On the role of geometry in scattering theory for nonlinear Schrödinger equations Rémi Carles (CNRS & Université Montpellier 2) Orléans, April 9, 2008
Free Schrödinger equation on R n : i t u + 1 2 u = 0 ; u t=0 = u 0 L 2 (R n ). Solution known explicitly, and long time asymptotics available: e i t 2 u 0 (x) ei x 2 2t t ± 1 (2iπt) n/2f (u 0) Defocusing nonlinear Schrödinger equation on R n : ( ) x i t u + 1 2 u = u 2σ u ; u t=0 = u 0 H 1 F(H 1 )(R n ). If 0 < σ < 2 n 2, u C(R; H1 F(H 1 )). t.
Large time behavior: If n 1 < σ < n 2 2, there is scattering (at least in this weak sense): (1) u ± L 2 (R n ), u(t) ei t 2 u ± L 2 t ± 0. If 0 < σ 1 n : (1) is possible only in the trivial case u ± = u = 0. Long range effect. Tool: dispersive property of the free group, ei t 2 L 1 L 1 t n/2.
Goal of this talk. What happens if we add a potential in NLS? What happens in the case of non-euclidean geometries?
Introducing an external potential. i t u + 1 2 u = u 2σ u + V u, with V = V (x) R. We mention four particular cases: Linear potential: V (x) = E x, for E R n. Harmonic potential: V (x) = +ω 2 x 2. Repulsive harmonic potential: V (x) = ω 2 x 2. Mixture of the above two: e.g., V (x) = ω1 2x2 1 + ω2 2 x2 2, n 2.
Linear potential: V (x) = E x. (With Yoshihisa Nakamura.) This case is trivial, in view of Avron Herbst formula. Set v(t, x) = u (t, x t22 ) ( E e i te x t3 3 E 2). Then the equation for u is equivalent to: i t v + 1 2 v = v 2σ v. The potential does not change the scattering theory.
Harmonic potential: V (x) = + ω2 2 x 2. Local in time dispersive properties: set H = 1 2 + ω2 2 x 2. ( e ith L 1 L ω sin(ωt) ) n/2. H has eigenvalues: no dispersion for large time, hence no scattering (at least in the usual sense).
Repulsive harmonic potential: V (x) = ω2 2 x 2. Global in time dispersive properties: set H = 1 2 ω2 2 x 2. ( e ith L 1 L ω sinh(ωt) Exponential decay, instead of algebraic decay. ) n/2. Scattering in Σ = H 1 F(H 1 ), with no long range effect: i t u + 1 2 u = u 2σ u ω2 2 x 2 u, for all 0 < σ < n 2 2 : u(t) t ± e ith u ±.
Repulsive harmonic potential: sketch of the proof. ( ) n/2 e ith L 1 L ω C(n, d) 1 sinh(ωt) t d/2, d n. Same Strichartz estimates as on R d, d n. On R d, wave operators in H 1 for 2/d σ < 2/(d 2). Since d n is arbitrary, 0 < σ < 2/(n 2). Wave operators in Σ. replaced by J(t) = e ith ( e ith ) = iω sinh(ωt)x + cosh(ωt). Evolution law for J(t)u 2 L 2 (// pseudo-conformal law): asymptotic completeness in Σ. NB: The (free) energy is not signed: J(t) is more geometric.
A hybrid model: V (x) = ω2 1 2 x 2 1 + ω2 2 2 x 2 2. Mehler: H = 1 2 + V, ( e ith L 1 L ω 1 sinh(ω 1 t) ) 1/2 ( ω 2 sin(ω 2 t) ) 1/2 ( 1 t ) (n 2)/2 = w(t) n/2. w L 1 w(r): HLS+usual proof for Strichartz estimates Strichartz estimates like in R n with V 0 w n/d L 1 w(r), d n: Strichartz estimates like in R d, d n. Example. Global existence and scattering in L 2 for small data for all 0 < σ 2/n: no long range. If ω 1 is large enough (compared to 1, and to ω 2 ), global existence and scattering in Σ. Otherwise??
