DFG–JSPS Joint Seminar
Infinite Dimensional Harmonic Analysis IV
(University of Tokyo,
Graduate School of Mathematical Sciences,
10–14 September 2007)
The Schrödinger equation
on symmetric spaces
Jean–Philippe Anker
(Université d’Orléans)
Joint work in progress with
Vittoria Pierfelice (Université d’Orléans)
1. Introduction
Schrödinger equation :

 i ∂ u(t, x) + ∆ u(t, x) = g(t, x)
x
t
 u(0, x) = f (x)
(1)
Wave equation :

 ∂ 2u(t, x) − ∆ u(t, x) = g(t, x)
x
t
 u(0, x) = f (x), ∂t |
0
t=0 u(t, x) = f1(x)
Homogeneous equations : g = 0
(2)
linear
Typical nonlinearities
:

g(t, x) +
 u(t, x) |u(t, x)|γ

|u(t, x)|1+γ
Setting : Riemannian manifold

n


R
n sphere
e.g. symmetric space
S


 n
H hyperbolic space
2. Schrödinger in Rn
Solution to the Schrödinger equation (1) :
u(t, x) = e it∆f (x) +
|
{z
}
F (t,x)
Z t
|0
e i(t−s)∆x g(s, x)ds
{z
G(t,x)
Homogeneous solution
F (t, x) = e it∆f (x) = (f ∗ st )(x)
given by explicit kernel
}
|x|2
n
n
π
i
st (x) = 2−nπ − 2 e sign(t)i 4 n |t|− 2 e− 4 t
(heat kernel with imaginary time)
Two main estimates :
• Dispersive estimates :
keit∆k
−n
2
L1→L∞ = kstkL∞ . |t|
− 1q )
−n( 1
it∆
2
∀ 2≤q≤∞
⇒ ke
kLq′→Lq . |t|
−n
(1− 1q − 1q̃ )
it∆
2
⇒ ke
kLq′→Lq̃ . |t|
∀ 2 ≤ q, q̃ ≤ ∞
• Strichartz type estimates :
ku(t, x)k
p̃ q̃
Lt Lx
. kf (x)k
L2
x
+ kg(t, x)k
p′ q ′
Lt Lx
(3)
∀ (p, q), (p̃, q̃) in the admissible interval
1/q
1/p + n/2 1/q = n/4
1/2
1/2 − 1/n
1/2
1/p
In dimension n > 2, the estimate () holds true
1 − 1)
at the endpoint ( 1
,
2 2 n
References : [Ginibre–Velo], [Keel–Tao]
Wellposedness (existence and uniqueness)
for NLS (NonLinear Schrödinger equation) :
• g(t, x) + u(t, x)|u(t, x)|γ
(γ > 0)
 L2 mass
conservation of
 H 1 energy
wellposedness
in L2(Rn )
in H 1(Rn )
global
4
γ<n
4
γ < n−2
global
4
γ=n
s.i.d.
4
γ = n−2
s.i.d.
s.i.d. = small initial data
• g(t, x) + |u(t, x)|1+γ
(γ > 0)
no conservation !
wellposedness
in L2(Rn )
in H 1(Rn )
local
4
γ<n
global
4
γ=n
s.i.d.
4
γ < n−2
s.i.d.
4
γ = n−2
s.i.d.
Reference : [Cazenave]
Scattering for NLS
Strichartz estimates
p q
global solution u(t, x) ∈ Lt Lx
p q
t → ±∞ : u(t, x) → 0 in Lt Lx
NLS approaches HS
u(t, x) close to e it∆ u±(x)
Scattering for NLS in L2(Rn) :
For any global solution u ∈ C(R, L2(Rn)),
do there exist u± ∈ L2(Rn ) such that
k u(t, x) − e it∆ u±(x) k
L2
x
→ 0 as t → ±∞ ?
Idem for H 1 scattering
• g(t, x) + u(t, x) |u(t, x)|γ
(γ > 0)

2 scattering for γ = 4 and s.i.d

L

n

4
4
H 1 scattering for n < γ < n−2



2
no scattering for 0 < γ < n
• g(t, x) + |u(t, x)|1+γ
(γ > 0)
no result (to our knowledge)
Reference : [Tao–Visan–Zhang]
3. NLS on manifolds
Aim : extend theory from Euclidean space to
Riemannian manifold e.g. symmetric space
Motivations :
• understand influence of geometry
◦ compact case :
expect less dispersion
weaker results
◦ noncompact case :
expect more dispersion
stronger results
• transfer results (in both directions)
between the flat case and the curved case
Parenthesis :
◦ [Georgiev–Lindblat–Sogge]
GWP for NLW in Rn
◦ [Tataru] shortcut via Hn

 t2 − x2 − · · · − x2 > 0
n
1
Light cone Ω
t > 0
Hn
es
t
Ω
Rn
= ∂ 2t − ∆Rn
− n+3
2 s
=e
n−1 2o
n−1 s
2
◦e 2
◦ ∂s − ∆Hn −
{z 2
}
|
n
−L
dispersive estimates on Hn
with exponential decay
GWP for NLW on Rn
weighted Strichartz estimates on Hn
Rn
NLS on compact manifolds M
◦ [Burq–Gérard–Tzvetkov]
local Strichartz estimates
with loss of derivatives
ke it∆f (x)kLp(I;Lq (M )) ≤ CI kf kH 1/p(M )
◦ [Bourgain]
LWP on H s(Tn) when s > n−2
2
◦
 [Burq–Gérard–Tzvetkov]
 LWP on H s (M ) when s >
 LWP on H s (Sn) when s >
◦ [Gérard–Pierfelice]
GWP on H 1(M )
for 4–dimensional manifolds
n−1
2
n−2
2
NLS on hyperbolic space Hn
Previous results :
◦ [Fontaine]
◦ [Tataru]
◦
[Banica] weighted dispersive estimate


dimension n ≥ 3



radial functions



 weight w(x) = sinh |x|
n |x| n
3o
−2
−2
kw(x)u(t, x)k ∞ . |t| +|t|
kw(x)−1f (x)k 1
Lx
Lx
◦
[Pierfelice] weighted Strichartz estimate



Damek–Ricci spaces




 radial functions
m m
1
2
sinh
|x|
sinh
2|x|

weight
w(x)
=


|x|
2|x|



 same admissible interval as Rn
1−1
2
kw(x) q̃ u(t, x)k
p̃ q̃
Lt Lx
. kf (x)k
+ kw(x)
L2
x
1−1
q 2
g(t, x)k
◦ [Banica–Carles–Staffilani]
application to NLS and scattering
for radial functions
◦ [Chanillo]
partial results for G/K with G complex
p′ q ′
Lt Lx
Kernel expression : r = |x|
• n odd :
n i sign(t) π
1 i( n−1 )2 t
−
−
− n+1
4 |t| 2 e
2
st (r) = 2 2 π 2 e
i r2
1 ∂ n−1
−
2
4 t
× −
sinh r ∂r
e
• n even :
n+1
π
1 i( n−1 )2 t
−
i
sign(t)
−
− n+1
4 |t| 2 e
2
st(r) = 2 2 π 2 e
Z ∞
sinh s
1 ∂ n2 − i s2
×
ds √
−
e 4 t
sinh s ∂s
cosh s−cosh r
r
Kernel estimates : t 6= 0
|st(r)| .

 |t|−n/2 j(r)−1/2
 |t|−3/2 ϕ (r)
0
if |t| ≤ 1+r
if |t| ≥ 1+r

n−1
 j(r)−1/2 ≍ (1+r) n−1
2 e− 2 r
where

− n−1
ϕ0(r) ≍ (1+r) e 2 r

 |t|−n/2 if 0 < |t| ≤ 1
∀ q>2
⇒ kstk q .
 −3/2
L
|t|
Idem for kstk
Lq,a
if |t| ≥ 1
∀ q > 2 and ∀ a ≥ 1
Dispersive estimates :
• 0 < |t| ≤ 1 :
ke it∆ k
′
Lq →Lq̃
.

1
.

1
• |t| ≥ 1 :
ke it∆ k
′
Lq →Lq̃
if q = q̃ = 2
1
1 1
1
 |t|− max{ 2 − q , 2 − q̃ }n
if q = q̃ = 2
 |t|− 23
if 2 < q, q̃ ≤ ∞
∼
1/q
1/2
1/2
1/q’
if 2 < q, q̃ ≤ ∞
Proof : Interpolation
3

 |t| ≥ 1
Crucial estimate : ke it∆kLq′→Lq . |t|− 2
 q>2
follows from
3
−2
• kstkLq,1 . |t|
• the sharp Kunze–Stein phenomenon :
[Ionescu 2000] L2,1(G)∗L2,1(G) ⊂ L2,∞(G)
′,a
′,b
′,c
q
q
q
⇒ [Cowling 1997] L (G)∗L (G) ⊂ L (G)
if 1 ≤ q ′ < 2 and 1a + 1b − 1c ≥ 1
′
′
′ ′
⇒ Lq (K\G/K)∗Lq (G/K) ⊂ Lq ,c (K\G/K)
if 1 ≤ q ′ < 2 and c1′ ≤ q2′ −1
′
q
⇒ L (G/K)∗Lq,c(K\G/K) ⊂ Lq (G/K)
if q > 2 and c ≤ 2q
Strichartz estimates :
ku(t, x)k
p̃ q̃
Lt Lx
. kf (x)k
L2
x
+ kg(t, x)k
p′ q ′
Lt Lx
∀ (p, q), (p̃, q̃) in the admissible triangle
1/q
1/2
1/2−1/n
1/2
1/p
Wellposedness
Same results as in Rn
with the following improvement
Theorem :
Assume |g(t, x)| . |u(t, x)|1+γ with γ > 0
Then global wellposedness for s.i.d.

 in L2 (Hn) if γ ≤ 4
n
 in H 1 (Hn) if γ ≤ 4
n−2
Scattering
Theorem
: Same assumption. Then

 L2 scattering for γ ≤ 4 and s.i.d.
n
 H 1 scattering for γ ≤ 4 and s.i.d.
n−2
In contrast with Rn, no long range effect
scattering for γ > 0 arbitrarily small
4. Further results and problems
NLW (NonLinear Wave equation)
Smoothing properties
NLS and NLW
• Damek–Ricci spaces
• higher rank noncompact symmetric spaces
• compact symmetric spaces
• discrete setting
◦ trees, hyperbolic groups, ...
◦ buildings
• special functions related to root systems
◦ Dunkl (rational)
◦ Heckman–Opdam (trigonometric)
◦ Macdonald–Cherednik (q–theory)
Conjectural kernel estimate in higher rank :
Let t 6= 0 and x ∈ a+

 |t| ≥ 1+hα, xi
Assume
 |t| ≤ 1+hα, xi
∀ α∈I
∀ α ∈ S rI
_
βC
S I
_
αC
Ι
Then |st(x)| .
|t|−dI /2 ϕI,0(xI )
n
o
jI (x) 1/2
jS (x)
Here : I ⊂ S (simple roots)
+
|−
dI = ℓ/2 − |RI,red
P
α∈R+rRI+
mα/2
ϕI,0 = basic spherical function of (GI , KI )
jI = jacobian of the exponential map
pI −→ GI /KI