Resonances for r -normally hyperbolic trapped sets
Semyon Dyatlov
University of California, Berkeley
June 17, 2013
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
1 / 20
Motivation
Resonances are a discrete set of complex frequencies describing decay
of waves/classical correlations for open systems
Motivated by decay of linear waves on black holes, we study
resonances for r -normally hyperbolic trapped sets
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
2 / 20
Motivation
Resonances are a discrete set of complex frequencies describing decay
of waves/classical correlations for open systems
Motivated by decay of linear waves on black holes, we study
resonances for r -normally hyperbolic trapped sets
We obtain two resonance free strips in terms of the transversal
expansion rates
νmin ≤ νmax
and a Weyl law in the band in between:
λ
−
νmin −ε
2
+ε
− νmax
2
−(νmin −ε)
This is one of the very few known asymptotics for counting resonances.
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
2 / 20
Motivation
Resonances are a discrete set of complex frequencies describing decay
of waves/classical correlations for open systems
Motivated by decay of linear waves on black holes, we study
resonances for r -normally hyperbolic trapped sets
We obtain two resonance free strips in terms of the transversal
expansion rates
νmin ≤ νmax
and a Weyl law in the band in between:
λ
−
νmin −ε
2
+ε
− νmax
2
−(νmin −ε)
This is one of the very few known asymptotics for counting resonances.
Our result relies on the construction of a microlocal projector
corresponding to resonances in the band, a Fourier integral operator
We also get new information on microlocalization of resonant states
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
2 / 20
Motivation
Motivation: ringdown on black holes
Chandrasekhar '83:
. . . we may expect that any initial perturbation will, during its last stages,
decay in a manner characteristic of the black hole itself and independent of
the cause. In other words, we may expect that during these last stages, the
black hole emits gravitational waves with frequencies and rates of damping
that are characteristic of the black hole itself, in the manner of a bell
sounding its last dying notes.
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
3 / 20
Motivation
Motivation
Linear waves on (slowly) rotating Kerrde
Sitter black holes
e , g̃ )
(M
g̃ u = f ∈ C0∞ ,
u
|t <0 = 0.
r−
Open system: most energy escapes through
the event horizons.
r+
Two radial timelike geodesics,
with light cones shown
→ +∞?
What is the behavior of u (t ) as t
Theorem [D '10, '11]
There is a discrete set of resonances
{λj } ⊂ {Im λ ≤ 0}
such that
∀ν > 0,
∗ holds
the following resonance expansion
X
( , )=
u t x
Im
Semyon Dyatlov (UC Berkeley)
e
−it λj
Πj f (x ) + O(e −ν t ).
λj >−ν
Resonances for r -NH trapped sets
June 17, 2013
4 / 20
Motivation
Motivation
Linear waves on (slowly) rotating Kerrde
Sitter black holes
e , g̃ )
(M
g̃ u = f ∈ C0∞ ,
u
|t <0 = 0.
r−
Open system: most energy escapes through
the event horizons.
r+
Two radial timelike geodesics,
with light cones shown
→ +∞?
What is the behavior of u (t ) as t
Theorem [D '10, '11]
There is a discrete set of resonances
{λj } ⊂ {Im λ ≤ 0}
such that
∀ν > 0,
∗ holds
the following resonance expansion
X
( , )=
u t x
Im
Semyon Dyatlov (UC Berkeley)
e
−it λj
Πj f (x ) + O(e −ν t ).
λj >−ν
Resonances for r -NH trapped sets
June 17, 2013
4 / 20
Motivation
Theorem [D '10, '11]
Moreover, resonances satisfy a quantization condition, that is they lie
asymptotically on a distorted lattice.
The quantization condition agrees with the exact values computed in
BertiCardosoStarinets '09 (shown here for a
= 0, 0.05, . . . , 0.25)
−0.095
−0.0955
−0.096
−0.0965
l=4
l=3
l=2
−0.097
−0.0975
l=1
−0.098
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−0.285
−0.29
l=4
l=3
−0.295
l=2
−0.3
−0.305
[BeCaSt]
N=2
l=1
−0.31
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Previous work: Sá BarretoZworski '97, BonyHäfner '07,
DafermosRodnianski '07, MelroseSá BarretoVasy '08,
WunschZworski '10, Vasy '10. . . + numerous results for
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
Λ=0
June 17, 2013
5 / 20
Motivation
Theorem [D '10, '11]
Moreover, resonances satisfy a quantization condition, that is they lie
asymptotically on a distorted lattice.