Another open question. If V (x) = x α or x α, for 0 < α < 2. Local existence of solution to the nonlinear equation: OK. Scattering theory? With J.-F. Bony, D. Häfner and L. Michel: in the linear case, complete answer. The usual short range assumption V pert (x) x γ with γ > 1 becomes V pert (x) x γ, with γ > 1 α 2. Similar interpolation for nonlinear equations, when changing the geometry instead of introducing an external potential.
Changing the geometry. Many results on compact manifolds: Case of T n : J. Bourgain (Strichartz estimates). Compact manifold without boundary: N. Burq-P. Gérard-N. Tzvetkov. More recently, R. Anton, M. Blair-H. Smith-C. Sogge. No scattering theory available (more complicated dynamics): there are eigenvalues.
Nonlinear Schrödinger on non-compact manifolds (except R n... ). Asymptotically flat metric: D. Tataru-G. Staffilani, N. Burq, L. Robbiano- C. Zuily, D. Tataru, BGT, J.-M. Bouclet-N. Tzvetkov... Asymptotically conic metric: A. Hassell-T. Tao-J. Wunsch. Case of the hyperbolic space H n : V. Banica, V. Banica-RC-G. Staffilani, A. Ionescu-G. Staffilani, J.-P. Anker-V. Pierfelice. More general Damek Ricci spaces: V. Pierfelice. Asymptotically hyperbolic manifolds: J.-M. Bouclet. Variations on hyperbolic: N. Burq-C. Guillarmou-A. Hassell. More reasonable to expect scattering.
An intuitive parallel. Potential Manifold 0 R n : non-perturbed case + x 2 S n : strongly trapping geometry x 2 H n : strong dispersion
Scattering on hyperbolic space. H n = { R n+1 Ω = (x 0,..., x n ) = (cosh r, sinh r ω), r 0, ω S n 1}. Def. Radial functions: f(ω) independent of ω S n 1, f(ω) = f(r). NB: r = dist H n(0, Ω). We denote U(t) = e it H n. We consider the nonlinear equation: i t u + H nu = u 2σ u ; U( t)u(t) = ϕ. t=t0 NB: The nonlinearity is defocusing. Strichartz estimates global existence in H 1 (H n ) for 0 < σ < n 2 2 (V. Banica).
Weighted global in time Strichartz estimates in H n. ( )n 1 sinh r 2 w(r) := ; U(t) = e it H n. r Def. Let d 2: (p, q) is d-admissible if 2 q 2d 2 p = δ(q) := d ( 1 2 1 q ) d 2 and, (p, q) (2, ). Proposition (V. Banica, V. Pierfelice). Let n 3. 1. For any n-admissible pair (p, q), there exists C q such that w1 2 q U( )φ C q φ L p (R;L q L 2 ) for every radial function φ L 2 rad (Hn ). 2. Similar weighted estimates for inhomogeneous equations.
Corollary (with V. Banica and G. Staffilani). Let d n 3. Strichartz estimates hold for d-admissible pairs and radial functions on H n : 1. For any d-admissible pair (p, q), there exists C q = C q (n, d) s.t. U( )φ L p (R;L q ) C q φ L 2, φ L 2 rad (Hn ). 2. For any d-admissible pairs (p 1, q 1 ) and (p 2, q 2 ) and any interval I, there exists C q1,q 2 = C q1,q 2 (n, d) independent of I such that I {s t} for every F L p 2 U(t s)f (s)ds C q1,q 2 F L p 1 (I;L q p 1) L 2 ( ) I; L q 2 rad (H n ). ( I;L q ), 2 Remark. True also when n = 2 ([BCS]). Remark. Extended recently to the non-radial case: [AP], [IS].
How does one prove scattering in H 1 (R d )? Strichartz estimates for d-admissible pairs. Hölder inequality. Sobolev embedding: H 1 (R d ) L p (R d ) for 2 p 2d d 2. Yields the existence of wave operators in H 1 (R d ), for 2 d σ < 2 d 2. Claim: We have everything we need in H 1 rad (Hn ), n 2!
