Some results on time-dependent scattering theory without positive - - PowerPoint PPT Presentation

Some results on time-dependent scattering theory without positive conserved energy

Dietrich H¨ afner joint work with V. Georgescu, C. G´ erard Institut Fourier, Universit´ e de Grenoble 1 Spectral and scattering theories in Quantum Field Theory Porquerolles, June 2014

1 Introduction

1.1 The Klein-Gordon equation coupled to an electric field

1.1.1 The equation We consider on Rd the Klein-Gordon equation minimally coupled to an electric field. (∂t − iv(x))2φ(t, x) − ∆xφ(t, x) + m2φ(t, x) = 0. v ∈ C∞

0 (Rd) is the electric potential and m the mass of the Klein-Gordon

• field. Conserved energy :
• Rd |∂tφ(t, x)|2 + |∇xφ(t, x)|2 + (m2 − v 2(x))|φ(t, x)|2dx.

f(t) =

• φ(t)

i−1∂tφ(t)

• , f(t) = eitHf(0), H =
• 1

l −∆x + m2 − v 2 2v

• .

Energy : E(f, f) =

• Rd |f1|2(x) + ((−∆x + m2 − v 2(x))f0(x))f 0(x)dx, f =

f0 f1

• .

Pb: −∆x + m2 − v 2 might acquire negative spectrum.

1.1.2 Results on the Klein-Gordon equation

Energy space : E = H1(Rd) ⊕ L2(Rd). Results : σess(H) =] − ∞, −m] ∪ [m, +∞[ σ(H)\R = ∪1≤j≤n{λj, λj}, where λj, λj are eigenvalues of finite Riesz index.

   

For s > 1/2 : sup

Rez∈I,0<|ℑz|≤δ

x−s(H − z)−1x−sB(E) < ∞, where I ⊂ R is a compact interval disjoint from ±m, containing no real eigenvalues of H, nor so called critical points of H. Mourre estimate ? Pb : H is not a selfadjoint operator on a Hilbert space ! → Mourre theory on Krein spaces.

1.2 The wave equation on the De Sitter Kerr metric

1.2.1 De Sitter Kerr metric in Boyer-Lindquist coordinates MBH = Rt × Rr × S2

ω, with spacetime metric

g = ∆r − a2 sin2 θ∆θ λ2ρ2 dt2 + 2a sin2 θ((r 2 + a2)2∆θ − a2 sin2 θ∆r) λ2ρ2 dtdϕ − ρ2 ∆r dr 2 − ρ2 ∆θ dθ2 − sin2 θσ2 λ2ρ2 dϕ2, ρ2 = r 2 + a2 cos2 θ, ∆r =

• 1 − Λ

3 r 2

• (r 2 + a2) − 2Mr,

∆θ = 1 + 1 3Λa2 cos2 θ, σ2 = (r 2 + a2)2∆θ − a2∆r sin2 θ, λ = 1 + 1 3Λa2. Λ > 0: cosmological constant, M > 0 : masse, a : angular momentum per unit masse.

◮ ρ2 = 0 is a curvature singularity, ∆r = 0 are coordinate singularities.

∆r > 0 on some open interval r− < r < r+. r = r−: black hole horizon, r = r+ cosmological horizon.

◮ ∂ϕ and ∂t are Killing. There exist r1(θ), r2(θ) s. t. ∂t is

◮ timelike on {(t, r, θ, ϕ) : r1(θ) < r < r2(θ)}, ◮ spacelike on

{(t, r, θ, ϕ) : r− < r < r1(θ)}∪{(t, r, θ, ϕ : r2(θ) < r < r+} =: E−∪E+.

The regions E−, E+ are called ergospheres.

1.2.2 The wave equation on the De Sitter Kerr metric

We now consider the unitary transform U : L2(M;

σ2 ∆r ∆θ drdω)

→ L2(M; drdω) ψ →

σ

√

∆r ∆θ ψ

If ψ fulfills (✷g + m2)ψ = 0, then u = Uψ fulfills (∂2

t − 2ik∂t + h)u

= 0. with (1) k = a(∆r − (r 2 + a2)∆θ) σ2 Dϕ, h = −(∆r − a2 sin2 θ∆θ) sin2 θσ2 ∂2

ϕ −

√∆r∆θ λσ ∂r∆r∂r √∆r∆θ λσ − √∆r∆θ λ sin θσ ∂θ sin θ∆θ∂θ √∆r∆θ λσ + ρ2∆r∆θ λ2σ2 m2. h is not positive inside the ergospheres. This entails that the natural conserved quantity ˜ E(u) = ∂tu2 + (hu|u) is not positive. Pb : k has two different limits for r → r±. No realization as selfadjoint operator on a Krein space possible.

1.3 References

Previous work :

◮ Spectral analysis on Krein spaces : Langer, Najman, Tretter, Jonas. ◮ Scattering theory for the Klein-Gordon equation coupled to an

electric field : Kako (short range), C. G´ erard (long range).

◮ different limits for k, dimension 1 : Bachelot. ◮ Scattering theory on the Kerr metric for non superradiant situations :

H, H-Nicolas.

◮ Decay of the local energy for the wave equation on the (De Sitter)

Kerr metric : Andersson-Blue, Dafermos-Rodnianski, Dyatlov, Finster-Kamran-Smoller-Yau,Tataru-Tohaneanu, Vasy,... Papers these lectures are based on : [1] Vladimir Georgescu, Christian G´ erard, Dietrich H¨ afner,Boundary values of resolvents of self-adjoint operators in Krein spaces, J. Funct.

• Anal. 265 (2013), no. 12, 3245-3304.

[2] Vladimir Georgescu, Christian G´ erard, Dietrich H¨ afner, Resolvent and propagation estimates for Klein-Gordon equations with non-positive energy, 47pp, arXiv:1303.4610. [3] Vladimir Georgescu, Christian G´ erard, Dietrich H¨ afner, Asymptotic completeness for superradiant Klein-Gordon equations and applications to the De Sitter Kerr metric, 62pp, arXiv1405.5304.

Part 1 : Boundary values of resolvents of self-adjoint

• perators in Krein spaces and applications to

Klein-Gordon equations

Plan of part 1

Krein spaces Basic definitions Operators on Krein spaces Functional calculus Smooth and Borel functional calculus on Banach spaces C0− groups Mγ functional calculus Boundary value estimates Main theorem Virial theorem An important proposition Proof of the main theorem Definitizable operators on Krein spaces Definitizable operators Pontryagin spaces Abstract Klein Gordon equation Energy spaces Klein-Gordon operators Limiting absorption principle Example : Klein-Gordon equation on scattering manifolds

Krein spaces

Definition

A Krein space is a hilbertizable vector space K equipped with a bounded hermitian sesquilinear form ·|· such that for any continuous linear form ϕ on K there is a unique u ∈ K such that ϕ = u|·. The form ·|· is called the Krein structure. Let J : K → K∗ be the linear continuous map defined by Ju = ·|u, so that u|v = u, Jv (., . is the anti-duality bracket). J is bijective. Thus the Krein structure ·|· allows us to identify K∗ and K with the help of J. We say that a linear subspace H is a Hilbert subspace of K if

• H, ·|·|H×H
• is a Hilbert space.

Proposition

A Krein space is a reflexive Banach space.

Operators on Krein spaces

• 1. Adjoints on Krein spaces

◮ If T ∈ B(K), T ∗ ∈ B(K∗) defined in the Banach space sense. ◮ Transport it on K with the help of J. ◮ Involution T → T ∗ on B(K) such that T ∗u|v = u|Tv. ◮ Definition extends to closed densely defined operators. ◮ An operator S is selfadjoint if S∗ = S. ◮ An operator S is positive if u|Su ≥ 0 for all u ∈ D(S).

• 2. Projections on Krein spaces

◮ A projection on K is an element Π ∈ B(K) such that Π2 = Π,

• rthogonal if self-adjoint.

◮ A positive projection is a projection Π such that Π ≥ 0.

Proposition (Bognar)

The range of a positive projection is a Hilbert subspace of K. Reciprocally, if H is a Hilbert subspace of K then there is a unique self-adjoint projection Π such that ΠK = H and this projection is positive.

Smooth and Borel functional calculus on Banach spaces

Let K be a Banach space, H be a closed densely defined operator on K and R(z) its resolvent.

Definition

Let β(H) be the set of λ ∈ R such that there is a real open neighborhood I of λ and there are numbers ν > 0, n ∈ N, C > 0 such that R(z) ≤ C|Imz|1−n if Rez ∈ I, 0 < |Imz| ≤ ν. If I ⊂ β(H) is an open interval and χ ∈ C∞

0 (I), then we can define χ(H)

be the Helffer-Sj¨

• strand formula.

χ(H) = − 1 2πi

• C

R(z)∂ ˜ χ(z)dz ∧ dz. We shall say that the smooth functional calculus extends to a C0− functional calculus on I if χ(H) ≤ C supλ∈I |χ(λ)| for all χ ∈ C∞

c (I).

Unique continuous extension to an algebra morphism C0(I) → B(K).

Theorem

Assume that K is a reflexive Banach space and let F0 : C0(I) → B(K) be a norm continuous algebra morphism. Then there is a unique algebra morphism F : B(I) → B(K) which extends F0 and such that: b-limn ϕn = ϕ ⇒ F(ϕn) → F(ϕ) weakly.

C0− groups

◮ Let Wt = eitA be a C0-group on a Banach space K.

◮ There are numbers M ≥ 1 and γ ≥ 0 such that

Wt ≤ Meγ|t| for all t ∈ R.

◮ The spectrum of the operator A is included in the strip |Imz| ≤ γ and it

could be equal to this strip.

◮ S ∈ B(K) is of class Cα(A) if the map

R ∋ t → S(t) = e−itASeitA ∈ B(K) is Cα for the strong operator topology. For an unbounded operator S we say that S ∈ Cα(A) if R(z0) ∈ Cα(A) for some z0 ∈ ρ(S).

◮ If K is a Krein space we say that the Krein structure is of class

C1(A) if the conditions in the next proposition are verified.

Proposition

The following assertions are equivalent: the function t → Wtu|Wtu is derivable at zero for each u ∈ H; the function t → Wtu|Wtu is of class C1 for each u ∈ H; the map t → W ∗

t Wt is locally Lipschitz;

A∗ = A + B where B is a bounded operator.

Mγ functional calculus

◮ Mγ : set of functions f : R → C whose Fourier transforms are

complex measures such that: fMγ :=

• eγ|t||

f(t)|dt < ∞.

◮ Mγ is a unital Banach ∗-algebra for the usual operations of addition

and multiplication and f ∗(τ) = f(−τ) as involution.

◮ If

Wt ≤ Meγ|t|, t ∈ R, then it follows that we can define f(A) for f ∈ Mγ by the formula f(A) =

• Wt

f(t)dt.

◮ Mγ ∋ f → f(A) ∈ B(H) is a linear multiplicative map such that

f(A) ≤ MγfMγ .

◮ We have for σ > 0, .−σ ∈ Mγ if γ < 1/2.

Boundary value estimates

Theorem

Let K be a Krein space and A the generator of a C0-group of operators

• n K such that the Krein structure is of class C1(A). Let H be a

self-adjoint operator on K and Π a positive projection which commutes with H such that the following conditions are satisfied:

◮ H is of class Cα(A) for some α > 3/2, in particular H′ = [H, iA] is

well defined;

◮ there is ϕ ∈ C∞ c (β(H)) real with ϕ(λ) = 1 on a neighborhood of a

compact interval J such that ϕ(H)Π = ϕ(H) and ϕ(H)(ReH′)ϕ(H) ≥ aϕ(H)2, a > 0. Then if s > 1/2 and ε > 0 is small enough, we have supz∈J±i]0,ν]εA−sR(z)εA−s < ∞, for some ν > 0.

Remark

◮ Generalization of the Mourre theorem. ◮ The condition α > 3/2 is the condition which comes out naturally in

the proof, but it is certainly not optimal.

Virial Theorem

We assume that H admits a Borel functional calculus on I, that λ ∈ I and that H ∈ C1(A).

Lemma (Virial Theorem)

1λ(H)[iH, A]1λ(H) = 0.

Corollary

Assume that for some J ⊂ I we have 1 lJ(H) ≥ 0 and that there is a number a > 0 and a compact operator K such that 1 lJ(H)(ReH′)1 lJ(H) ≥ a1 lJ(H) + K. Then the point spectrum of H in J is finite and consists of eigenvalues of finite multiplicity. Moreover, if λ ∈ J is not an eigenvalue of H and b < a then there is a compact neighborhood J of λ in J such that 1 l

J(H)(ReH′)1

l

J(H) ≥ b1

l

J(H).

An important proposition

Proposition

Let H be a self-adjoint operator with σ(H) = C on the Krein space K. Let Π be a positive projection which commutes with H and let B, C, D be bounded operators such that (1) B = B∗, C = ΠC, (2) BC = CD, (3) CC∗ ≤ Π[H, iB]Π as quadratic forms on D(H). Then the operator L(z) = C∗R(z)C satisfies L(z)u|L(z)u ≤ c(B + D)L(z)uu for u ∈ K, z ∈ σ(H), where c depends only on K and Π. Uses ideas of Putnam, earlier paper of C. G´ erard.

• Proof. Let Imz ≥ 0.

R∗[H, iB]R = R∗[H − z, i(B + b)]R = i(B + b)R − R∗i(B + b) + (2Imz)R∗(B + b)R = 2Im

• R∗(B + b)
• + (2Imz)R∗(B + b)R.

Since (B + b)C = C(D + b) we get C∗R∗[H, iB]RC = 2Im

• C∗R∗C(D + b)
• + (2Imz)C∗R∗(B + b)RC. (2)

Since C = ΠC and Π commutes with H we have C∗R∗(B + b)RC = C∗R∗Π(B + b)ΠRC. Using ±Πu|SΠu ≤ SΠKΠu|Πu we may choose b = −BΠK such that (2Imz)C∗R∗(B + b)RC ≤ 0, hence from (2) we get: C∗R∗[H, iB]RC ≤ 2Im

• L∗(D + b)
• .

Now observe that C∗R∗[H, iB]RC = C∗R∗Π[H, iB]ΠRC hence from hypothesis (3), we get L∗L = C∗R∗CC∗RC ≤ 2Im

• L∗(D + b)
• .

Now for u ∈ K, with a constant m depending only on K: Lu|Lu ≤ 2ImLu|(D + b)u ≤ mLu(D+b)u ≤ mLu(D+BΠK)u using that b = −BΠK. Since BΠK ≤ dB, for some constant d depending only on Π, this gives the required estimate for c = max(m, md).

Proof of the main theorem

Let I be an open neighborhood of J on which ϕ(λ) = 1. We notice that it suffices to show sup

z / ∈R

εA−sR(z)ξ(H)2εA−s < ∞ for each real ξ ∈ C∞

c (I). In the following ξ = ξ(H).

g(τ) = τ−s, f such that f ′ = g2, and gε = g(εA), fε = f(εA). Fix φ ∈ C∞

c (R) real such that φ(λ) = λ on a neighborhood of the support of

ϕ and set S = φ(H). Fε := ε−1Refε. Then [S, iξFεξ] ∼ gεξ(ReH′)ξg∗

ε ≥ a

2ξgεg∗

εξ

for ε small enough. We now apply the preceding Proposition with B = ξFεξ, C = ξgε and D = g−1

ε Fεξ2gε. For Lǫ = g∗ ǫ ξ2Rgǫ we obtain:

Lεu|Lεu ≤ K(Bε+Dε)Lεuu ≤ δLεu2+(4δ)−1(Bε+Dε)2u2. Let η ∈ C∞

0 (I) such that ηξ = ξ. We have Πη = η, N−1v2 ≤ v|v for

v ∈ ΠK and (1 − η)Lǫ = [g∗

ε, η]ξ2Rgǫ = O(ǫ)Lǫ.

Definitizable operators I : basic definition

Definition

A self-adjoint operator H is definitizable if ρ(H) = ∅ and there exists a real polynomial p = 0 such that u|p(H)u ≥ 0, ∀ u ∈ DomHk, k := degp.

Remark

p(z) in can be replaced by (z0 ∈ ρ(H)) : q(z) = p(z) (z − z0)k(z − z0)k , k = deg p/2 if deg p even, k = (deg p + 1)/2 if deg p odd. If λ is an isolated point of σ(H) we define the Riesz spectral projection E(λ, H) := 1 2iπ

• C

(z − H)−1dz where C is a small curve in ρ(H) surrounding λ.

Definitizable operators II : non real spectrum

Proposition (Langer, Jonas)

Let H be a definitizable self-adjoint operator. Then:

◮ If z ∈ σ(H)\R then p(z) = 0 for each definitizing polynomial p. ◮ There is a definitizing polynomial p such that σ(H) \ R is exactly the

set of non-real zeroes of p.

◮ σ(H)\R is the union of pairs {λi, λi} of eigenvalues of finite Riesz

index. Set now EC

pp =

• λ∈σ(H), Imλ>0

E(λ, H) + E(λ, H), KC

p := EC ppK.

Then EC

pp is a projection, EC pp = (EC pp)∗, hence KC pp is a Krein space and

K = KC

pp ⊕ (KC pp)⊥ =: KC pp ⊕ K1.

Remark

When we discuss functional calculus for definitizable operators it is enough to suppose σ(H) ⊂ R.

Definitizable operators III: critical points

Definition

Let H, p as above. Set cp(H) := p−1({0}) ∩ σ(H) ∩ R.

◮ the set cfin(H) equal to the intersection of the cp(H) for all definitizing

polynomials for H is called the set of (finite) critical points of H.

◮ the set c(H) := cfin(H) ∪ {∞} considered as a subset of the

• ne-point compactification ˆ

C := C ∪ {∞} is called the set of critical points of H.

Definition

Let H be a definitizable operator on K. For λj ∈ cfin(H) we denote by kj the minimum over all definitizing polynomials p with p(λj) = 0, of the multiplicity of λj as a zero of p. For λ = ∞, we set κ = 0 if H is even definitizable and κ = 1 otherwise. We denote by C(H) the set C(H) = {(λj, kj)} ∪{ (∞, κ)},

• btained with these conventions.

Definition

Let k ∈ N and f : R → C.

◮ we say that f is of class Ck at λ ∈ R if there is a polynomial p with

deg p ≤ k such that: f(x) = p(x) + o((x − λ)k);

◮ we say that f is of class C0 at λ = ∞ if f is bounded in a

neighborhood of ±∞ in R.

◮ we say that f is of class C1 at λ = ∞ if there exists a constant f∞

such that f(x) = f∞ + o(x−1) near ±∞. For l ≤ k we denote by pl(x) the part of p of degree less or equal l − 1 so that f(x) − pl(x) ∈ 0((x − λ)l).

Definition

We denote by BC(H)(R) the ∗−algebra of bounded Borel functions f on R such that f is of class Ckj at each λj and of class Cκ at ∞. We equip BC(H)(R) with the norm: fC(H) := sup

x∈R

|f(x)| +

• (λj ,kj )∈C
• 0≤l≤kj

sup

x∈R

• f(x) − pl(x)

(x − λj)l

• + sup

|x|≥1

κ |x(f(x) − f∞)| .

Definitizable operators IV : Borel functional calculus

Let H be a definitizable operator with σ(H) ⊂ R and R be the set of bounded rational functions ϕ : R → C. We can easily define a rational functional calulus ϕ(H), ϕ ∈ R. For this functional calculus we obtain the estimate, see Jonas : ϕ(H) ≤ CϕC(H), ∀ϕ ∈ R. (3) It thus extends to BC(H)(R):

Theorem

Let H be a self-adjoint definitizable operator on the Krein space K with σ(H) ⊂ R.Then there is a unique linear continuous map ϕ → ϕ(H) from BC(H)(R) into B(K) with the two following properties

◮ if ϕ(λ) = (λ − z)−1 for some non-real z then ϕ(H) = (H − z)−1 ◮ if b−limn ϕn = ϕ for ϕn ∈ BC(H)(R) then ϕ(H) = w−lim ϕn(H).

This map is a morphism of unital ∗-algebras.

Corollary

If H is in addition even definitizable, then it is the generator of a unitary C0− group on K.

Lemma

Let I ⊂ R a bounded interval such that there exists a definitizing polynomial p with ±p(x) > 0 for x ∈ I. Then ±u|1 lI(H)u ≥ 0, u ∈ K. Let H be a definitizable operator. Let σc be the set of all complex eigenvalues of H. If λj ∈ σc is a complex eigenvalue of H we define kj as the order of the pole of (H − z)−1. We denote CC(H) the set CC(H) = {(λj, kj)}

• btained with these conventions.

Proposition

(H − z)−1

• (λj ,kj )∈CC

|z − λj|−kj + |Imz|−1  1 +

• (λj ,kj )∈C\{(∞,κ)}

|z − λj|−kj + |z|κ   for all z / ∈ σc ∪ R. To prove the proposition we apply the estimate (3) to ϕ(x) = (x − z)−1.

Pontryagin spaces

Let K be a Krein space.

◮ Fix a scalar product (·|·) on K endowing K with its hilbertizable

topology.

◮ Riesz theorem :

u|v = (u|Mv), u, v ∈ K, M bounded, invertible self-adjoint.

◮ Polar decomposition of M, M = J|M| where J = J∗, J2 = 1

l.

◮ Equivalent scalar product :

(u|v)M := (u||M|v), so that u|v = (u|Jv)M, u, v ∈ K.

Definition

A Krein space (K, ·|·) is a Pontryagin space if either 1 lR−(J) or 1 lR+(J) has finite rank.

Theorem

A self-adjoint operator H on a Pontryagin space is definitizable with an even definitizing polynomial p.

Abstract Klein-Gordon equation : non homogeneous energy space

We consider ∂2

t φ(t) − 2ik∂tφ(t) + hφ(t) = 0,

where φ : R → H, H is a (complex) Hilbert space, h selfadjoint, k : h− 1

2 H → H symmetric, bounded.

Conserved energy : ∂tφ2 + (hφ|φ). Non homogenous energy space E: E := h− 1

2 H ⊕ H.

Lemma

◮ If 0 ∈ ρ(h) then E equipped with the hermitian sesquilinear form:

(f|f)E := (f0|hf0) + (f1|f1) is a Krein space.

◮ if in addition Tr1

l]−∞,0](h) < ∞, then (E, (·|·)E) is Pontryagin.

Homogeneous energy space and charge spaces

Homogeneous energy space ˙ E ˙ E := |h|− 1

2 H ⊕ H.

E ⊂ ˙ E continuously and densely. E = ˙ E iff 0 ∈ ρ(h).

Lemma

Assume that Kerh = {0}. Then ˙ E equipped with (·|·)E is a Krein space. If in addition Tr1 l]−∞,0](h) < ∞, then ˙ E is Pontryagin.

Remark (Charge spaces)

If we put f(t) =

• φ(t)

i−1∂tφ(t) − kφ(t)

• ,

then the charge q(f, f) = (f1|f0) + (f0|f1) is conserved. If we put Kθ = h−θH ⊕ hθH, 0 ≤ θ ≤ 1/2, then (Kθ, q) is a Krein space. We have K1/4 = [E, E∗]1/2 and analogous results to the results presented here hold on K1/4.

Klein-Gordon operators on energy spaces

H := ˙ H := 1 l h 2k

• ,

D(H) := h−1H ⊕ h− 1

2 H,

D( ˙ H) :=

• |h|− 1

2 H ∩ |h|−1H

• ⊕ h− 1

2 H.

E ⊂ ˙ E and D(H) ⊂ D( ˙ H) continuously and densely. H may also be considered as an operator acting in ˙ E, ˙ H is its closure in ˙ E.

Theorem

◮ Assume that 0 ∈ ρ(h). Then H is a self-adjoint operator on the Krein

space (E, (·|·)E) with ρ(H) = ∅.

◮ If in addition Tr1

l]−∞,0](h) < ∞, then H is even-definitizable.

Theorem

◮ Assume that there exists z ∈ ρ(h), z = 0. Then ˙

H is self-adjoint on ( ˙ E, (·|·)E) with ρ(H) = ∅.

◮ if in addition Tr1

l]−∞,0](h) < ∞, then ˙ H is even-definitizable.

Limiting absorption principle I: Hypotheses

◮ Energy (E):

(E1) Kerh = {0}, (E2) Tr1 l]−∞,0](h) < ∞, (E3) k|h|−1/2 ∈ B(H).

◮ Asymptotics (A): h = b2 − r

with (A1) b ≥ 0, self-adjoint on H, b2 ∼ |h|, (A2) r symmetric on h− 1

2 H,

(A3) kb−1, b−1rb−1 ∈ B∞(H).

◮ Conjugate operator (M): Let a be a selfadjoint operator on H such

that (M1) b2 ∈ C2(a), aχ = χ(b2)aχ(b2), (M2) kb−1, b−1rb−1 ∈ C2(aχ; H), b−1rb−1 ∈ C1(aχ; H), (M3) (i) aχx−1 ∈ B(H), ∀ χ ∈ C∞

c (R),

(ii) [b, x−δ]xδ ∈ B(H), 0 ≤ δ ≤ 1. Let τ(b2) be the set of thresholds for (b2, a): if λ∈ τ(b2) there exists an interval I ⊂ R, with λ ∈ I, a constant c0 > 0 and R ∈ B∞(H) such that 1 lI(b2)[b2, ia]1 lI(b2) ≥ c01 lI(b2) + R., τ(b) :=

• τ(b2).

Limiting absorption principle II : Result and consequences

Theorem

Assume (E), (A), (M). Let I ⊂ R± a compact interval such that i) I ∩ ±τ(b) = ∅, ii) I ∩ cp( ˙ H) = ∅, iii) 0 ∈ I. We also suppose χ ∈ C∞

0 (I). Then there exists ǫ0 > 0 s. t. for 1 2 < δ ≤ 1:

sup

Rez∈I, 0<|Imz|≤ǫ0

(x−δ)diag(H − z)−1(x−δ)diagB(E) < ∞,

• R

x−δeitHχ(H)x−δϕ2

E ϕ2.

Remark

We define the mass m2 = inf(σ(h) ∩ R+), m ≥ 0. In the massless case (m = 0), H admits a Borel functional calculus although (E, (.|.)E) is not a Krein space.

Scattering manifolds

N smooth, d − 1 dimensional compact manifold. M ≃ M0 ∪ ]1, +∞[s×Nω, M0 ⋐ M is relatively compact. Sm(M) : real valued functions f ∈ C∞(M) such that ∀ k ∈ N, α ∈ Nd−1, |∂k

s ∂α ωf(s, ω)| ≤ Ck,αsm−k

for (s, ω) ∈]1, ∞[×N.

Definition

A Riemannian metric g0 on M is called conic if there exists R > 0 and a Riemannian metric h on N such that g0 = ds2 + s2hjk(ω)dωjdωk for (s, ω) ∈ [R, ∞[×N. A Riemannian metric g on M is called a scattering metric if g = g0 + m, where g0 is a conic metric and m is of the form m = m0(s, ω)ds2 + sm1

j (s, ω)(dsdωj + dωjds) + s2m2 jk(s, ω)dωjdωk

with ml ∈ S−µl (M) for l = 0, 1, 2, µl > 0.

Klein-Gordon equation on scattering manifolds I

Klein-Gordon equation: (∂t − iv)2φ − (∇k − iAk)(∇k − iAk)φ + m2φ = 0. v: electric potential, Ak(s, ω)dxk: magnetic potential, m : mass. Local coordinates and unitary transformation (g = detg): (∂t − iv)2ψ − g−1/4(∂j − iAj)g1/2gjk(∂k − iAk)g−1/4ψ + m2(s, ω)ψ = 0, H = L2(M; dsdω), p = −g−1/4∂jg1/2gjk∂kg−1/4. Assumptions: Aj(s, ω), m(s, ω) − m∞ ∈ S−µ0(M), µ0 > 0, m∞ := lims→∞ m(s, ω) ≥ 0. (4) We assume that v = v(s, ω) is a multiplication operator and v(s, ω) = vl(s, ω) + vs(s, ω), vl(s, ω) ∈ S−µ0(M), vs(s, ω)p−1/2 ∈ B∞(H), s2vs(s, ω)p−1/2 ∈ B(H). (5) a = 1 2(η(s)sDs + Dssη(s)), where η ∈ C∞(R, R+)with η(s) = 1 for s ≥ 2 and η(s) = 0 for s ≤ 1.

Klein-Gordon equation on scattering manifolds II : results

Proposition (Massive case)

Assume (4), (5), m∞ > 0 and Ker h = {0}. Then

◮

σess(H) = σess( ˙ H) =] − ∞, −m∞] ∪ [m∞, +∞[.

◮ conditions (E), (A), (M) are satisfied; ◮ one has τ(b) = {m∞}.

For the massless case we require instead of (5) v(s, ω) = vl(s, ω) + vs(s, ω), ∃ R0 > 1, 0 ≤ δ < 1 such that |vl(s, ω)| ≤ δ d−2

2 s−1, for s ≥ R0,

svsp−1/2 ∈ B∞(H), s3vsp−1/2 ∈ B(H). (6) (6) permits to use Hardy’s inequality to deal with vl.

Proposition (Massless case)

Assume (4) with m∞ = 0, (6), Ker h = {0} and d ≥ 3. Then:

◮ σess(H) = σess( ˙

H) = R;

◮ conditions (E), (A), (M ) are satisfied; ◮ one has τ(b) = {0}.

Part 2 : Asymptotic completeness for superradiant Klein-Gordon equations and applications to the De Sitter Kerr metric.

Dietrich H¨ afner joint work with V. Georgescu, C. G´ erard Institut Fourier, Universit´ e de Grenoble 1 Spectral and scattering theories in Quantum Field Theory Porquerolles, June 2014

Plan of part 2

1 Abstract Klein-Gordon equation 2 Meromorphic extensions 3 Klein-Gordon operators with “two ends” 4 Propagation estimates 5 Uniform boundedness of the evolution 1 : Abstract setting 6 Asymptotic completeness 1 : Abstract setting 7 Geometric setting 8 Asymptotic completeness 2 : Geometric setting 9 The wave equation on the De Sitter Kerr metric 10 Asymptotic completeness 3 : The De Sitter Kerr case

1 Abstract Klein Gordon equation

1.1The abstract equation H Hilbert space. h, k selfadjoint, k ∈ B(H).    (∂2

t − 2ik∂t + h)u

= 0, u|t=0 = u0, ∂tu|t=0 = u1. (1) Hyperbolic equation (A1) h0 := h + k 2 ≥ 0. Formally u = eiztv solution if and only if p(z)v = 0 with p(z) = h0 − (k − z)2 = h + z(2k − z), z ∈ C. p(z) is called the quadratic pencil. Conserved quantities u|uℓ := u1 − ℓu02 + (p(ℓ)u0|u0), where p(ℓ) = h0 − (k − ℓ)2. Conserved by the evolution, but in general not positive definite, because none of the operators p(ℓ) is in general positive.

