On the scattering theory of asymptotically flat manifolds and - PowerPoint PPT Presentation

Strichartz and scattering estimates on the Euclidean space Scattering inequalities ◮ Resolvent estimates: give the behaviour with respect to λ ∈ R of δ → 0 ± ( − ∆ − λ − i δ ) − 1 R 0 ( λ ± i 0) = lim In general, the existence of the limit is called limiting absorption principle Intuition. R 0 ( λ + i δ ) is the Fourier multiplier by ( | ξ | 2 − λ − i δ ) − 1 . This multiplier has a limit as δ → 0 ± ( ∼ principal value) provided it is tested against smooth enough functions on the Fourier side

Strichartz and scattering estimates on the Euclidean space Scattering inequalities ◮ Resolvent estimates: give the behaviour with respect to λ ∈ R of δ → 0 ± ( − ∆ − λ − i δ ) − 1 R 0 ( λ ± i 0) = lim In general, the existence of the limit is called limiting absorption principle Intuition. R 0 ( λ + i δ ) is the Fourier multiplier by ( | ξ | 2 − λ − i δ ) − 1 . This multiplier has a limit as δ → 0 ± ( ∼ principal value) provided it is tested against smooth enough functions on the Fourier side ↔ decaying functions on the spatial side.

Strichartz and scattering estimates on the Euclidean space Scattering inequalities ◮ Resolvent estimates: give the behaviour with respect to λ ∈ R of δ → 0 ± ( − ∆ − λ − i δ ) − 1 R 0 ( λ ± i 0) = lim In general, the existence of the limit is called limiting absorption principle Intuition. R 0 ( λ + i δ ) is the Fourier multiplier by ( | ξ | 2 − λ − i δ ) − 1 . This multiplier has a limit as δ → 0 ± ( ∼ principal value) provided it is tested against smooth enough functions on the Fourier side ↔ decaying functions on the spatial side. Examples. 1. High energy estimates:

Strichartz and scattering estimates on the Euclidean space Scattering inequalities ◮ Resolvent estimates: give the behaviour with respect to λ ∈ R of δ → 0 ± ( − ∆ − λ − i δ ) − 1 R 0 ( λ ± i 0) = lim In general, the existence of the limit is called limiting absorption principle Intuition. R 0 ( λ + i δ ) is the Fourier multiplier by ( | ξ | 2 − λ − i δ ) − 1 . This multiplier has a limit as δ → 0 ± ( ∼ principal value) provided it is tested against smooth enough functions on the Fourier side ↔ decaying functions on the spatial side. Examples. 1. High energy estimates: if ν > 1 / 2, � � x � − ν R 0 ( λ ± i 0) � x � − ν � L 2 → L 2 � λ − 1 / 2 , � �� �� λ ≥ 1 �

Strichartz and scattering estimates on the Euclidean space Scattering inequalities ◮ Resolvent estimates: give the behaviour with respect to λ ∈ R of δ → 0 ± ( − ∆ − λ − i δ ) − 1 R 0 ( λ ± i 0) = lim In general, the existence of the limit is called limiting absorption principle Intuition. R 0 ( λ + i δ ) is the Fourier multiplier by ( | ξ | 2 − λ − i δ ) − 1 . This multiplier has a limit as δ → 0 ± ( ∼ principal value) provided it is tested against smooth enough functions on the Fourier side ↔ decaying functions on the spatial side. Examples. 1. High energy estimates: if ν > 1 / 2, � � x � − ν R 0 ( λ ± i 0) � x � − ν � L 2 → L 2 � λ − 1 / 2 , � �� �� λ ≥ 1 � 2. Low energy estimates:

Strichartz and scattering estimates on the Euclidean space Scattering inequalities ◮ Resolvent estimates: give the behaviour with respect to λ ∈ R of δ → 0 ± ( − ∆ − λ − i δ ) − 1 R 0 ( λ ± i 0) = lim In general, the existence of the limit is called limiting absorption principle Intuition. R 0 ( λ + i δ ) is the Fourier multiplier by ( | ξ | 2 − λ − i δ ) − 1 . This multiplier has a limit as δ → 0 ± ( ∼ principal value) provided it is tested against smooth enough functions on the Fourier side ↔ decaying functions on the spatial side. Examples. 1. High energy estimates: if ν > 1 / 2, � � x � − ν R 0 ( λ ± i 0) � x � − ν � L 2 → L 2 � λ − 1 / 2 , �� � �� λ ≥ 1 � 2. Low energy estimates: if ν = 1 and n ≥ 3 � � x � − 1 R 0 ( λ ± i 0) � x � − 1 � � �� �� L 2 → L 2 � 1 , | λ | ≤ 1 �

