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 R d the Klein-Gordon equation minimally coupled to an electric field. ( ∂ t − i v ( x )) 2 φ ( t , x ) − ∆ x φ ( t , x ) + m 2 φ ( t , x ) = 0 . v ∈ C ∞ 0 ( R d ) is the electric potential and m the mass of the Klein-Gordon field. Conserved energy : � R d | ∂ t φ ( t , x ) | 2 + |∇ x φ ( t , x ) | 2 + ( m 2 − v 2 ( x )) | φ ( t , x ) | 2 dx . � � � � φ ( t ) 0 1 l , f ( t ) = e i tH f ( 0 ) , H = f ( t ) = . − ∆ x + m 2 − v 2 i − 1 ∂ t φ ( t ) 2 v Energy : � f 0 � � R d | f 1 | 2 ( x ) + (( − ∆ x + m 2 − v 2 ( x )) f 0 ( x )) f 0 ( x ) dx , f = E ( f , f ) = . f 1 Pb: − ∆ x + m 2 − v 2 might acquire negative spectrum.

1.1.2 Results on the Klein-Gordon equation Energy space : E = H 1 ( R d ) ⊕ L 2 ( R d ) . Results : σ ess ( H ) =] − ∞ , − m ] ∪ [ m , + ∞ [ σ ( H ) \ R = ∪ 1 ≤ j ≤ n { λ j , λ j } , where λ j , λ j are eigenvalues of finite Riesz index.     �� x � − s ( H − z ) − 1 � x � − s � B ( E ) < ∞ , For s > 1 / 2 : sup Re z ∈ I , 0 < |ℑ z |≤ δ 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 M BH = R t × R r × S 2 ω , with spacetime metric ∆ r − a 2 sin 2 θ ∆ θ dt 2 + 2 a sin 2 θ (( r 2 + a 2 ) 2 ∆ θ − a 2 sin 2 θ ∆ r ) g = dtd ϕ λ 2 ρ 2 λ 2 ρ 2 ∆ θ d θ 2 − sin 2 θσ 2 ∆ r dr 2 − ρ 2 ρ 2 d ϕ 2 , − λ 2 ρ 2 � � 1 − Λ r 2 + a 2 cos 2 θ, ( r 2 + a 2 ) − 2 Mr , ρ 2 3 r 2 = ∆ r = 1 + 1 3 Λ a 2 cos 2 θ, σ 2 = ( r 2 + a 2 ) 2 ∆ θ − a 2 ∆ r sin 2 θ, λ = 1 + 1 3 Λ a 2 . ∆ θ = Λ > 0: cosmological constant, M > 0 : masse, a : angular momentum per unit masse. ◮ ρ 2 = 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 r 1 ( θ ) , r 2 ( θ ) s. t. ∂ t is ◮ timelike on { ( t , r , θ, ϕ ) : r 1 ( θ ) < r < r 2 ( θ ) } , ◮ spacelike on { ( t , r , θ, ϕ ) : r − < r < r 1 ( θ ) }∪{ ( t , r , θ, ϕ : r 2 ( 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 σ 2 L 2 ( M ; L 2 ( M ; drd ω ) ∆ r ∆ θ drd ω ) → U : √ σ ψ �→ ∆ r ∆ θ ψ If ψ fulfills ( ✷ g + m 2 ) ψ = 0, then u = U ψ fulfills ( ∂ 2 t − 2 ik ∂ t + h ) u = 0 . with (1) a (∆ r − ( r 2 + a 2 )∆ θ ) k = D ϕ , σ 2 √ ∆ r ∆ θ √ ∆ r ∆ θ − (∆ r − a 2 sin 2 θ ∆ θ ) ∂ 2 h = ϕ − ∂ r ∆ r ∂ r sin 2 θσ 2 λσ λσ √ ∆ r ∆ θ √ ∆ r ∆ θ + ρ 2 ∆ r ∆ θ m 2 . − λ sin θσ ∂ θ sin θ ∆ θ ∂ θ λσ λ 2 σ 2 h is not positive inside the ergospheres. This entails that the natural conserved quantity E ( u ) = � ∂ t u � 2 + ( 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, B oundary 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, R esolvent 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 operators in Krein spaces and applications to Klein-Gordon equations

Plan of part 1 Definitizable operators on Krein Krein spaces spaces Basic definitions Definitizable operators Operators on Krein spaces Pontryagin spaces Functional calculus Abstract Klein Gordon equation Smooth and Borel functional Energy spaces calculus on Banach spaces Klein-Gordon operators C 0 − groups Limiting absorption principle M γ functional calculus Example : Klein-Gordon Boundary value estimates equation on scattering Main theorem manifolds Virial theorem An important proposition Proof of the main theorem

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 Π 2 = Π , orthogonal 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 Π K = 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 | Im z | 1 − n if Re z ∈ 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¨ ostrand formula. � χ ( H ) = − 1 χ ( z ) dz ∧ dz . R ( z ) ∂ ˜ 2 π i C We shall say that the smooth functional calculus extends to a C 0 − functional calculus on I if � χ ( H ) � ≤ C sup λ ∈ I | χ ( λ ) | for all χ ∈ C ∞ c ( I ) . Unique continuous extension to an algebra morphism C 0 ( I ) → B ( K ) . Theorem Assume that K is a reflexive Banach space and let F 0 : C 0 ( I ) → B ( K ) be a norm continuous algebra morphism. Then there is a unique algebra morphism F : B ( I ) → B ( K ) which extends F 0 and such that: b- lim n ϕ n = ϕ ⇒ F ( ϕ n ) → F ( ϕ ) weakly.

C 0 − groups ◮ Let W t = e i tA be a C 0 -group on a Banach space K . ◮ There are numbers M ≥ 1 and γ ≥ 0 such that � W t � ≤ M e γ | t | for all t ∈ R . ◮ The spectrum of the operator A is included in the strip | Im z | ≤ γ and it could be equal to this strip. ◮ S ∈ B ( K ) is of class C α ( A ) if the map R ∋ t �→ S ( t ) = e − i tA Se i tA ∈ B ( K ) is C α for the strong operator topology. For an unbounded operator S we say that S ∈ C α ( A ) if R ( z 0 ) ∈ C α ( A ) for some z 0 ∈ ρ ( S ) . ◮ If K is a Krein space we say that the Krein structure is of class C 1 ( A ) if the conditions in the next proposition are verified. Proposition The following assertions are equivalent: the function t �→ � W t u | W t u � is derivable at zero for each u ∈ H ; the function t �→ � W t u | W t u � is of class C 1 for each u ∈ H ; the map t �→ W ∗ t W t is locally Lipschitz; A ∗ = A + B where B is a bounded operator.