1.2 Spaces

Hi: scale of Sobolev spaces associated to h0. (A2)      0 / ∈ σpp(h0); k, h1/2 kh−1/2 ∈ B(H); ∀z ∈ C \ R, (k − z)−1B(h−1/2

H) |Imz|−K0, K0 > 0.

∃M0 > 0, ∀|z| ≥ M0kB(H), (k − z)−1B(h−1/2

H) 1 |z|−kB(H) .

Homogeneous energy spaces ˙ E = Φ(k)h−1/2 H ⊕ H, ˙ E∗ = Φ(k)H ⊕ h1/2 H, Φ(k) = 1 l k 1 l

• .

˙ E is equipped with the norm (u0, u1)2

˙ E = u1 − ku02 + (h0u0|u0). We

identify ˙ E∗ with the dual of ˙ E with the help of the charge .|. = (u0|v1 − kv0) + (u1 − ku0|v0).

Lemma

For all ℓ ∈ R, .|.ℓ is continuous with respect to the norm . ˙

E if and only

if h0 (k − ℓ)2 in the sense of quadratic forms on D(h0).

Remark

In the case of the De Sitter Kerr metric this condition is not fulfilled, in particular ( ˙ E, .|.ℓ) is not a Krein space in this case.

1.4 Energy Klein Gordon operators

ˆ H = 1 l h 2k

• , ρ(h, k) := {z ∈ C | p(z) : h0− 1

2 H ˜

→h0

1 2 H}.

Klein Gordon operator on the homogeneous energy space D( ˙ H) = Φ(k)((h−1/2 H ∩ h−1

0 H) ⊕ h0−1/2H),

∃C0 > 0, ρ( ˙ H) ∩ (C \ (−C0, C0)) = ρ(h, k) ∩ (C \ (−C0, C0)), ˙ R(z) := ( ˙ H − z)−1. Gauge transformations v = e−itℓu (∂t − ik)2u + h0u = 0 ⇔ (∂t − i(k − ℓ))2v + h0v = 0. Φ(ℓ)HΦ−1(ℓ) =: Hℓ + ℓ1 l, Hℓ =

• 1

l p(ℓ) 2(k − ℓ)

• , p(ℓ) = h0 − (k − ℓ)2.

1.5 Basic resolvent estimates and existence of the dynamics

Lemma (Basic resolvent estimates)

Let ǫ > 0. We have p−1(z)u

• |z|−1|Imz|−1u,

h1/2 p−1(z)u

• |Imz|−1u.

uniformly in |z| ≥ (1 + ǫ)kB(H), |Imz| > 0.

Remark

i) Interpretation : superradiance does not occur for |z| ≥ (1 + ǫ)k. ii) Explanation : p(z) = h0 − (k − z)2, h0 ≥ 0.

Lemma (Existence of the dynamics)

( ˙ H, D( ˙ H)) is the generator of a C0− group e−it ˙

H on ˙

E.

2 Meromorphic extensions

2.1 Background

Definition

Let H be a Hilbert space. Let U be a neighborhood of z0 ∈ C, and let F : U \ {z0} → B(H) be a holomorphic function. F is finite meromorphic at z0 if in the Laurent expansion F(z) = +∞

n=m(z − z0)nAn,

m > −∞, the operators Am, ..., A−1 are of finite rank for m < 0. If in addition A0 is a Fredholm operator, then F is called Fredholm at z0.

Proposition

Let D ⊂ C be a connected open set, let Z ⊂ D be a discrete and closed subset of D, and let F : D → B(H) be a holomorphic function on D \ Z. Assume that

◮ F is finite meromorphic and Fredholm at each point of D; ◮ there exists z0 ∈ D \ Z such that F(z0) is invertible.

Then there exists a discrete closed subset Z ′ of D such that Z ⊂ Z ′ and :

◮ F(z) is invertible for z ∈ D \ Z ′; ◮ F −1 : D \ Z ′ → B(H) is finite meromorphic on D and Fredholm at

each point of D.

2.2 Meromorphic extensions of weighted resolvents

Assumptions (A3) h ≥ 0, 0 / ∈ σpp(h), ∀u ∈ D(h1/2), ku h1/2u. (w, D(w)) selfadjoint. (ME1)            a) wkw ∈ B(H). b) [k, w] = 0. c) h−1/2[h, w−ǫ]wǫ/2, [h, w−ǫ]wǫ/2h−1/2, [h, w−ǫ]h−1/2 ∈ B(H), ∀ǫ > 0 d) ∀ǫ > 0, w−ǫu h1/2u, ∀u ∈ h−1/2H, e) w−ǫh−˜

ǫ ∈ B∞(H), ∀ǫ, ˜

ǫ > 0. (ME2) For all ǫ > 0, w−ǫ(h − z2)−1w−ǫ extends from Imz > 0 to Imz > −δǫ, δǫ > 0 as a finite meromorphic function with values in B∞(H).

Proposition

Assume (A1)-(A3), (ME1)-(ME2). Then w−ǫ ˙ R(z)w−ǫ extends finite meromorphically to Imz > −δǫ/2 as an operator valued function with values in B∞( ˙ E).

3 Klein-Gordon operators with “two ends”

3.1 Assumptions (x, D(x)) selfadjoint, σ(x) = σac(x) = R, [k, x] = 0. Let χi ∈ C∞

b (R), i = 1, 2,

suppχ1 ∩ suppχ2 = ∅. We suppose (TE1)    w = w(x), w ∈ C∞(R), [x, k] = 0, χ1(x)h0χ2(x) = 0.  



 



  

   k± = k ∓ ℓj2

∓,

h± = h0 − k 2

±

˜ h− = h− + 2ℓk− − ℓ2 = h0 − (ℓ − k−)2. (TE2) There exists ℓ ∈ R, ǫ > 0 such that (h+, k+), (˜ h−, k− − ℓ) satisfy (A3). p±(z) := h± + z(2k± − z). Note that ˜ h− = p−(ℓ).

3.2 Asymptotic Hamiltonians

˙ E+ = h−1/2

+

H ⊕ H, ˙ E− = Φ(ℓ)˜ h−1/2

−

H ⊕ H. ˙ H± =

• 1

l h± 2k±

• .

are selfadjoint. We note ˙ R±(z) := ( ˙ H± − z)−1. (TE3)                      a) wi+ki+w, wi−(k − ℓ)i−w ∈ B(H), b) [h, i±] = ˜ i[h, i±]˜ i, c) (h+, k+, w) and (˜ h−, k− − ℓ, w) fulfill (ME1), (ME2), d) h1/2

± i±h−1/2 ±

, h1/2 i±h−1/2 ∈ B(H), e) w[h, i±]wh−1/2

±

, w[h, i±]wh−1/2 , [h, i±]h−1/2

±

, [h, i±]h−1/2 , h−1/2 [w−1, h0]w ∈ B(H), f) (h0, w) fulfill (ME1)d).

Proposition

Let ǫ > 0. Then w−ǫ ˙ R±(z)w−ǫ extends finite meromorphically to Imz > −δǫ/2 as an operator valued function with values in B( ˙ E±).