Algebraic conditions: d n 2 and 2 d σ < 2 d 2. Any 0 < σ < 2 Hrad 1 (Hn ), n 2. n 2 is admissible: existence of wave operators in No smallness condition, and no long range effect! Remark. Similar approach for small data in L 2 rad (Hn ), for 0 < σ 2 n : existence of wave operators, global existence, asymptotic completeness, in a small ball around 0 in L 2 rad (Hn ).
Asymptotic completeness in H 1 (H n ). Morawetz inequality: [BCS] for n = 3: 2 3 < σ < 2 (general), 0 < σ < 2 (radial). With V. Banica and T. Duyckaerts: n 4, radial. [Ionescu-Staffilani]: n 2, no radial assumption. Asymptotic completeness for 0 < σ < 2 n 2.
From Euclidean to hyperbolic. (With V. Banica and T. Duyckaerts.) For n, k 1, let Denote by M n k φ(r) = k j=0 1 (2j + 1)! r2j+1. the rotationally symmetric manifold, with metric ds 2 = dr 2 + φ(r) 2 dω 2. Remark. For k = 0: R n. For k = : H n. Laplace Beltrami : M = 2 r + (n 1) φ (r) φ(r) r + 1 φ(r) 2 S n 1.
NLS on M n k. i t u + M u = u 2σ u, x M n k, u(t, x) = ũ(t, r). Introduce N = (2k + 1)(n 1) + 1: N n for k 0. Existence of wave operators (V. Banica-T. Duyckaerts) and asymptotic completeness for 2 N < σ < 2 n 2. Remark. For k 1, 2/N < 1/n: the frontier short range/long range has already moved. Remark. For σ 1/N, formal proof of the presence of long range effects: study of the free dynamics.
Free dynamics. i t u + M u = 0, x M n k, u(0, x) = u 0(r). There exists L unitary from L 2 rad (M k n) to L2 rad (Rn ), such that u(t) v(t) L 2 (M) 0, t + with v(t, r) = eir2 /(4t) t n/2 ( r )n 1 2 (Lu0 ) ( r φ(r) t ).
Idea of the proof. New unknown v(t, r) = u(t, r) F (r) = n 1 φ (r) 2 φ(r) Linear scattering for v: Standard result: e it R n ϕ Λ(t) ( φ(r) r )n 1 2 (n 1)(n 3) + 4 L 2 (R n ) : i t v + R nv = F v, ( φ ) 2 (r) 1 φ(r) r 2 = O v(t) e it R n v + L 2 (R n ) 0. t + ei x 2/(4t) 0, with Λ(t, x) = t + t n/2 ( (Fϕ) 1 1 + r 2 ( x 2t ) )..
Existence of long range effects. Suppose n 2 and 0 < σ 1/N, where N = (2k + 1)(n 1) + 1. Let u C([T, [; L 2 rad (M n k )) sol. NLS, u + L 2 rad (M n k ) with u(t) e it Mu + L 2 0. t + Let ψ C 0 (M) be radial, and t 2 t 1 T. By assumption, ψ, e it 2 M u(t 2 ) e it 1 M u(t 1 ) = i t2 e it M ψ, ( u 2σ u ) (t) dt goes to zero as t 1, t 2 +. Free dynamics: e it M ψ, ( u 2σ u ) (t) 1 t nσ+n for ϕ = Lψ Lu + 2σ Lu +. ( r 0 φ(r) t 1 ) (n 1)(σ+1) ( ) r ϕ t φ(r) n 1 dr
Change of variable r tr: 1 t nσ+n 1 0 ( tr φ(tr) ) (n 1)(σ+1) ϕ (r) φ(tr) n 1 dr. For r 1 and large t, the function at stake behaves like 1 t nσ+n 1 ( ) (n 1)(σ+1) tr (tr) 2k+1 ϕ(r) (tr) (n 1)(2k+1) = r (N n)σ+n 1 t Nσ ϕ(r). This function of t is not integrable, unless ϕ 0. This means that Lu + = 0 = u + (Ker L = {0}). The assumption and the conservation of mass then imply u 0.