The quantization condition agrees with the exact values computed in
BertiCardosoStarinets '09 (shown here for a
= 0, 0.05, . . . , 0.25)
−0.095
−0.0955
−0.096
−0.0965
l=4
l=3
l=2
−0.097
−0.0975
l=1
−0.098
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−0.285
−0.29
l=4
l=3
−0.295
l=2
−0.3
−0.305
[BeCaSt]
N=2
l=1
−0.31
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Previous work: Sá BarretoZworski '97, BonyHäfner '07,
DafermosRodnianski '07, MelroseSá BarretoVasy '08,
WunschZworski '10, Vasy '10. . . + numerous results for
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
Λ=0
June 17, 2013
5 / 20
Motivation
How about (stationary) perturbations of Kerrde Sitter?
Strategy for exact KdS:
Resonance
Quantization
expansion
condition
Symmetries
Perturbations destroy symmetries and our ability to handle individual
resonances (similarly to the compact case).
Strategy for perturbations:
Long-time behavior
of waves
Semyon Dyatlov (UC Berkeley)
Quantitative
Dynamical
information about
assumptions stable
resonances in bulk
under perturbations
Resonances for r -NH trapped sets
June 17, 2013
6 / 20
Motivation
How about (stationary) perturbations of Kerrde Sitter?
Strategy for exact KdS:
Resonance
Quantization
expansion
condition
Symmetries
Perturbations destroy symmetries and our ability to handle individual
resonances (similarly to the compact case).
Strategy for perturbations:
Long-time behavior
of waves
Semyon Dyatlov (UC Berkeley)
Quantitative
Dynamical
information about
assumptions stable
resonances in bulk
under perturbations
Resonances for r -NH trapped sets
June 17, 2013
6 / 20
Results
Setup
The structure of open ends/event horizons does not change our result
(Vasy '10 for event horizons of KdS)
=⇒
use a simpler model at innity:
(Mx , g ) ' Rn
e , g̃ ) = (Mx , g ) × Rt ,
(M
outside of a compact set
Dene resonances as poles of the meromorphic continuation of the
resolvent of the Laplacian
R
∆ ≥ 0:
(
(λ) = (∆ − λ2 )−1 :
Long-living resonances: Re λ
the geodesic ow e
2
(M ) → L2 (M ),
2
2
Lcomp (M ) → L
loc (M ),
Im λ
> 0;
Im λ ≤ 0.
L
→ ∞, | Im λ| ≤ C .
tH , p (x , ξ) = |ξ|g :
Depend on trapping for
p
Γ± = {ρ ∈ T ∗ M | e tH (ρ) 6→ ∞
p
Semyon Dyatlov (UC Berkeley)
as t
→ ∓∞},
Resonances for r -NH trapped sets
K
= Γ+ ∩ Γ− .
June 17, 2013
7 / 20
Results
Dynamical assumptions
Γ±
K
are C
r
hypersurfaces (r
= Γ+ ∩| Γ−
1)
is symplectic
νmin ≤ νmax are the expansion rates
tH along Γ± transversally to K
e
of
p
µmax ≥ 0
rate of e
is the maximal expansion
tH
r -normal
pinching:
p
along K
hyperbolicity:
νmin > r · µmax
Normally hyperbolic trapping,
with Γ+ , Γ− , and K .
νmin > 12 νmax
Stable under small smooth perturbations: HirschPughShub '77
True for slowly rotating KdS (for SdS, K is the photonsphere)
More general normally hyperbolic trapping (r
= 0)
appears for Ruelle
resonances for Anosov ows [FaureSjöstrand '11] and quantum
chemistry [GoussevSchubertWaalkensWiggins '10]
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
8 / 20
Results
Dynamical assumptions
Γ±
K
are C
r
hypersurfaces (r
= Γ+ ∩| Γ−
1)
is symplectic
νmin ≤ νmax are the expansion rates
tH along Γ± transversally to K
e
of
p
µmax ≥ 0
rate of e
is the maximal expansion
tH
r -normal
pinching:
p
along K
hyperbolicity:
νmin > r · µmax
Normally hyperbolic trapping,
with Γ+ , Γ− , and K .