3.3 Construction of the resolvent

Proposition

If the conditions (A1)-(A2) and (TE1)-(TE3) are satisfied then there is a finite set Z ⊂ C \ R with Z = Z such that the spectra of H and ˙ H are included in R ∪ Z and such that the resolvents R and ˙ R are finite meromorphic functions on C \ R. Moreover, the point spectrum of H coincides with the point spectrum of ˙ H and the set Z consists of eigenvalues of finite multiplicity of H and ˙ H. Proof. Q(z) := i−( ˙ H− − z)−1i− + i+( ˙ H+ − z)−1i+. ( ˙ H − z)Q(z) = 1 l + [ ˙ H, i−]( ˙ H− − z)−1i− + [ ˙ H, i+]( ˙ H+ − z)−1i+ = 1 l + K−(z) + K+(z) = 1 + K(z). 1 + K(z) = (1 + K−(z)j− + K+j+) × (1 + K−(z)(1 − j−) + K+(z)(1 − j+)), (1 + K−(z)j− + K+(z)j+)−1 = 1 − K−(z)j− − K+(z)j+. Thus ˙ R(z) := Q(z)(1 + K(z))−1 = Q(z)(1 + K−(z)(1 − j−) + K+(1 − j+))−1 × (1 − K−(z)j− − K+j+).

3.4 Smooth functional calculus

fm := sup

λ∈R, α≤m

|f (α)(λ)|.

Proposition

Assume (A1), (A2), (TE1)-(TE3). (i) Let f ∈ C∞

0 (R). Let ˜

f be an almost analytic extension of f such that supp˜ f ∩ σC

pp( ˙

H) = ∅. Then the integral f( ˙ H) :=

1 2πi

• C

∂˜ f ∂z (z) ˙

R(z)dz ∧ dz is norm convergent in B( ˙ E) and independent of the choice of the almost analytic extension of f. (ii) The map C∞

0 (R) ∋ f → f( ˙

H) ∈ B( ˙ E) is a homomorphism of algebras with f( ˙ H)∗ = f( ˙ H∗), f( ˙ H)B( ˙

E) ≤ fm

for some m ∈ N.

Proposition

Assume σC

pp( ˙

H) = ∅. Let χ ∈ C∞

0 (R), χ ≡ 1 in a neighborhood of zero.

Then s − limL→∞ χ ˙

H L

• = 1

l.

Proof. We show that (2) lim

L→∞ χ

• L−1 ˙

H

• − i−χ

˙ H− L

• i− − i+χ

˙ H+ L

• i+ = 0.

χ

• L−1 ˙

H

• − i−χ

˙ H− L

• i− − i+χ

˙ H+ L

• i+

= 1 2πi

• ∂ ˜

χ(z)L( ˙ R(Lz) − Q(Lz))dz ∧ dz. Now recall that ˙ R(z) = Q(z)(1 + K(z))−1, thus ˙ R(Lz) − Q(Lz) = − ˙ R(Lz)K(Lz). We have the estimate for L ≥ 1 ∂ ˜ χ(z)L ˙ R(Lz)K(Lz) 1 L → 0. This implies (2). As ˙ H± is selfadjoint in ˙ E± we find : s- lim

L→∞ i±χ

• L−1 ˙

H±

• i± = i2

±.

Thus s- lim

L→∞ χ

• L−1 ˙

H

• = i2

− + i2 + = 1.

4.1 Resonances and boundary values of the resolvent

Lemma

w−ǫ ˙ R(z)w−ǫ can be extended meromorphically from the upper half plane to Imz > −δǫ, δǫ > 0 with values in B∞( ˙ E). poles: resonances. We have w−ǫ ˙ R(z)w−ǫ = (1 l + Aw(z))−1w−ǫQ(z)w−ǫ.

Proposition

Assume (A1)-(A2), (TE1)-(TE3). Let ǫ > 0. There exists a discrete closed set ˙ TH ⊂ R, ν > 0 such that for all χ ∈ C∞

0 (R \ ˙

TH) we have (3) sup

u ˙

E=1, ν≥δ>0

• R

(w−ǫ ˙ R(λ+iδ)χ( ˙ H)u2

˙ E+w−ǫ ˙

R(λ−iδ)χ( ˙ H)u2

˙ E)dλ < ∞.

Definition

We call λ ∈ R a regular point of ˙ H if there exists χ ∈ C∞

0 (R), χ(λ) = 1

such that (3) holds. Otherwise we call it a singular point.

Remark

Note that in the selfadjoint case ˙ TH is the set of real resonances by Kato’s theory of H-smoothness.

4.2 Propagation estimates

Proposition

Assume (A1)-(A2), (TE1)-(TE3). Let ǫ > 0. Then there exists a discrete closed set ˙ T ⊂ R such that for all χ ∈ C∞

0 (R \ ˙

T ) and all k ∈ N we have w−ǫe−it ˙

Hχ( ˙

H)w−ǫB( ˙

E) t−k.

Proposition

Assume (A1)-(A2), (TE1)-(TE3). Let ǫ > 0. Then we have for all χ ∈ C∞

0 (R \ ˙

TH):

• R

w−ǫe−it ˙

Hχ( ˙

H)ϕ2

˙ Edt ϕ2 ˙ E.

Theorem

Suppose that λ0 ∈ R is neither a resonance of w−ǫ ˙ R(λ)w−ǫ nor of w−ǫQ(λ)w−ǫ. Then λ0 is a regular point of ˙ H.

• Proof. w−ǫ ˙

R(z) = w−ǫQ(z) − w−ǫ ˙ R(z)w−ǫwǫK(z).

5 Uniform boundedness of the evolution 1 : Abstract setting

(B1)    a) For all ψ ∈ C∞

0 (R),

h1/2 ψ(x)h−1/2 ∈ B(H). b) If in addition ψ ≡ 1 in a neighborhood of 0, ψ ≥ 0, then s − limn→∞ ψ x

n

• = 1

l in h−1/2 H. (B2) [−ik, h] w−1h0w−1 in the sense of quadratic forms on D(h0). For χ ∈ C∞(R) and µ > 0 we put χµ(.) = χ

• .

µ

• .

Theorem

Assume (A1), (A2), (TE1)-(TE3), (B1), (B2). i) Let χ ∈ C∞(R), suppχ ⊂ R \ [−1, 1], χ ≡ 1 on R \ (−2, 2). Then there exists µ0 > 0, C1 > 0 such that we have for µ ≥ µ0 e−it ˙

Hχµ( ˙

H)u ˙

E ≤ C1χµ( ˙

H)u ˙

E

∀u ∈ ˙ E, ∀t ∈ R. ii) Let ϕ ∈ C∞

0 (R \ ˙

TH). Then there exists C2 > 0 such that for all u ∈ ˙ E and t ∈ R we have e−it ˙

Hϕ( ˙

H)u ˙

E ≤ C2ϕ( ˙

H)u ˙

E.

5.1 High frequency analysis

Lemma

Assume (A1), (A2), (TE1)-(TE3), (B1), (B2). If χ is as in the statement of

• Thm. 20 then for µ > 0 sufficiently large we have:

(χµ( ˙ H)u)0H 1 µχµ( ˙ H)u ˙

E.

• Proof. Let ˆ

χ be as χ, ˆ χχ = χ, ϕ = ˆ χ − 1. ˜ ϕ(x + iy) =

N

• r=0

ϕ(r)(x)(iy)r r! τ y δx

• with τ ∈ C∞

0 (R), τ(s) = 1 in |s| ≤ 1/2, τ(s) = 0 in |s| ≥ 1.

∂ ˜ ϕ(z) = ˆ χ(N+1)(x) (iy)(N+1) (N + 1)!τ y δx

• +

N

• r=0

ϕ(r)(x)(iy)r r!

• τ ′

y δx i δx + yx δx2

• .

µ ≥ µ0 = MkB(H), M large. Then for z ∈ supp∂ ˜ ϕ, |µz| ≥ (1 + ǫ)k.

1 = −(χ − 1)(0) =

1 2πi

• ∂ ˜

ϕ(z) 1

z dz ∧ dz.

ˆ χµ( ˙ H) = ϕµ( ˙ H) + 1 = 1 2πi

• ∂ ˜

ϕ(z)   ˙ H µ − z −1 + 1 z   dz ∧ dz = − 1 2πi

• ∂ ˜

ϕ(z)( ˙ H − µz)−1 ˙ H z dz ∧ dz. (4) Let v µ = χµ( ˙ H)u. We compute

• ( ˙

H − µz)−1 ˙ H z v µ v µ

1

• = 1

z p−1(µz)(µzv µ

1 + hv µ 0 ).

p−1(µz)zµv µ

1 H

• p−1(µz)zµ(v µ

1 − kv µ 0 )H + p−1(µz)zµkv µ 0 H

• 1

|Imz|µ(v µ

1 − kv µ 0 )H +

1 µ|Imz|v µ

0 H,

p−1(µz)hv µ

0 H

• p−1(z)h0v µ

0 H + p−1(µz)k2v µ 0 H

• 1

|Imz|µh1/2 v µ

0 H +

1 µ2|Imz|v µ

0 H

• 1

|Imz|µh1/2 v µ

0 H +

1 µ2|Imz|v µ

0 H

We obtain (χµ( ˙ H)u)0H 1 µχµ( ˙ H)u ˙

E + 1

µ2 (χµ( ˙ H)u)0H. This gives the lemma for µ sufficiently large.

Corollary

Assume (A1), (A2), (TE1)-(TE3), (B1), (B2). Let χ be as in Thm. 20. Then for µ > 0 sufficiently large there exists ε > 0 such that for all u ∈ E: χµ( ˙ H)u, χµ( ˙ H)u0 ≥ εχµ( ˙ H)u2

˙ E.

Corollary

Assume (A1), (A2), (TE1)-(TE3), (B1), (B2). Let χ be as in Thm. 20. Then there exists C1 > 0 such that for all u ∈ ˙ E, t ∈ R e−it ˙

Hχµ( ˙

H)u ˙

E ≤ C1χµ( ˙

H)u ˙

E.

• Proof. We use that e−it ˙

Hχµ( ˙

H)u, e−it ˙

Hχµ( ˙

H)u0 is conserved. By Corollary 22 we have e−itHχµ( ˙ H)u2

˙ E

• e−it ˙

Hχµ( ˙

H)u, e−it ˙

Hχµ( ˙

H)u0 = χµ( ˙ H)u, χµ( ˙ H)u0 χµ( ˙ H)u2

˙ E,

which finishes the proof.

6 Asymptotic completeness 1 : Abstract setting

Definition

We call χ ∈ C∞(R) an admissible energy cut-off function for ˙ H if

◮ χ ≡ 0 in a neighborhood of ˙

TH and

◮ (χ ≡ 0 or χ ≡ 1) on R \ (−R, R) for some R > 0.

We note CH the set of all admissible energy cut-offs for ˙ H.

Definition

The spaces of scattering data are defined by ˙ Escatt = {χ( ˙ H)u; u ∈ ˙ E, χ ∈ CH}, ˙ E±

scatt = {χ( ˙

H±)u; u ∈ ˙ E±, χ ∈ CH}

Theorem

Assume (A1), (A2), (TE1)-(TE3), (B1)-(B2). (i) For all ϕ± ∈ ˙ E±

scatt there exist ψ± ∈ ˙

Escatt such that e−it ˙

Hψ± − i±e−it ˙ H±ϕ± → 0, t → ∞

in ˙ E. (ii) For all ψ± ∈ ˙ Escatt there exist ϕ± ∈ ˙ E±

scatt such that

e−it ˙

H±ϕ± − i±e−it ˙ Hψ± → 0, t → ∞

in ˙ E±.

7 Geometric setting

7.1 Separable hamiltonian M = R(r−,r+) × Sd−1

ω

, P = d−1

ij=1 D∗ i αijDj ≥ 0.

(G1) L2(Sd−1

ω

; dω) = ⊕n∈ZY n, Dθ1|Y n = n, P leaves Y n invariant. q(r) :=

• (r+ − r)(r − r−),

T σ = {f ∈ C∞(M); ∂α

r ∂β ωf ∈ O(q(r)σ−2α)}.

hs

0 = α1Drα2 2Drα1 + α2 3P + α2 4,

αi = αi(r). (G2)

• αi − q(r)(i−α−

i + i+α+ i )

∈ T 1+δ, i = 1, 2, 3, 4, αi

• q(r),

i = 1, 2, 3, 4. ks = ks,rDθ1 + ks,v. (G3)    i+ks,r, i+ks,v ∈ T 2, i−(ks,r − k −

s,r)

∈ T 2, i−(ks,v − k −

s,v)

∈ T 2. α±

i , k − s,r, k− s,v ∈ R.

hs = hs

0 − k2 s , H = L2(R(r−,r+) × Sd−1 ω

; drdω), Hn = H ∩ Y n.

7.2 Perturbed hamiltonian

(∂2

t − 2iks∂t + hs)u = 0. Perturbation: (∂2 t − 2ik∂t + h)u = 0.

h0|C∞

(M)

= hs

0 +

• i,j∈{1,...,d−1}

D∗

i gijDj +

• i∈{1,...,d−1}

(giDi + D∗

i gi)

+ DrgrrDr + grDr + Drgr + f =: hs

0 + hp.

(G4) The functions gij, gi, grr, gr, f are independent of θ1 (G5) h0

• q(r)(Drq2(r)Dr + P + 1)q(r),

hs

• q(r)(Drq2(r)Dr + P + 1)q(r).

(G6)            gij ∈ T 2+δ, i, j ∈ {1, ..., d − 1}, grr ∈ T 4+δ gr ∈ T 2+δ gi ∈ T 1+δ, i ∈ {1, ..., d − 1} f ∈ T 2. k = krDθ1 + kv, kr = ks,r + kp,r, kv = ks,v + kp,v (G7) kp,v, kr,v ∈ T 2. h := h0 − k 2.

8 Asymptotic completeness 2 : Geometric setting

h+∞ := hs

0,

h−∞ := h+∞ − ℓ2, k+∞ := 0, k−∞ := ℓ. Define ˙ H±∞, ˙ E±∞, ˙ E±∞ in the usual manner.

Theorem

Assume (G1)-(G7). (i) For all ϕ± ∈ ˙ Escatt

±∞ there exist ψ± ∈ ˙

Escatt such that e−it ˙

Hψ± − i±e−it ˙ H±∞ϕ± → 0, t → ∞

in ˙ E. (ii) For all ψ± ∈ ˙ Escatt there exist ϕ± ∈ ˙ Escatt

±∞ such that

e−it ˙

H±∞ϕ± − i±e−it ˙ Hψ± → 0, t → ∞

in ˙ E±∞.

Proposition

(G1)-(G7) entail (A1), (A2), (TE1)-(TE2), (B1), (B2).

Remark

h+, ˜ h− are similar to the Laplacian on an asymptotically hyperbolic

• manifold. Meromorphic extension : Mazzeo-Melrose ’87.

9.1 The wave equation on the De Sitter Kerr metric

9.1.1 De Sitter Kerr metric in Boyer-Lindquist coordinates MBH = Rt × Rr × S2

ω, with spacetime metric

g = ∆r − a2 sin2 θ∆θ λ2ρ2 dt2 + 2a sin2 θ((r 2 + a2)2∆θ − a2 sin2 θ∆r) λ2ρ2 dtdϕ − ρ2 ∆r dr 2 − ρ2 ∆θ dθ2 − sin2 θσ2 λ2ρ2 dϕ2, ρ2 = r 2 + a2 cos2 θ, ∆r =

• 1 − Λ

3 r 2

• (r 2 + a2) − 2Mr,

∆θ = 1 + 1 3Λa2 cos2 θ, σ2 = (r 2 + a2)2∆θ − a2∆r sin2 θ, λ = 1 + 1 3Λa2. Λ > 0: cosmological constant, M > 0 : masse, a : angular momentum per unit masse.

