Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Some results about scattering theory for the
massive Dirac elds in the Schwarzschild-Anti-de
Sitter spacetime
Guillaume Idelon-Riton
AARG Final Meeting
Tuesday 4th October 2016
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Plan
1
Introduction
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Plan
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Plan
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Plan
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Plan
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
2
Asymptotic completeness
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Plan
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
2
Asymptotic completeness
Results
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Plan
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
2
Asymptotic completeness
Results
Ideas of the proof
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Plan
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
2
Asymptotic completeness
Results
Ideas of the proof
3
Logarithmic lower bound on the local energy decay and resonances
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Plan
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
2
Asymptotic completeness
Results
Ideas of the proof
3
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Plan
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
2
Asymptotic completeness
Results
Ideas of the proof
3
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
2
Asymptotic completeness
3
Logarithmic lower bound on the local energy decay and resonances
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
The Schwarzschild-Anti-de Sitter spacetime
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
2
Asymptotic completeness
Results
Ideas of the proof
3
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
- The Schwarzschild-Anti-de Sitter spacetime is a solution of the
Einstein vacuum equations with a cosmological constant
Λ = − l32 < 0.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
- The Schwarzschild-Anti-de Sitter spacetime is a solution of the
Einstein vacuum equations with a cosmological constant
Λ = − l32 < 0.
2
- Let F (r ) = 1 − 2rM + rl 2 with M > 0. We have :
gSAdS = F (r ) dt 2 − F (r )
Guillaume Idelon-Riton
−1
dr 2 − r 2 dθ2 + sin2 (θ) dϕ2 .
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
- The Schwarzschild-Anti-de Sitter spacetime is a solution of the
Einstein vacuum equations with a cosmological constant
Λ = − l32 < 0.
2
- Let F (r ) = 1 − 2rM + rl 2 with M > 0. We have :
gSAdS = F (r ) dt 2 − F (r )
−1
dr 2 − r 2 dθ2 + sin2 (θ) dϕ2 .
- Singularities at r = 0 and at r = rSAdS .
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
- The Schwarzschild-Anti-de Sitter spacetime is a solution of the
Einstein vacuum equations with a cosmological constant
Λ = − l32 < 0.
2
- Let F (r ) = 1 − 2rM + rl 2 with M > 0. We have :
gSAdS = F (r ) dt 2 − F (r )
−1
dr 2 − r 2 dθ2 + sin2 (θ) dϕ2 .
- Singularities at r = 0 and at r = rSAdS .
- We consider :
2
SAdS = Rt ×]rSAdS , +∞[r ×Sθ,ϕ
.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Black hole spacetime
Guillaume Idelon-Riton
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Radial null curves
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
Along radial null curves, we have :
−1 2
F (r ) ṫ 2 − F (r )
Guillaume Idelon-Riton
r˙ = 0.
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Radial null curves
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
Along radial null curves, we have :
−1 2
F (r ) ṫ 2 − F (r )
Thus :
r˙ = 0.
dt
−1
= ±F (r ) .
dr
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Radial null curves
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
Along radial null curves, we have :
−1 2
F (r ) ṫ 2 − F (r )
r˙ = 0.
Thus :
dt
−1
= ±F (r ) .
dr
We introduce a new coordinate x such that :
dx
−1
= F (r ) .
dr
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Radial null curves
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
The variable x satises :
x
→
r →rSAdS
−∞;
Guillaume Idelon-Riton
x
→
r →+∞
0
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Radial null curves
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
The variable x satises :
x
→
r →rSAdS
−∞;
x
→
r →+∞
0
and :
t = x + constant,
t = −x + constant
along those curves. We go to innity in nite time. The SAdS spacetime
is not globally hyperbolic.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Non globally hyperbolic
Guillaume Idelon-Riton
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Some known results
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
• P. Breitenlohner et D. Z. Freedman study well-posedness of the
Klein-Gordon equation in Anti-de Sitter spacetime.
• A. Bachelot study the same problem for the massive Dirac equation
•
•
•
•
in Anti-de Sitter spacetime.
A. Vasy study the problem in asymptotically Anti-de Sitter
spacetime and gives some results concerning the scattering for the
wave equation.
G. Holzegel et J. Smulevici investigate the stability of the
Schwarzschild-Anti-de Sitter spacetime and the local energy decay of
the Klein-Gordon elds in the Kerr-Anti-de Sitter spacetime.
O. Gannot localizes resonances exponentially close to the real axis in
Schwarzschild-Anti-de Sitter spacetime and denes resonances in the
Kerr-Anti-de Sitter space-time.
C. Warnick shows the existence of resonances in asymptotically
Anti-de Sitter spacetimes.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The massive Dirac equation
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
2
Asymptotic completeness
Results
Ideas of the proof
3
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The equation
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
The massive Dirac equation can be written as :
1
γ 0 ∂t ψ + γ 1 ∂x ψ +
+ imF
1
(r ) 2
F (r ) 2
r
1
γ 2 ∂θ + cot (θ) + γ 3
2
1
∂ ψ
sin (θ) ϕ
ψ = 0.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The equation
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
The massive Dirac equation can be written as :
1
γ 0 ∂t ψ + γ 1 ∂x ψ +
+ imF
1
(r ) 2
F (r ) 2
r
1
γ 2 ∂θ + cot (θ) + γ 3
2
1
∂ ψ
sin (θ) ϕ
ψ = 0.
We then formulate this equation as a spectral problem :
∂t ψ = iHm ψ
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The equation
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
The massive Dirac equation can be written as :
1
γ 0 ∂t ψ + γ 1 ∂x ψ +
+ imF
1
(r ) 2
F (r ) 2
r
1
γ 2 ∂θ + cot (θ) + γ 3
2
1
∂ ψ
sin (θ) ϕ
ψ = 0.
We then formulate this equation as a spectral problem :
∂t ψ = iHm ψ
where :
0 1
Hm = iγ γ ∂x + i
0
− mγ F
F
1
(r ) 2
1
(r ) 2
r
D
S2
}|
z
1
0 2
γ γ ∂θ + cot (θ) + γ 0 γ 3
2
1
∂
sin (θ) ϕ
{
.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Dirac matrices
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
We consider this operator on the Hilbert space
4
H = L2 ] − ∞, 0[x ×S2θ,ϕ , dxdω
.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Dirac matrices
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
We consider this operator on the Hilbert space
4
H = L2 ] − ∞, 0[x ×S2θ,ϕ , dxdω
.
We denote Γ1 = −γ 0 γ 1 = diag (1, −1, −1, 1), Dx = −i∂x ,
1
1
2
A (x) = F (rr (x))
and B (x) = F (r (x)) 2 .
(x)
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Dirac matrices
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
We consider this operator on the Hilbert space
4
H = L2 ] − ∞, 0[x ×S2θ,ϕ , dxdω
.
We denote Γ1 = −γ 0 γ 1 = diag (1, −1, −1, 1), Dx = −i∂x ,
1
1
2
A (x) = F (rr (x))
and B (x) = F (r (x)) 2 .
(x)
The Dirac matrices satisfy the following relations :
∗
∗
- γ 0 = γ 0 , γ j = −γ j , 1 6 j 6 3,
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Dirac matrices
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
We consider this operator on the Hilbert space
4
H = L2 ] − ∞, 0[x ×S2θ,ϕ , dxdω
.
We denote Γ1 = −γ 0 γ 1 = diag (1, −1, −1, 1), Dx = −i∂x ,
1
1
2
A (x) = F (rr (x))
and B (x) = F (r (x)) 2 .
(x)
The Dirac matrices satisfy the following relations :
∗
∗
- γ 0 = γ 0 , γ j = −γ j , 1 6 j 6 3,
- γ µ γ ν + γ ν γ µ = 2g µν 1, 0 6 µ, ν 6 3.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
Spinoidal spherical harmonics decomposition
We can decompose the Dirac operator D S2 on the sphere with spherical
spinoidal harmonics to obtain :
1
s,n
Hm
= Γ1 Dx + γ 0 γ 2 s +
A (x) − mγ 0 B (x) ,
2
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
Spinoidal spherical harmonics decomposition
We can decompose the Dirac operator D S2 on the sphere with spherical
spinoidal harmonics to obtain :
1
s,n
Hm
= Γ1 Dx + γ 0 γ 2 s +
A (x) − mγ 0 B (x) ,
2
that we consider on the Hilbert space :
4
Hs,n = L2 (] − ∞, 0[, dx) ⊗ Ys,n
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
Spinoidal spherical harmonics decomposition
We can decompose the Dirac operator D S2 on the sphere with spherical
spinoidal harmonics to obtain :
1
s,n
Hm
= Γ1 Dx + γ 0 γ 2 s +
A (x) − mγ 0 B (x) ,
2
that we consider on the Hilbert space :
4
Hs,n = L2 (] − ∞, 0[, dx) ⊗ Ys,n
where Ys,n are the eigenvectors of D S2 .
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The functions A and B
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
The potentials are smooth and satisfy :
A (x) =
1
+ x 2 + o x 2 , x ∼ 0,
e κx + o (e κx ) , x ∼ −∞,
(
l
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The functions A and B
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
The potentials are smooth and satisfy :
A (x) =
1
+ x 2 + o x 2 , x ∼ 0,
e κx + o (e κx ) , x ∼ −∞,
(
l
(
− xl + x + o (x) , x ∼ 0,
B (x) =
e κx + o (e κx ) , x ∼ −∞,
where κ is the surface gravity.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The semi-classical Dirac operator
Let h =
1
s+ 12
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The semi-classical Dirac operator
Let h =
1
s+ 12
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
. The semi-classical Dirac operator is given by :
s,n
Hm,h = hHm
= Γ1 hDx + γ 0 γ 2 A (x) − hmγ 0 B (x) .
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The semi-classical Dirac operator
Let h =
1
s+ 12
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
. The semi-classical Dirac operator is given by :
s,n
Hm,h = hHm
= Γ1 hDx + γ 0 γ 2 A (x) − hmγ 0 B (x) .
The square is given by :
2
Hm,h
= −h2 ∂x2 + A2 (x) + h2 m2 B 2 (x)
− ihγ 1 γ 2 A0 (x) + ih2 mγ 1 B 0 (x) .
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The semi-classical Dirac operator
Let h =
1
s+ 12
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
. The semi-classical Dirac operator is given by :
s,n
Hm,h = hHm
= Γ1 hDx + γ 0 γ 2 A (x) − hmγ 0 B (x) .
The square is given by :
2
Hm,h
= −h2 ∂x2 + A2 (x) + h2 m2 B 2 (x)
− ihγ 1 γ 2 A0 (x) + ih2 mγ 1 B 0 (x) .
The Klein-Gordon operator is not a good approximation of the Dirac
operator in a semi-classical sense.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
The Cauchy problem
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
2
Asymptotic completeness
Results
Ideas of the proof
3
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Asymptotic behavior at x = 0
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
We introduce the natural domain of our operator :
s,n
s,n
Dnat (Hm
) = {ϕ ∈ Hs,n |Hm
ϕ ∈ Hs,n }.
Using a cut-o function and a unitary transform, we are able to use the
precise result obtained by A. Bachelot concerning the asymptotic
behavior near the boundary of the elements in the natural domain of the
Dirac equation in the Anti-de Sitter space.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Asymptotic behavior at x = 0
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
Theorem
For
ml >
1
2,
we have
||Φ(x, .)||L2 (S 2 ) = O
For
ml =
1
2,
√ −x , x → 0.
we have
||Φ(x, .)||L2 (S 2 ) = O
p
(−x) ln (−x) , x → 0.
1
1
2 2
2
2
0 < ml < 12 , there exist functions
ψ− ∈ W− , χ− ∈ W+ , ψ+ , χ+ ∈ L (S )
0
and φ ∈ C
] − ∞, 0]x ; L2 (S 2 ; C4 ) satisfying




ψ− (θ, ϕ)
ψ+ (θ, ϕ)
 χ− (θ, ϕ) 


ml  χ+ (θ, ϕ) 

Φ(x, θ, ϕ) = (−x)−ml 
−iψ− (θ, ϕ) + (−x)  iψ+ (θ, ϕ)  + φ(x, θ, ϕ),
iχ− (θ, ϕ)
−iχ+ (θ, ϕ)
For
||φ(x, .)||L2 (S 2 ) = o
Guillaume Idelon-Riton
√
−x , x → 0.
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Self-adjointness
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
Proposition
For 2ml > 1, the operator Hms,n is essentially self-adjoint on
C0∞ (]−∞, 0[). Moreover, if 2ml > 1, the domain of this operator is
H01 (] − ∞, 0[).
Proposition
MIT
For 2ml <
1, the operator Hs,n
is self-adjoint
on
the domain
√
MIT
s,n
D Hs,n
= {ϕ ∈ Hs,n |Hm
ϕ ∈ Hs,n , γ 1 + i ϕ (x) = o
−x }.
Proposition
The operator Hm , equipped with the appropriate domain, is self-adjoint
on D (Hm ).
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Asymptotic completeness
1
Introduction
2
Asymptotic completeness
Results
Ideas of the proof
3
Logarithmic lower bound on the local energy decay and resonances
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Motivation
- We want to show that the elds are going to the distant regions of
our spacetime.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Motivation
- We want to show that the elds are going to the distant regions of
our spacetime.
- In these regions, we wish to approach the dynamic by a simpler one.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Results
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
2
Asymptotic completeness
Results
Ideas of the proof
3
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Absence of eigenvalue
Proposition
For all m > 0 , the Dirac operator Hm has no real eigenvalue.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Asymptotic dynamic
We dene the operator :
Hc = Γ1 Dx
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Asymptotic dynamic
We dene the operator :
with domain :
Hc = Γ1 Dx
D (Hc ) = {ϕ ∈ H|Hc ϕ ∈ H, ϕ1 (0) = −ϕ3 (0) , ϕ2 (0) = ϕ4 (0)}.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Asymptotic completeness
Theorem
For all ϕ ∈ H and for all m > 0, the limits :
lim e itHc e −itHm ϕ
t→∞
lim e itHm e −itHc ϕ
t→∞
exist. We write :
Ωϕ = lim e itHc e −itHm ϕ
t→∞
W ϕ = lim e itHm e −itHc ϕ
t→∞
for all ϕ ∈ H. Then we have :
Ω∗ = W .
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Asymptotic velocity
We dene C∞ (R) = {f ∈ C 0 (R) |f (y ) → 0}.
y →±∞
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Asymptotic velocity
We dene C∞ (R) = {f ∈ C 0 (R) |f (y ) → 0}. We obtain :
y →±∞
Theorem
Let A = Γ1 x . Then, for all m > 0 and J ∈ C∞ (R), we have :
s − lim e
t→∞
itHm
J
Guillaume Idelon-Riton
A
t
e −itHm = J (1) .
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Ideas of the proof
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
2
Asymptotic completeness
Results
Ideas of the proof
3
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
The asymptotic velocity is a consequence of the asymptotic completeness.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
The asymptotic velocity is a consequence of the asymptotic completeness.
Indeed, the dynamic of Hc is given by a transport at speed 1 in the
direction of the black hole.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
A formal calculus
We introduce the operator
e itH Ae −itH .
Formally, we have :
d e itH Ae −itH
= e itH [H, iA] e −itH .
dt
If [H, iA] > 1 − δ , then
dynamic.
A
t
is necessary larger than 1 − δ along the
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Minimal velocity estimates
Proposition
Let A = Γ1 x . For all m > 0, all g ∈ C0∞ (R), supp (g ) ⊂ (−∞, 1 − δ)
with δ > 0, and all f ∈ C0∞ (R), we have :
Z
∞
2
s,n dt
2
s,n
−itHm
g A f (Hm
)e
u
t 6 C kuk ,
t
∀u ∈ Hs,n ,
1
s,n
A
s − lim g
e −itHm = 0.
t→∞
t
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
The Mourre method
We consider two self-adjoint operators H and A. We will call A a
conjugate operator.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
The Mourre method
We consider two self-adjoint operators H and A. We will call A a
conjugate operator.
Denition
We say that H ∈ C 2 (A) if there exists z ∈ ρ (H) such that :
−1
t → e itA (H − z)
e −itA
is of class C 2 for the strong topology of L (H).
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
The Mourre method
We consider two self-adjoint operators H and A. We will call A a
conjugate operator.
Denition
We say that H ∈ C 2 (A) if there exists z ∈ ρ (H) such that :
−1
t → e itA (H − z)
e −itA
is of class C 2 for the strong topology of L (H).
Denition
Suppose that H ∈ C 2 (A). We say that the couple (H, A) satisfy a
Mourre estimate on a compact interval I ⊂ R if there exists a constant
c > 0 and a compact operator K such that :
1I (H) [H, iA] 1I (H) > c1I (H) + K .
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Consequences
If there is no eigenvalue on I , we can suppose that K = 0 by decreasing
the size of I .
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Consequences
If there is no eigenvalue on I , we can suppose that K = 0 by decreasing
the size of I .
Proposition
Suppose that H ∈ C 2 (A) and that a Mourre estimate with K = 0 is
satised on an interval I . Then, for all g ∈ C0∞ (R), supp (g ) ⊂ (−∞, c)
and all f ∈ C0∞ (I ), we have :
2
g A f (H) e −itH u dt 6 C kuk2 , ∀u ∈ H,
t
t
1
A
s − lim g
f (H) e −itH = 0.
t→∞
t
Z
∞
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Mourre estimates
In our case, the conjugate operator is A = Γ1 x . This conjugate operator
rst appears in the work of T. Daudé.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Mourre estimates
In our case, the conjugate operator is A = Γ1 x . This conjugate operator
rst appears in the work of T. Daudé.
Here, we obtain :
1
s,n
xA (x) + 2imγ 1 xB (x) .
[Hm
, iA] = 1 + 2iγ 2 γ 1 s +
2
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Mourre estimates
In our case, the conjugate operator is A = Γ1 x . This conjugate operator
rst appears in the work of T. Daudé.
Here, we obtain :
1
s,n
xA (x) + 2imγ 1 xB (x) .
[Hm
, iA] = 1 + 2iγ 2 γ 1 s +
2
Proposition
Let f ∈ C 0 (] − ∞, 0]) tending to 0 at −∞. Let z ∈ ρ(Hms,n )) where
s,n
s,n
ρ(Hm
) is the resolvent set of Hm
. Then the operator f (x)(Hms,n − z)−1
is compact on H for all m > 0.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Results
Ideas of the proof
Idea of the proof
Operator H− :
Operator H+ :
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Logarithmic lower bound on the local energy decay and resonances
1
Introduction
2
Asymptotic completeness
3
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Local energy decay
- We want to study how the energy of the eld decreases in bounded
region of the space.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Local energy decay
- We want to study how the energy of the eld decreases in bounded
region of the space.
- The conning potential implies a slower decay.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Lower bound
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
2
Asymptotic completeness
Results
Ideas of the proof
3
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Lower bound on the local energy decay
Theorem
For all compact set K ⊂] − ∞, 0[, there exists a constant C > 0 such
that :
lim sup
sup
t→+∞ ϕ∈H,kϕk=1
ln (t) e itH ϕL2 (K ) > C .
For the Klein-Gordon equation in Kerr-Anti-de Sitter, G.Holzegel and J.
Smulevici obtained a logarithmic decay for the local energy.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Idea of the proof
Let ϕh such that k(Hm,h − (E + (h))) ϕkL2 (]−∞,0[) 6 e − h and
+
ψh = e iE (h)t ϕh .
D
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Idea of the proof
Let ϕh such that k(Hm,h − (E + (h))) ϕkL2 (]−∞,0[) 6 e − h and
+
ψh = e iE (h)t ϕh . We consider the equation :
D
−i∂t χ − Hχ = F
where kF kL2 (]−∞,0[) 6 e − h .
D
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Idea of the proof
Let ϕh such that k(Hm,h − (E + (h))) ϕkL2 (]−∞,0[) 6 e − h and
+
ψh = e iE (h)t ϕh . We consider the equation :
D
−i∂t χ − Hχ = F
where kF kL2 (]−∞,0[) 6 e − h . Let ψ̃h = e itH ϕh . Using Duhamel's formula,
we have :
D
D
kψh k 2 − ψ̃h 2 6 Cte − h .
L (K )
L (K )
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Idea of the proof
Let ϕh such that k(Hm,h − (E + (h))) ϕkL2 (]−∞,0[) 6 e − h and
+
ψh = e iE (h)t ϕh . We consider the equation :
D
−i∂t χ − Hχ = F
where kF kL2 (]−∞,0[) 6 e − h . Let ψ̃h = e itH ϕh . Using Duhamel's formula,
we have :
D
D
kψh k 2 − ψ̃h 2 6 Cte − h .
L (K )
L (K )
We deduce that :
itH D
e ϕh 2
> kϕh kL2 (K ) − Cte − h .
L (K )
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Idea of the proof
Let ϕh such that k(Hm,h − (E + (h))) ϕkL2 (]−∞,0[) 6 e − h and
+
ψh = e iE (h)t ϕh . We consider the equation :
D
−i∂t χ − Hχ = F
where kF kL2 (]−∞,0[) 6 e − h . Let ψ̃h = e itH ϕh . Using Duhamel's formula,
we have :
D
D
kψh k 2 − ψ̃h 2 6 Cte − h .
L (K )
L (K )
We deduce that :
itH D
e ϕh 2
> kϕh kL2 (K ) − Cte − h .
L (K )
Take th such that kϕh kL2 (K ) − Cth e − h = h.
D
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Idea of the proof
Let ϕh such that k(Hm,h − (E + (h))) ϕkL2 (]−∞,0[) 6 e − h and
+
ψh = e iE (h)t ϕh . We consider the equation :
D
−i∂t χ − Hχ = F
where kF kL2 (]−∞,0[) 6 e − h . Let ψ̃h = e itH ϕh . Using Duhamel's formula,
we have :
D
D
kψh k 2 − ψ̃h 2 6 Cte − h .
L (K )
L (K )
We deduce that :
itH D
e ϕh 2
> kϕh kL2 (K ) − Cte − h .
L (K )
Take th such that kϕh kL2 (K ) − Cth e − h = h. For h suciently small, we
obtain :
D
it H e h ϕh 2
>
L (K )
Guillaume Idelon-Riton
D
.
2 ln (th )
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Quasimodes
Recall that :
Hm,h = Γ1 hDx + γ 0 γ 2 A (x) − hmγ 0 B (x) .
Theorem
We suppose that ml > 1. Let S > 0 such that l12 + S < A2 (x+ ). There
1
exist constants h0 > 0, D > 0, a real number (E + (h)) 2 > 0 such that
E + (h) < l12 + S and a function ϕ ∈ D (Hm,h ) with kϕkL2 (]−∞,0[) = 1
such that :
1 Hm,h − E + (h) 2 ϕ
L2 (]−∞,0[)
D
6 e− h ,
for all h ∈ (0, h0 ).
A similar result has been obtained by O. Gannot in the
Schwarzschild-Anti-de Sitter spacetime.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Exponential decay
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Ideas of proof
2 , P =P
- We introduce P = Hm,h
+
|[x+ ,0[ and H+ = Hm,h|[x+ ,0[ with
Dirichlet boundary condition at x+ .
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Ideas of proof
2 , P =P
- We introduce P = Hm,h
+
|[x+ ,0[ and H+ = Hm,h|[x+ ,0[ with
Dirichlet boundary condition at x+ .
- The
an eigenvalue E + (h) such that
+ operator1 P+ admits
E (h) − 2 + h 6 Ch 12 .
l
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Ideas of proof
2 , P =P
- We introduce P = Hm,h
+
|[x+ ,0[ and H+ = Hm,h|[x+ ,0[ with
Dirichlet boundary condition at x+ .
- The
an eigenvalue E + (h) such that
+ operator1 P+ admits
E (h) − 2 + h 6 Ch 12 .
l
1
- Thus, H+ admits an eigenvalue E + (h) 2 . Let ϕ+ be an associated
eigenvector. Remark that ϕ+ is also an eigenvector of P+ for E + (h).
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Ideas of proof
2 , P =P
- We introduce P = Hm,h
+
|[x+ ,0[ and H+ = Hm,h|[x+ ,0[ with
Dirichlet boundary condition at x+ .
- The
an eigenvalue E + (h) such that
+ operator1 P+ admits
E (h) − 2 + h 6 Ch 12 .
l
1
- Thus, H+ admits an eigenvalue E + (h) 2 . Let ϕ+ be an associated
eigenvector. Remark that ϕ+ is also an eigenvector of P+ for E + (h).
- Agmon Estimates :
kϕ+ kL2 (Σ1 ) 6 Ce − h kϕ+ kL2 (Σ2 ) .
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Ideas of proof
2 , P =P
- We introduce P = Hm,h
+
|[x+ ,0[ and H+ = Hm,h|[x+ ,0[ with
Dirichlet boundary condition at x+ .
- The
an eigenvalue E + (h) such that
+ operator1 P+ admits
E (h) − 2 + h 6 Ch 12 .
l
1
- Thus, H+ admits an eigenvalue E + (h) 2 . Let ϕ+ be an associated
eigenvector. Remark that ϕ+ is also an eigenvector of P+ for E + (h).
- Agmon Estimates :
kϕ+ kL2 (Σ1 ) 6 Ce − h kϕ+ kL2 (Σ2 ) .
- We introduce χ ∈ C0∞ (]x+ , 0]) such that χ = 1 on [x+ , 0[\Σ1 .
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Ideas of proof
2 , P =P
- We introduce P = Hm,h
+
|[x+ ,0[ and H+ = Hm,h|[x+ ,0[ with
Dirichlet boundary condition at x+ .
- The
an eigenvalue E + (h) such that
+ operator1 P+ admits
E (h) − 2 + h 6 Ch 12 .
l
1
- Thus, H+ admits an eigenvalue E + (h) 2 . Let ϕ+ be an associated
eigenvector. Remark that ϕ+ is also an eigenvector of P+ for E + (h).
- Agmon Estimates :
kϕ+ kL2 (Σ1 ) 6 Ce − h kϕ+ kL2 (Σ2 ) .
- We introduce χ ∈ C0∞ (]x+ , 0]) such that χ = 1 on [x+ , 0[\Σ1 .
- We obtain :
1
Hm,h − E + (h) 2 χϕ+ L2 (]−∞,0[)
Guillaume Idelon-Riton
= iγ 0 γ 1 h∂x , χ ϕ+ L2 (]−∞,0[)
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Resonances
1
Introduction
The Schwarzschild-Anti-de Sitter spacetime
The massive Dirac equation
The Cauchy problem
2
Asymptotic completeness
Results
Ideas of the proof
3
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Formally
(−i∂t + H) ϕ = F .
We use the Fourier transform in time to obtain :
(H − λ) ϕ̂ = F̂ .
When (H − λ) is invertible, we have :
1
ϕ=
2π
Z
Im(λ)=c
Guillaume Idelon-Riton
−1
e itλ (H − λ)
F̂ dλ,
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Existence of resonances
Dene the function :
(
e x , x ∼ −∞,
f (x) =
C , x ∼ 0,
where C ∈ R is a constant.
Proposition
The operator f (Hm,h − λ)−1 f dened for Im (λ) > 0 extend
meromorphically to {λ ∈ C | Im (λ) > −} for all 0 < < κ2 where κ is
the surface gravity. The poles of this extension are called the resonances.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Existence of resonances
Opérator H− :
Opérator H+ :
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Formula for the resolvent
This formula is based on the one obtained in the non massive case on the
half-line by A. Iantchenko and E. Korotyaev.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Formula for the resolvent
This formula is based on the one obtained in the non massive case on the
half-line by A. Iantchenko and E. Korotyaev. Consider the equation :
(Hm,h − λ) ϕ = 0.
We can construct Jost solutions satisfying :
0
 0 


0
 0 


ψh (x) =  −i λx  + o 
e −i λxh 
e h
0
0




at −∞.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Formula for the resolvent
This formula is based on the one obtained in the non massive case on the
half-line by A. Iantchenko and E. Korotyaev. Consider the equation :
(Hm,h − λ) ϕ = 0.
We can construct Jost solutions satisfying :
0
 0 


0
 0 


ψh (x) =  −i λx  + o 
e −i λxh 
e h
0
0




at −∞. We obtain another solution ψ̃h = (−i) γ 0 γ 1 γ 2 ψ .
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Formula for the resolvent
This formula is based on the one obtained in the non massive case on the
half-line by A. Iantchenko and E. Korotyaev. Consider the equation :
(Hm,h − λ) ϕ = 0.
We can construct Jost solutions satisfying :
0
 0 


0
 0 


ψh (x) =  −i λx  + o 
e −i λxh 
e h
0
0




at −∞. We obtain another solution ψ̃h = (−i) γ 0 γ 1 γ 2 ψ . We can also
construct two solutions ϕh and ϕ̃h satisfying the boundary conditions.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Formula for the resolvent
We write α (λ, h) = ϕ1,h ψ2,h − ψ1,h ϕ2,h + ϕ3,h ψ4,h − ψ3,h ϕ4,h ,
β (λ, h) = ϕ1,h ψ3,h − ψ1,h ϕ3,h + ϕ2,h ψ4,h − ψ2,h ϕ4,h and :


0
α
β
0
−α 0
0 β
.
Mα,β = 
−β
0
0 α
0 −β −α 0
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Formula for the resolvant
Proposition
Consider the function dened by :
−1
iΓ1 1]−∞,x[ (y )
Rm,h (x, y , λ) = ϕh (x) ψht (y ) + ϕ̃h (x) ψ̃ht (y ) Mα,β
−1
+ ψh (x) ϕth (y ) + ψ̃h (x) ϕ̃th (y ) Mα,β
iΓ1 1]x,0[ (y ) .
The integral operator given by :
Z
0
Rm,h (x, y , λ) f (y ) dy .
Rm,h (λ) f (x) =
−∞
satises
(Hm,h − λ) (Rm,h (λ) f ) (x) = f (x) .
2
4
for all f ∈ L (] − ∞, 0[) .
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Method for localizing the resonances
- We are looking for λ which cancel
α (λ, h) = ϕ1,h ψ2,h − ψ1,h ϕ2,h + ϕ3,h ψ4,h − ψ3,h ϕ4,h .
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Method for localizing the resonances
- We are looking for λ which cancel
α (λ, h) = ϕ1,h ψ2,h − ψ1,h ϕ2,h + ϕ3,h ψ4,h − ψ3,h ϕ4,h .
- Suppose that we have two solutions ϕ and ψ admitting development
in term of the parameter h when h → 0.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Method for localizing the resonances
- We are looking for λ which cancel
α (λ, h) = ϕ1,h ψ2,h − ψ1,h ϕ2,h + ϕ3,h ψ4,h − ψ3,h ϕ4,h .
- Suppose that we have two solutions ϕ and ψ admitting development
in term of the parameter h when h → 0.
- This gives development for α which are typically of the form :
S(λ)
α (λ, h) = 1 + e h (1 + O (h)) ,
where S (λ) =
R xj
xi
A2 (y ) + h2 m2 B 2 (y ) − λ2
Guillaume Idelon-Riton
12
dy .
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Method for localizing the resonances
- We are looking for λ which cancel
α (λ, h) = ϕ1,h ψ2,h − ψ1,h ϕ2,h + ϕ3,h ψ4,h − ψ3,h ϕ4,h .
- Suppose that we have two solutions ϕ and ψ admitting development
in term of the parameter h when h → 0.
- This gives development for α which are typically of the form :
S(λ)
α (λ, h) = 1 + e h (1 + O (h)) ,
1
where S (λ) = xxi j A2 (y ) + h2 m2 B 2 (y ) − λ2 2 dy .
- The resonances λ satisfy S (λ) = πih + O h2 .
R
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Method for localizing the resonances
- We are looking for λ which cancel
α (λ, h) = ϕ1,h ψ2,h − ψ1,h ϕ2,h + ϕ3,h ψ4,h − ψ3,h ϕ4,h .
- Suppose that we have two solutions ϕ and ψ admitting development
in term of the parameter h when h → 0.
- This gives development for α which are typically of the form :
S(λ)
α (λ, h) = 1 + e h (1 + O (h)) ,
1
where S (λ) = xxi j A2 (y ) + h2 m2 B 2 (y ) − λ2 2 dy .
- The resonances λ satisfy S (λ) = πih + O h2 .
- We study the zeros of S (λ) − πih and use the Rouché theorem
to
show that they correspond to zeros of S (λ) = πih + O h2 .
R
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Method for localizing the resonances
- We are looking for λ which cancel
α (λ, h) = ϕ1,h ψ2,h − ψ1,h ϕ2,h + ϕ3,h ψ4,h − ψ3,h ϕ4,h .
- Suppose that we have two solutions ϕ and ψ admitting development
in term of the parameter h when h → 0.
- This gives development for α which are typically of the form :
S(λ)
α (λ, h) = 1 + e h (1 + O (h)) ,
1
where S (λ) = xxi j A2 (y ) + h2 m2 B 2 (y ) − λ2 2 dy .
- The resonances λ satisfy S (λ) = πih + O h2 .
- We study the zeros of S (λ) − πih and use the Rouché theorem
to
show that they correspond to zeros of S (λ) = πih + O h2 .
- By carefully studying the action S (λ) in term of λ, we obtain
developments of λ in term of h.
This method was used by T. Ramond and S. Fujiié for the Schrödinger
equation.
R
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Summary
1) We have studied the existence and unicity of a solution to the
massive Dirac equation in the Schwarzschild-Anti-de Sitter
spacetime.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Summary
1) We have studied the existence and unicity of a solution to the
massive Dirac equation in the Schwarzschild-Anti-de Sitter
spacetime.
2) We obtained an asymptotic completeness result that shows that the
particules are going to the black hole at speed 1.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Summary
1) We have studied the existence and unicity of a solution to the
massive Dirac equation in the Schwarzschild-Anti-de Sitter
spacetime.
2) We obtained an asymptotic completeness result that shows that the
particules are going to the black hole at speed 1.
3) We obtain a lower bound on the local energy decay by constructing
exponentially accurate quasimodes.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Summary
1) We have studied the existence and unicity of a solution to the
massive Dirac equation in the Schwarzschild-Anti-de Sitter
spacetime.
2) We obtained an asymptotic completeness result that shows that the
particules are going to the black hole at speed 1.
3) We obtain a lower bound on the local energy decay by constructing
exponentially accurate quasimodes.
4) We proved the existence of resonances and introduce some tools to
localize them.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
What is left
1) Construct solutions having development in term of h.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
What is left
1) Construct solutions having development in term of h.
2) Extend the WKB solutions in order to obtain developments of the
Wronskians in term of h.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
What is left
1) Construct solutions having development in term of h.
2) Extend the WKB solutions in order to obtain developments of the
Wronskians in term of h.
3) Localize the resonances using this development.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
What is left
1) Construct solutions having development in term of h.
2) Extend the WKB solutions in order to obtain developments of the
Wronskians in term of h.
3) Localize the resonances using this development.
4) Finally, we wish to prove a logarithmic decay for the local energy.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Thank you !
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Exact WKB method
We are interested in solving the equation :
(Hm,h − λ) ϕ = 0
where :
Hm,h = Γ1 hDx + γ 0 γ 2 A (x) − hmγ 0 B (x) .
We rst extend holomorphically our potential to a domain in the left half
of the complex plane.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Change of variable
We introduce the phase
z (x) =
Rx
xα
A2 (y ) + h2 m2 B 2 (y ) − λ2
Guillaume Idelon-Riton
21
dy .
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Change of variable
We introduce the phase
z (x) =
Rx
xα
A2 (y ) + h2 m2 B 2 (y ) − λ2
21
dy .
We choose a determination of the square root which gives 3 turning
points.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Change of variable
We introduce the phase
z (x) =
Rx
xα
A2 (y ) + h2 m2 B 2 (y ) − λ2
21
dy .
We choose a determination of the square root which gives 3 turning
points.
z(x)
The function x → e h is almost a solution to the eigenvalue
problem.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Change of variable
We introduce the phase
z (x) =
Rx
xα
A2 (y ) + h2 m2 B 2 (y ) − λ2
21
dy .
We choose a determination of the square root which gives 3 turning
points.
z(x)
The function x → e h is almost a solution to the eigenvalue
problem.
z
We introduce the change of variable w (z) = Pe − h ϕ.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Modication of the equation
Lemma
Letting w (z) =
(
d
dz
d
dz
+∞
P
n=0
U2n
, we obtain the equations :
U2n+1
(U2n+1 ) = − h2 I2 + hNh (z) U2n+1 + MH,h (z) U2n
(U2n+2 ) = MH,h (z) U2n+1 + hNh (z) U2n ,
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Modication of the equation
Lemma
Letting w (z) =
(
d
dz
d
dz
+∞
P
n=0
U2n
, we obtain the equations :
U2n+1
(U2n+1 ) = − h2 I2 + hNh (z) U2n+1 + MH,h (z) U2n
(U2n+2 ) = MH,h (z) U2n+1 + hNh (z) U2n ,
We wish to solve our equations on a simply connected open bounded
subset Ω of our domain such that Ω contains no turning point.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Time-ordering
Denition
Let Γ (z̃, z) be a path from z̃ to z where z̃ is a point in C. We dene :
T
e
R
Γ(z̃,z)
f (ζ)dζ
=
+∞ Z
X
n=0
Γ(z̃,z)
Z
Z
···
Γ(z̃,zn )
Guillaume Idelon-Riton
Γ(z̃,z2 )
f (zn ) · · · f (z1 ) dz1 · · · dzn .
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Solutions of the system
Proposition
The solution of the system satisfying Un (z̃) = 0 for a choice of base
point z̃ is written as :
Z
U2n+1 (z) =
2
e h (ζ−z)I2 Υ (z, ζ) MH,h (ζ) U2n (ζ) dζ,
Γ(z̃,z)
Z
U2n+2 (z) =
MH,h (ζ) U2n+1 (ζ) + hNh (ζ) U2n (ζ) dζ,
Γ(z̃,z)
where Γ (z̃, z) is a path joining z̃ to z and Υ (z, z̃) = T e
Guillaume Idelon-Riton
R
Γ(z̃,z)
hNh (ζ)dζ
.
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
- We introduce the Stokes lines that are the curves satisfying :
Re (z (x) − z (y )) = 0.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
- We introduce the Stokes lines that are the curves satisfying :
Re (z (x) − z (y )) = 0.
- We choose a path Γ (z̃, z) such that Re (ζ) increases along this path,
that is, the path is tranverse to the Stokes lines.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Lower bound
Resonances
Development of solutions
We dene the following sums :
U even (x, h) =
X
U2n (z (x)) ,
U odd (x, h) =
n>0
X
U2n+1 (z (x)) .
n>0
Proposition
For all N ∈ N, we have the developments :
U even (x, h) −
N
X
U2n (z (x)) = O hN+1
n=0
U odd (x, h) −
N
X
U2n+1 (z (x)) = O hN+2
n=0
uniformly on every compact of Ω.
Guillaume Idelon-Riton
Some results about scattering theory for the massive Dirac elds
Introduction
Asymptotic completeness
Logarithmic lower bound on the local energy decay and resonances
Guillaume Idelon-Riton
Lower bound
Resonances
Some results about scattering theory for the massive Dirac elds