Strichartz and scattering estimates on the Euclidean space Scattering inequalities ◮ Resolvent estimates: give the behaviour with respect to λ ∈ R of δ → 0 ± ( − ∆ − λ − i δ ) − 1 R 0 ( λ ± i 0) = lim In general, the existence of the limit is called limiting absorption principle Intuition. R 0 ( λ + i δ ) is the Fourier multiplier by ( | ξ | 2 − λ − i δ ) − 1 . This multiplier has a limit as δ → 0 ± ( ∼ principal value) provided it is tested against smooth enough functions on the Fourier side ↔ decaying functions on the spatial side. Examples. 1. High energy estimates: if ν > 1 / 2, � � x � − ν R 0 ( λ ± i 0) � x � − ν � L 2 → L 2 � λ − 1 / 2 , �� � �� λ ≥ 1 � 2. Low energy estimates: if ν = 1 and n ≥ 3 � � x � − 1 R 0 ( λ ± i 0) � x � − 1 � � �� �� L 2 → L 2 � 1 , | λ | ≤ 1 � 3. One may (actually, one has to) also consider estimates on R 0 ( λ ± i 0) k

Strichartz and scattering estimates on the Euclidean space Scattering inequalities ◮ Resolvent estimates: give the behaviour with respect to λ ∈ R of δ → 0 ± ( − ∆ − λ − i δ ) − 1 R 0 ( λ ± i 0) = lim In general, the existence of the limit is called limiting absorption principle Intuition. R 0 ( λ + i δ ) is the Fourier multiplier by ( | ξ | 2 − λ − i δ ) − 1 . This multiplier has a limit as δ → 0 ± ( ∼ principal value) provided it is tested against smooth enough functions on the Fourier side ↔ decaying functions on the spatial side. Examples. 1. High energy estimates: if ν > 1 / 2, � � x � − ν R 0 ( λ ± i 0) � x � − ν � L 2 → L 2 � λ − 1 / 2 , � �� �� λ ≥ 1 � 2. Low energy estimates: if ν = 1 and n ≥ 3 � � x � − 1 R 0 ( λ ± i 0) � x � − 1 � � �� �� L 2 → L 2 � 1 , | λ | ≤ 1 � 3. One may (actually, one has to) also consider estimates on d k − 1 1 R 0 ( λ ± i 0) k = d λ k − 1 R 0 ( λ ± i 0) ( k − 1)!

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (continued) ◮ Propagation / time decay estimates:

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (continued) ◮ Propagation / time decay estimates: given a (spectral) cutoff ϕ ∈ C ∞ 0 (0 , + ∞ ), understand the time decay of ϕ ( − ∆ /λ ) e it ∆

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (continued) ◮ Propagation / time decay estimates: given a (spectral) cutoff ϕ ∈ C ∞ 0 (0 , + ∞ ), understand the time decay of ϕ ( − ∆ /λ ) e it ∆ as t → ∞ ,

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (continued) ◮ Propagation / time decay estimates: given a (spectral) cutoff ϕ ∈ C ∞ 0 (0 , + ∞ ), understand the time decay of ϕ ( − ∆ /λ ) e it ∆ as t → ∞ , in term of the parameter λ > 0.

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (continued) ◮ Propagation / time decay estimates: given a (spectral) cutoff ϕ ∈ C ∞ 0 (0 , + ∞ ), understand the time decay of ϕ ( − ∆ /λ ) e it ∆ as t → ∞ , in term of the parameter λ > 0. Intuition. For λ = 1, the Schwartz kernel of ϕ ( − ∆) e it ∆ is the oscillatory integral d ξ � e i ( x − y ) · ξ − it | ξ | 2 ϕ ( | ξ | 2 ) (2 π ) n

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (continued) ◮ Propagation / time decay estimates: given a (spectral) cutoff ϕ ∈ C ∞ 0 (0 , + ∞ ), understand the time decay of ϕ ( − ∆ /λ ) e it ∆ as t → ∞ , in term of the parameter λ > 0. Intuition. For λ = 1, the Schwartz kernel of ϕ ( − ∆) e it ∆ is the oscillatory integral (2 π ) n = i d ξ � � 2 | ξ | 2 · ∂ ξ e − it | ξ | 2 � ξ d ξ � e i ( x − y ) · ξ − it | ξ | 2 ϕ ( | ξ | 2 ) e i ( x − y ) · ξ ϕ ( | ξ | 2 ) (2 π ) n 2 t

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (continued) ◮ Propagation / time decay estimates: given a (spectral) cutoff ϕ ∈ C ∞ 0 (0 , + ∞ ), understand the time decay of ϕ ( − ∆ /λ ) e it ∆ as t → ∞ , in term of the parameter λ > 0. