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 Rn:
1
i∂tu + ∆u = 0 ; u|t=0 = u0 ∈ L2(Rn).
2
Solution known explicitly, and long time asymptotics available:
i 2t ∆
e
u0(x)
∼
t→±∞
|x|2
i 2t
e
1
x
.
F (u0)
n/2
t
(2iπt)
Defocusing nonlinear Schrödinger equation on Rn:
1
i∂tu + ∆u = |u|2σ u
2
;
u|t=0 = u0 ∈ H 1 ∩ F (H 1)(Rn).
2 , u ∈ C(R; H 1 ∩ F (H 1 )).
If 0 < σ < n−2
Large time behavior:
1 < σ < 2 , there is scattering (at least in this weak sense):
• If n
n−2
(1)
∃u± ∈ L2(Rn),
t
u(t) − ei 2 ∆u±
−→ 0.
2 t→±∞
L
1 : (1) is possible only in the trivial case u = u = 0.
• If 0 < σ 6 n
±
Long range effect.
Tool: dispersive property of the free group,
i t ∆
e 2 1
L →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.
1
i∂tu + ∆u = |u|2σ u + V u,
2
with V = V (x) ∈ R. We mention four particular cases:
• Linear potential: V (x) = E · x, for E ∈ Rn.
• Harmonic potential: V (x) = +ω 2|x|2.
• Repulsive harmonic potential: V (x) = −ω 2|x|2.
2 x2 , n > 2.
• Mixture of the above two: e.g., V (x) = −ω12x2
+
ω
1
2 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 −
t2
2
!
3
t
2
i tE·x− 3 |E|
E e
.
Then the equation for u is equivalent to:
1
i∂tv + ∆v = |v|2σ v.
2
The potential does not change the scattering theory.
2
Harmonic potential: V (x) = + ω2 |x|2.
2
ω
1
Local in time dispersive properties: set H = − 2 ∆ + 2 |x|2.
!n/2
ω
−itH .
.
e
1
L →L∞
| sin(ωt)|
H has eigenvalues: no dispersion for large time, hence no scattering
(at least in the usual sense).
2
Repulsive harmonic potential: V (x) = − ω2 |x|2.
2
1
ω
Global in time dispersive properties: set H = − 2 ∆ − 2 |x|2.
!n/2
ω
−itH .
.
e
1
L →L∞
| sinh(ωt)|
Exponential decay, instead of algebraic decay.
Scattering in Σ = H 1 ∩ F (H1), with no long range effect:
2
1
ω
|x|2u,
i∂tu + ∆u = |u|2σ u −
2
2
2 : u(t) ∼ e−itH u .
for all 0 < σ < n−2
±
t→±∞
Repulsive harmonic potential: sketch of the proof.
−itH e
1
L →L∞
.
ω
| sinh(ωt)|
!n/2
6 C(n, d)
1
|t|d/2
,
∀d > n.
Same Strichartz estimates as on Rd, ∀d > n.
On Rd, wave operators in H 1 for 2/d 6 σ < 2/(d − 2). Since d > n
is arbitrary, 0 < σ < 2/(n − 2). Wave operators in Σ.
−itH
itH
∇ replaced by J(t) = e
∇ e · = −iω sinh(ωt)x + cosh(ωt)∇.
Evolution law for kJ(t)uk2
(// pseudo-conformal law):
L2
asymptotic completeness in Σ.
NB: The (free) energy is not signed: J(t) is more geometric.
ω12 2
ω22 2
A hybrid model: V (x) = − 2 x1 + 2 x2.
1∆ + V ,
Mehler: H = − 2
−itH e
1
L →L∞
.
ω1
| sinh(ω1t)|
!1/2
ω2
| sin(ω2t)|
!1/2
1
|t|
!(n−2)/2
= w(t)n/2.
w ∈ L1w (R): HLS+usual proof for Strichartz estimates ⇒ Strichartz
estimates like in Rn with V ≡ 0
wn/d ∈ L1w (R), ∀d > n: Strichartz estimates like in Rd, ∀d > n.
Example. Global existence and scattering in L2 for small data for all
0 < σ 6 2/n: no long range.
If ω1 is large enough (compared to 1, and to ω2), global existence
and scattering in Σ. Otherwise??
Another open question.
If V (x) = |x|α or hxiα, 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 |Vpert(x)| . hxi−γ
with γ > 1 becomes
α
|Vpert(x)| . hxi−γ , with γ > 1 − .
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 Tn: 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 Rn. . . ).
• Asymptotically flat metric: D. Tataru-G. Staffilani, N. Burq, L. RobbianoC. Zuily, D. Tataru, BGT, J.-M. Bouclet-N. Tzvetkov. . .
• Asymptotically conic metric: A. Hassell-T. Tao-J. Wunsch.
• Case of the hyperbolic space Hn: 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
Rn
: non-perturbed case
+|x|2
Sn
: strongly trapping geometry
−|x|2
Hn
: strong dispersion
Scattering on hyperbolic space.
Hn =
n
Rn+1 3 Ω = (x0, . . . , xn) = (cosh r, sinh r ω), r
>
o
n−1
0, ω ∈ S
.
Def. Radial functions: f (Ω) independent of ω ∈ Sn−1, f (Ω) = f (r).
NB: r = distHn (0, Ω).
We denote U (t) = eit∆Hn .
We consider the nonlinear equation:
i∂tu + ∆Hn u = |u|2σ u
;
U (−t)u(t)
t=t0
= ϕ.
NB: The nonlinearity is defocusing.
2
Strichartz estimates
global existence in H 1 (Hn) for 0 < σ < n−2
(V. Banica).
Weighted global in time Strichartz estimates in Hn.
n−1
sinh r 2
w(r) :=
; U (t) = eit∆Hn .
r
2d and
Def. Let d > 2: (p, q) is d-admissible if 2 6 q 6 d−2
!
2
1 1
= δ(q) := d
−
,
p
2 q
(p, q) 6= (2, ∞) .
Proposition (V. Banica, V. Pierfelice). Let n > 3.
1. For any n-admissible pair (p, q), there exists Cq such that
1− 2
w
q U (·)φ
p
L (R;Lq )
n
for every radial function φ ∈ L2
rad (H ).
6 Cq kφkL2
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 Hn:
1. For any d-admissible pair (p, q), there exists Cq = Cq (n, d) s.t.
kU (·)φkLp(R;Lq ) 6 Cq kφkL2 ,
n ).
∀φ ∈ L2
(
H
rad
2. For any d-admissible pairs (p1, q1) and (p2, q2) and any interval I,
there exists Cq1,q2 = Cq1,q2 (n, d) independent of I such that
Z
U (t − s)F (s)ds
I∩{s6t}
p
L 1 (I;Lq1 )
0
0
q
2 (Hn ) .
for every F ∈ Lp2 I; Lrad
6 Cq1,q2 kF k
p0
L 2
q0 ,
I;L 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(Rd)?
• Strichartz estimates for d-admissible pairs.
• Hölder inequality.
2d .
• Sobolev embedding: H 1(Rd) ⊂ Lp(Rd) for 2 6 p 6 d−2
2 .
Yields the existence of wave operators in H 1(Rd), for 2d 6 σ < d−2
1 (Hn ), n > 2!
Claim: We have everything we need in Hrad
2 .
Algebraic conditions: d > n > 2 and 2d 6 σ < d−2
2 is admissible: existence of wave operators in
Any 0 < σ < n−2
1 (Hn ), n > 2.
Hrad
No smallness condition, and no long range effect!
n ), for 0 < σ 6 2 :
Remark. Similar approach for small data in L2
(
H
rad
n
• existence of wave operators,
• global existence,
• asymptotic completeness,
n
in a small ball around 0 in L2
rad (H ).
Asymptotic completeness in H 1(Hn).
Morawetz inequality:
2 < σ < 2 (general), 0 < σ < 2 (radial).
• [BCS] for n = 3: 3
• With V. Banica and T. Duyckaerts: n > 4, radial.
• [Ionescu-Staffilani]: n > 2, no radial assumption.
2 .
Asymptotic completeness for 0 < σ < n−2
From Euclidean to hyperbolic.
(With V. Banica and T. Duyckaerts.)
For n, k > 1, let
φ(r) =
k
X
1
r2j+1.
j=0 (2j + 1)!
Denote by Mkn the rotationally symmetric manifold, with metric
ds2 = dr2 + φ(r)2dω 2.
Remark. For k = 0: Rn. For k = ∞: Hn.
Laplace–Beltrami :
φ0(r)
2
∆M = ∂r + (n − 1)
∂r +
φ(r)
1
∆Sn−1 .
2
φ(r)
NLS on Mkn.
i∂tu + ∆M u = |u|2σ u,
e
x ∈ Mkn, u(t, x) = u(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
2
<σ<
.
N
n−2
Remark. For k > 1, 2/N < 1/n: the frontier short range/long range
has already moved.
Remark. For σ 6 1/N , “formal” proof of the presence of long range
effects: study of the free dynamics.
Free dynamics.
i∂tu + ∆M u = 0,
x ∈ Mkn, u(0, x) = u0(r).
n ) to L2 (Rn ), such that
There exists L unitary from L2
(M
rad
k
rad
ku(t) − v(t)kL2(M ) −→ 0,
with v(t, r) =
2
eir /(4t)
tn/2
t→+∞
! n−1
r
φ(r)
2
r
.
(Lu0)
t
Idea of the proof.
New unknown
φ(r)
v(t, r) = u(t, r)
r
! n−1
2
:

i∂tv + ∆Rn v = F v,
n − 1 φ00(r)
(n − 1)(n − 3)  φ0(r)
F (r) =
+
2 φ(r)
4
φ(r)
!2

1
1
− 2 =O
.
2
r
1+r
it∆
n
Linear scattering for v: v(t) − e R v+ 2 n −→ 0.
L (R ) t→+∞
Standard result:
2 /(4t)
i|x|
x
e
it∆Rn
F
ϕ
.
ϕ − Λ(t) 2 n −→ 0, with Λ(t, x) =
e
(
)
n/2
L (R ) t→+∞
2t
t
Existence of long range effects.
Suppose n > 2 and 0 < σ 6 1/N , where N = (2k + 1)(n − 1) + 1. Let
n
2
n
u ∈ C([T, ∞[; L2
rad (Mk )) sol. NLS, u+ ∈ Lrad (Mk ) with
it∆
Mu u(t) − e
+ L2 −→ 0.
t→+∞
Let ψ ∈ C0∞(M ) be radial, and t2 > t1 > T . By assumption,
D
ψ, e−it2∆M u(t2) − e−it1∆M u(t1) = −i
E
Z t D
2
t1
E
it∆
2σ
e M ψ, |u| u (t) dt
goes to zero as t1, t2 → +∞. Free dynamics:
D
E
1
Z ∞
eit∆M ψ, |u|2σ u (t) ≈ nσ+n
t
0
for ϕ = Lψ|Lu+|2σ Lu+.
r
φ(r)
!(n−1)(σ+1)
r
φ(r)n−1dr
t
ϕ
Change of variable r 7→ tr:
1
Z ∞
tnσ+n−1 0
tr
φ(tr)
!(n−1)(σ+1)
ϕ (r) φ(tr)n−1dr.
For r > 1 and large t, the function at stake behaves like
1
tnσ+n−1
tr
(tr)2k+1
!(n−1)(σ+1)
r−(N −n)σ+n−1
(n−1)(2k+1)
ϕ(r) (tr)
=
ϕ(r).
N
σ
t
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.