νmin > 12 νmax
Stable under small smooth perturbations: HirschPughShub '77
True for slowly rotating KdS (for SdS, K is the photonsphere)
More general normally hyperbolic trapping (r
= 0)
appears for Ruelle
resonances for Anosov ows [FaureSjöstrand '11] and quantum
chemistry [GoussevSchubertWaalkensWiggins '10]
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
8 / 20
Results
Main theorem [D '13]
For each xed
ε > 0,
there exist two resonance free strips
{Re λ 1, Im λ ∈ [−(νmin − ε), − 12 (νmax + ε)] ∪ [− 21 (νmin − ε), ∞)}
and in these strips,
kχR (λ)χkL2 →L2 = O(1)
for
χ ∈ C0∞ (M ).
Moreover, resonances between the two strips satisfy a Weyl law as h
#{0 ≤ Re λ ≤ h−1 , Im λ > − 12 (νmax + ε)} =
→ 0:
∩ B ∗M )
+ o (h1−n ).
(2π h)n−1
Volσ (K
h−1
λ
−
νmin −ε
2
+ε
− νmax
2
−(νmin −ε)
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
9 / 20
Results
What about exact Kerrde Sitter?
M
a
>0
mass of the black hole
speed of rotation, |a|
Λ > 0 cosmological
2
9 ΛM < 1
<M
constant,
The metric is r -normally hyperbolic
Λ1
µmax = 0.
for
or |a|
1: νmin > 0,
However, the pinching condition
ε 1:
νmin > 12 νmax
breaks down for a
there exist two closed trajectories (equators) E±
local expansion rate
ν∼
ν satises
√
3
3
28M
Semyon Dyatlov (UC Berkeley)
as
⊂K
= M − ε,
such that the
ε→0
on E+ ,
ν∼
p
ε/2M
M
Resonances for r -NH trapped sets
on E− .
June 17, 2013
10 / 20
Results
What about exact Kerrde Sitter?
0.25
a
>0
mass of the black hole
speed of rotation, |a|
Λ > 0 cosmological
2
9 ΛM < 1
<M
Λ1
µmax = 0.
or |a|
constant,
1
2 νmax
0.1
0.05
1: νmin > 0,
0
0.2
0.4
0.6
0.8
1
a/M
However, the pinching condition
ε 1:
νmin
0.15
The metric is r -normally hyperbolic
for
νmax
0.2
Mν
M
νmin > 12 νmax
breaks down for a
there exist two closed trajectories (equators) E±
local expansion rate
ν∼
ν satises
√
3
3
28M
Semyon Dyatlov (UC Berkeley)
as
⊂K
= M − ε,
such that the
ε→0
on E+ ,
ν∼
p
ε/2M
M
Resonances for r -NH trapped sets
on E− .
June 17, 2013
10 / 20
Results
Previous work
Resonant free strips for NH trapped sets: GérardSjöstrand '88,
Dolgopyat '98, Liverani '04, Tsujii '10, WunschZworski '11,
NonnenmacherZworski '13. Our gap
{Im λ > − 21 (νmin − ε)}
coincides with the one of NZ '13 (valid in more general situations)
Our Weyl law saturates general upper bounds in strips: Sjöstrand '90,
GuillopéLinZworski '04, SjöstrandZworski '07, Nonnenmacher
SjöstrandZworski '11,'12, DatchevD '12, DatchevDZworski '12.
Completely integrable cases: GérardSjöstrand '87, Christianson '07,
Sá BarretoZworski '97, D '11
Bands of resonances/Weyl laws: Zworski '87, SjöstrandVodev '97,
SjöstrandZworski '99, Sjöstrand '00,'11, FaureTsujii '12,'13
FT '13 shares some ideas with presented work. It applies in settings with
lower regularity, but does not recover microlocal structure away from K .
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
11 / 20
Proofs
Ingredients of the proof
Fourier integral operator
Π
projecting microlocally onto resonant
states for the band of resonances
Left/right ideals of pseudodierential operators annihilating
Π
Local reduction to Taylor expansion
Positive commutator estimates (replaced in this talk by wave
propagation estimates)
Grushin problems: resonances as zeros of a Fredholm determinant
Trace formulas above and below the band of resonances, using
microlocal analysis in
λ
Complex analysis (almost analytic continuation and argument
principle) to obtain the Weyl law
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
12 / 20
Proofs
Model case
M
= R,
= xDx +
P0
1
2i
( , ξ) = x ξ,
p x
,
( ) (x ) = e −t /2 f (e −t x )
U0 t f
νmin = νmax = 1
The Taylor expansion
f
(e −t x ) = f (0) + O(e −t )L2
loc
can be viewed as a resonance expansion:
( ) = U0 (t )Π0 f + O(e −3t /2 ),
U0 t f
Π0 f (x ) := f (0),
( )Π0 f = e −t /2 Π0 f
U0 t
A resonance at
λ = − 2i
and
Π0 = δ0 ⊗ 1
is the projector onto the
corresponding resonant state.
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
13 / 20
Proofs
General case
√
P
=
∆,
( ) = e −itP
U t
Projector:
Π : C0∞ (M ) → C ∞ (M ),
Resonance expansion: for f
χ∈
∞ (M ),
C
0
χ=1
near K ,
Fourier integral operator
∈ C0∞ (M ) living at frequencies ∼ h−1
∞
modulo O(h ) errors,
and
kχU (t )(1 − Π)f kL2 = O(h−1 e −(νmin −ε)t )kf kL2 ,
e
+ε
t
− νmax
2
kχΠf kL2 . kχU (t )Πf kL2 . e −
νmin −ε
2
t kχΠf k
L2
This is enough for two resonance free strips
+ u
= Πu + O(h∞ )
for resonant states in the band.
Also true for Kerr black holes, using VasyZworski '00 (note: high
frequency régime!)
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
14 / 20
Proofs
The projector Π
Construct unique h -Fourier integral
Γ−
Π, associated to a canonical
Λ ⊂ T ∗ M × T ∗ M , such that:
operator
relation
ρ−
K
ρ+
Γ+
2
∞
Π = Π + O(h ),
∞
[∆, Π] = O(h ),
microlocally near K
V± ⊂ T Γ±
π± : Γ± → K
If
(1)
(2)
∩ S ∗M .
The canonical relation Λ
(ow lines of V± dashed).
∗
are the symplectic complements of T Γ± in T (T M ) and
are the projections along the ow lines of
V± ,
then
Λ = {(ρ− , ρ+ ) | ρ± ∈ Γ± , π− (ρ− ) = π+ (ρ+ )}.
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
15 / 20
Proofs
Transport equations
To construct
Π,
we need to solve transport equations of the form
p =f,
H a
where f
a
∈ C ∞ (Γ± ), a|K = 0
∈ C ∞ (Γ± ), f |K = 0.
The unique solution is given by the integral
Z
a
=±
∞
f
◦ e ∓tH
p
dt
.
0
The integral converges:
|f ◦ e ∓tH | = O(e −(νmin −ε)t ),
p
but is a smooth?
Dierentiate along K :
∂k a = ±
Z
∞
∂ k f ◦ e ∓tH · ∂ k e ∓tH
p
p
dt
.
0
Therefore, r -normal hyperbolicity guarantees that a
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
∈ Cr.
June 17, 2013
16 / 20
Proofs
Transport equations
To construct
Π,
we need to solve transport equations of the form
p =f,
H a
where f
a
∈ C ∞ (Γ± ), a|K = 0
∈ C ∞ (Γ± ), f |K = 0.
The unique solution is given by the integral
Z
a
=±
∞
f
◦ e ∓tH
p
dt
.
0
The integral converges:
|f ◦ e ∓tH | = O(e −(νmin −ε)t ),
p
but is a smooth?
Dierentiate along K :
∂k a = ±
Z
∞
∂ k f ◦ e ∓tH · ∂ k e ∓tH
p
p
dt
.
0
Therefore, r -normal hyperbolicity guarantees that a
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
∈ Cr.
June 17, 2013
16 / 20
Proofs
√
P
=
∆,
( ) = e −itP ,
U t
( ) (x ) = e −t /2 f (e −t x ),
U0 t f
Resonance expansion: for f
χ∈
∞ (M ),
C
0
χ=1
near K ,
Π ∈ I (Λ);
Π0 f (x ) = f (0).
∈ C0∞ (M ) living at frequencies ∼ h−1
∞
modulo O(h ) errors,
kχU (t )(1 − Π)f kL2 = O(h−1 e −(νmin −ε)t )kf kL2 ,
e
−
νmax +ε
2
t kχΠf k
Semyon Dyatlov (UC Berkeley)
−
L2 . kχU (t )Πf kL2 . e
Resonances for r -NH trapped sets
νmin −ε
2
t kχΠf k 2 .
L
June 17, 2013
and
(3)
(4)
17 / 20
Proofs
To prove (3) for the model case, write
(1 − Π0 )f = xf1 ,
( )=
f1 x
f
(x ) − f (0)
x
χU0 (t )(1 − Π0 )f = χU0 (t )xU0 (−t )U0 (t )f1 ,
To prove (4) for the model case, note that hD
ha(x )Πf , Πf i
and we
depends only on
R
( ) dx .
a x
kf1 kL2 . kf kH 1 ;
,
( )
U0 t xU0
x Πf = 0
(−t ) = e −t x .
and thus
Then
kχU0 (t )Πf k2L2 = hχ2t (x )Πf , Πf i, χt = U0 (−t )χU0 (t ),
R 2
R
t
use that χt (x ) = χ(e x ) =⇒
χt dx = e −t χ2 dx .
The general case is handled using the same ideas, but replacing x and hD
by pseudodierential operators
Θ−
and
ΠΘ− = O(h∞ ),
Such
Θ±
Θ+
x
solving
Θ+ Π = O(h∞ ).
(5)
are not unique, but the sets of solutions to the equations form
one-sided ideals invariant under the propagator U (t ).
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
18 / 20
Proofs
To prove (3) for the model case, write
(1 − Π0 )f = xf1 ,
( )=
f1 x
f
(x ) − f (0)
x
χU0 (t )(1 − Π0 )f = χU0 (t )xU0 (−t )U0 (t )f1 ,
To prove (4) for the model case, note that hD
ha(x )Πf , Πf i
and we
depends only on
R
( ) dx .
a x
kf1 kL2 . kf kH 1 ;
,
( )
U0 t xU0
x Πf = 0
(−t ) = e −t x .
and thus
Then
kχU0 (t )Πf k2L2 = hχ2t (x )Πf , Πf i, χt = U0 (−t )χU0 (t ),
R 2
R
t
use that χt (x ) = χ(e x ) =⇒
χt dx = e −t χ2 dx .
The general case is handled using the same ideas, but replacing x and hD
by pseudodierential operators
Θ−
and
ΠΘ− = O(h∞ ),
Such
Θ±
Θ+
x
solving
Θ+ Π = O(h∞ ).
(5)
are not unique, but the sets of solutions to the equations form
one-sided ideals invariant under the propagator U (t ).
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
18 / 20
Proofs
To prove (3) for the model case, write
(1 − Π0 )f = xf1 ,
( )=
f1 x
f
(x ) − f (0)
x
χU0 (t )(1 − Π0 )f = χU0 (t )xU0 (−t )U0 (t )f1 ,
To prove (4) for the model case, note that hD
ha(x )Πf , Πf i
and we
depends only on
R
( ) dx .
a x
kf1 kL2 . kf kH 1 ;
,
( )
U0 t xU0
x Πf = 0
(−t ) = e −t x .
and thus
Then
kχU0 (t )Πf k2L2 = hχ2t (x )Πf , Πf i, χt = U0 (−t )χU0 (t ),
R 2
R
t
use that χt (x ) = χ(e x ) =⇒
χt dx = e −t χ2 dx .
The general case is handled using the same ideas, but replacing x and hD
by pseudodierential operators
Θ−
and
ΠΘ− = O(h∞ ),
Such
Θ±
Θ+
x
solving
Θ+ Π = O(h∞ ).
(5)
are not unique, but the sets of solutions to the equations form
one-sided ideals invariant under the propagator U (t ).
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
18 / 20
Open problems
Open problems
Show existence of further gaps and bands, below the rst band of
resonances. The main question is how to construct Fourier integral
Π1 , Π2 , . . .
operators
projecting onto resonant states for each band.
Obtain a quantization condition if the ow is completely integrable on
the trapped set, recovering GérardSjöstrand '87,
Sá BarretoZworski '97, and D '11.
Get an optimal remainder
and
O(h
O(h2−n )
in the Weyl law (we prove o (h
−n )
1
−n− ) should not be much harder), perhaps using the work of
2
Sjöstrand '00 on the damped wave equation.
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
19 / 20
Thank you for your attention!
Semyon Dyatlov (UC Berkeley)
Resonances for r -NH trapped sets
June 17, 2013
20 / 20