e ith jg v j (0) exist, and L 2 =Ran(Ω) k j=1 Ran(Ω j )asanorthogonaldirect sum. Dynamically, one has the following: 17

18 Theorem 3. Let {u s,j } S j s=1 be the efs of H j = 1 2 +V j,correspondingtothepositiveevs{λ s,j } S j s=1. Then ψ 0 L 2 the solution U(t)ψ 0 of ( ) can be written in the form U(t)ψ 0 = k S j j=1 s=1 A s,j e itλ s,j g v j (t)u s,j +e it 2 φ 0 + o(1) L 2, for some choice of constants A s,j and φ 0 L 2. Write ψ(t) := U(t)ψ 0. Project the equation via P b (H j,t) P b (H j,t)ψ(t) = S j s=1 a s,j(t) e itλ s,j u s,j where a s,j (t) A s,j as t. Let g v j (t)u(t)v s,j e itλ s,j u s,j 2 0 and define φ 1 := ψ 0 k j=1 S j s=1 A s,j v s,j. Since P b (H j,t)p b (H m,t) 0 one checks that P b (H j,t)u(t)φ 1 0. Hence Theorem 2 and Cook s method from above imply that s lim t e it 2 U(t)φ 1 =: φ 0 exists, at least if ψ 0 L 1 L 2. Hence U(t)ψ 0 = e it 2 φ 0 + kj=1 S j s=1 A s,j U(t)v s,j + o(1), and we are done. 18

19 For the nonlinear applications we need decay estimates for charge transfer models with matrix potentials. This requires estimates for single matrix potentials: e ita P s f 2+ t 3 2 f 1 2 (d =3) ( ) 12 + µ + U W with A = W 1 2 µ U, and P s is a suitable projection. Matrix charge transfer models require both Galilei transforms and modulations (see p. 3) acting on different A s. Impose the following spectral assumptions on A acting on H := L 2 (R 3 ) L 2 (R 3 ): U, W decay exponentially spec(a) R and spec(a) ( µ, µ) ={ω l 0 l M}, whereω 0 =0andallω j are distinct eigenvalues There are no eigenvalues in spec ess (A) For 1 l M, L l := ker(a ω l ) 2 =ker(a ω l ), and ker(a) ker(a 2 )=ker(a 3 )=:L 0. Moreover, these spaces are finite dimensional. 19

20 Ran(A ω l ), 1 l M and Ran(A 2 )areclosed. The spaces L l are spanned by exponentially decreasing functions in H (say with bound e ε0 x ), and their derivatives decay similarly. The points ±µ are not resonances of A. All these assumptions hold as well for the adjoint A. We denote the corresponding (generalized) eigenspaces by L l. There is a direct sum decomposition H = M j=0 L j + ( M j=0 L j ), and let P s denote the corresponding (nonorthogonal) projection onto the second summand. It plays the role of the continuous subspace from the self-adjoint case. In addition to the previous conditions we need the linear stability assumption e ita P s f C f

21 Under all these conditions we prove the desired bound (SysD) e ita P s f 2+ t 3 2 f 1 2 by means of an old technique of Rauch, based on analytic continuation of the resolvent across the spectrum. This only works for exponentially decreasing potentials. Cuccagna (2001) obtained (SysD) without the L 2 addition/intersection for potentials with power-like decay by proving L boundedness of the wave operators in this context (Yajima s approach in the scalar case). Rauch s method is much more transparent, albeit more restrictive. 21

22 Outline of the method: Laplace transform and inverse (Rez >0andd>0) 1 2πi Note: d+i 0 eita P s e tz dt = (ia z) 1 P s d i etτ (ia τ) 1 P s dτ = e ita P s. there is no spectral representation (unlike the selfadjoint case e ith = e itλ de(λ)), but not needed for these formulas. Idea: Shift contour in the second integral to the left across the spectrum of ia which is ir. Price: Resolvent remains bounded only in exponentially weighted L 2. Reason: ( + z) 1 (x) = e x z 4π x for z C \ (, 0]. To continue across the slit set z = ζ 2,Reζ>0. Then analytic continuation of ( + ζ 2 ) 1 (x) = e x ζ 4π x Reζ < 0 has exponential growth. Let (E ε f)(x) = e ε x f(x). Resolvent identity + analytic Fredholm alternative E ε (ia τ) 1 E ε (for some fixed ε>0) to 22

23 extends as a meromorphic L 2 -operator valued function to the right of the following contour Γ ε, with finitely many poles that all lie inside some disk. The two points are ±iµ (we set µ = 1). i i Therefore shifting the contour yields e ita P s = 1 2πi Γ ε e tτ (ia τ) 1 P s dτ +contribution from the residues. Exclude the residues coming from the imaginary axis by means of the following version of von Neumann s ergodic theorem: 1 T T0 e iωt e ita P s dt 0asT. 23

24 The proof of this result depends strongly on our spectral assumptions as well as the linear stability assumption. The contribution from the curved pieces of the contour Γ ε is exponentially decaying in time t, whereasthehorizontalsegmentsgivethe main contribution of t 3 2. The latter requires making sure that the analytically continued, exponentially weighted, resolvent is regular at the points ±iµ. Thisisensuredsince ±µ are neither eigenvalues nor resonances of A. 24

25 Cuccagna, C. Stabilization of solutions to nonlinear Schrödinger equations. Comm. Pure Appl. Math. 54 (2001), no. 9, Graf, J. M. Phase Space Analysis of the Charge transfer Model. Helv. Physica Acta 63 (1990), Journé, J.-L., Soffer, A., Sogge, C. D. Decay estimates for Schrödinger operators. Comm. Pure Appl. Math. 44 (1991), no. 5, Rauch, J. Local decay of scattering solutions to Schrödinger s equation. Comm. Math. Phys. 61 (1978), Yajima, K. A multichannel scattering theory for some time dependent Hamiltonians, charge transfer problem. Comm. Math. Phys. 75 (1980), no. 2,