◮ ρ2 = 0 is a curvature singularity, ∆r = 0 are coordinate singularities.

∆r > 0 on some open interval r− < r < r+. r = r−: black hole horizon, r = r+ cosmological horizon.

◮ ∂ϕ and ∂t are Killing. There exist r1(θ), r2(θ) s. t. ∂t is

◮ timelike on {(t, r, θ, ϕ) : r1(θ) < r < r2(θ)}, ◮ spacelike on

{(t, r, θ, ϕ) : r− < r < r1(θ)}∪{(t, r, θ, ϕ : r2(θ) < r < r+} =: E−∪E+.

The regions E−, E+ are called ergospheres.

9.2 The wave equation on the De Sitter Kerr metric

We now consider the unitary transform U : L2(M;

σ2 ∆r ∆θ drdω)

→ L2(M; drdω) ψ →

σ

√

∆r ∆θ ψ

If ψ fulfills (✷g + m2)ψ = 0, then u = Uψ fulfills (∂2

t − 2ik∂t + h)u = 0.

(5) with k = a(∆r − (r 2 + a2)∆θ) σ2 Dϕ, h = −(∆r − a2 sin2 θ∆θ) sin2 θσ2 ∂2

ϕ −

√∆r∆θ λσ ∂r∆r∂r √∆r∆θ λσ − √∆r∆θ λ sin θσ ∂θ sin θ∆θ∂θ √∆r∆θ λσ + ρ2∆r∆θ λ2σ2 m2. h is not positive inside the ergospheres. This entails that the natural conserved quantity ˜ E(u) = ∂tu2 + (hu|u) is not positive.

9.3 3+1 decomposition, energies, Killing fields

Let v = e−iktu. Then u is solution of (5) if and only if v is solution of (∂2

t + h(t))v = 0,

h(t) = e−ikth0eikt, h0 = h + k 2 ≥ 0. Natural energy : ∂tv2 + (h(t)v|v). Rewriting for u: ˙ E(u) = (∂t − ik)u2 + (h0u|u). This energy is positive, but may grow in time → superradiance.

Remark

k = ΩDϕ and Ω has finite limits Ωl/r when r → r∓. These limits are called angular velocities of the horizons. The Killing fields ∂t − Ωl/r∂ϕ on the De Sitter Kerr metric are timelike close to the black hole (l) resp. cosmological (r) horizon. Working with these Killing fields rather than with ∂t leads to the conserved energies : ˜ El/r(u) = (∂t − Ωl/r∂ϕ)u2 + (h0 − (k − Ωl/rDϕ)2u|u). Note that in the limit k → Ωl/rDϕ the expressions of ˙ E(u) and ˜ El/r(u) coincide.

9.4 Asymptotic dynamics

Regge-Wheeler type coordinate dx

dr = r2+a2 ∆r .

x ± t = const. along principal null geodesics. Unitary transform: V : L2(R(r−,r+) × S2) → L2(R × S2, dxdω), v(r, ω) →

• ∆r

r2+a2 v(r(x), ω).

Asymptotic equations : (∂2

t − 2Ωl/r∂ϕ∂t + hl/r)ul/r = 0,

(6) hl/r = Ω2

l/r∂2 ϕ − ∂2 x.

The conserved quantities : (∂t − iΩl/rDϕ)ul/r2 + ((hl/r − Ω2

l/r∂2 ϕ)ul/r|ul/r)

= (∂t − iΩl/rDϕ)ul/r2 + (−∂2

x ul/r|ul/r)

are positive. Question : Can we compare the solutions of (5) to solutions of (6) for large times ?

10 Asymptotic completeness 3 : The De Sitter Kerr case

10.1 Uniform boundedness of the evolution (7) Hn = {u ∈ L2(R × S2) : (Dϕ − n)u = 0}, n ∈ Z. We construct the energy spaces ˙ En, En as well as the Klein-Gordon

• perators Hn, ˙

Hn as in Sect. 1.

Theorem

There exists a0 > 0 such that for |a| < a0 the following holds: for all n ∈ Z, there exists Cn > 0 such that (8) e−it ˙

Hnu ˙ En ≤ Cnu ˙ En, u ∈ ˙

En, t ∈ R. Note that for n = 0 the Hamiltonian ˙ Hn = ˙ H0 is selfadjoint, therefore the

• nly issue is n = 0.

Proof : Results of Dyatlov about the absence of complex eigenvalues and real resonances for n = 0, hypoellipticity argument, general result about the link between real resonances and singular points.

10.2 Separable comparison dynamics

Let ℓ± = Ω±n. Also let i± ∈ C∞(R), i− = 0 in a neighborhood of ∞, i+ = 0 in a neighborhood of −∞ and i2

− + i2 + = 1. We put:

h±∞ := −ℓ2

± − ∂2 x +

∆r λ2(r 2 + a2)P + ∆rm2, k±∞ := ℓ±, where P := − λ2 sin2 θ ∂2

ϕ −

1 sin θ ∂θ sin θ∆θ∂θ. Let ˙ En

±∞, ˙

Hn

±∞ be the homogeneous energy spaces and operators

associated to (h±∞, k±∞) according to Sect. 2.

Theorem

There exists a0 > 0 such that for |a| < a0 and n ∈ Z \ {0} the following holds:

◮ i) The wave operators W ± = s-limt→∞ eit ˙ Hni±e−it ˙ Hn

±∞ exist as

bounded operators W ± ∈ B( ˙ En

±∞; ˙

En).

◮ ii) The inverse wave operators Ω± = s-limt→∞ eit ˙ Hn

±∞i±e−it ˙

Hn exist

as bounded operators Ω± ∈ B( ˙ En; ˙ En

±∞).

i) and ii) also hold for n = 0 if m > 0.

10.3 Asymptotic profiles

Let us denote the cutoffs i+/− by ir/l. Let hn

r/l = −∂2 x − ℓ2 +/−, kr/l = ℓ+/−,

acting on Hn defined in (7). We associate to these operators the natural homogeneous energy spaces ˙ En

l/r and Hamiltonians ˙

Hn

l/r. Let {λq : q ∈ N} = σ(P) and

Zq = 1 l{λq}(P)H. Then D(h0) = D(h0,s) = {u ∈ H :

• q∈N

hs,q

0 1

l{λq}(P)u2 < ∞}, where hs,q is the restriction of h0,s to L2(R) ⊗ Zq. Let Wq := (L2(R) ⊗ Zq) ⊕ (L2(R) ⊗ Zq), Eq,n

l/r := En r/l ∩ Wq,

Efin,n

l/r

:=

• u ∈ En

l/r : ∃Q > 0, u ∈ ⊕q≤QEq,n l/r

• .

Theorem

There exists a0 > 0 such that for all |a| < a0 and n ∈ Z \ {0} the following holds:

◮ i) For all u ∈ Efin,n r/l

the limits Wr/lu = lim

t→∞ eit ˙ Hni2 r/le−it ˙ Hn

r/l u

exist in ˙

• En. The operators Wr/l extend to bounded operators

Wr/l ∈ B( ˙ En

r/l; ˙

En).

◮ ii) The inverse wave operators

Ωr/l = s- lim

t→∞ eit ˙ Hn

r/l i2

r/le−it ˙ Hn

exist in B( ˙ En; ˙ En

r/l).

i), ii) also hold for n = 0 if m > 0.

Some open problems

◮ General setting

◮ Regularity of the truncated resolvent in the Krein space setting. ◮ Situations with two different limits when now meromorphic extension of

the resolvent in a strip below the real axis is available (typically asymptotically euclidean manifolds).

◮ (De Sitter) Kerr

◮ Uniform result in n should be ok using “trapping estimates”. The

trapping is normally hyperbolic.

Thank you for your attention !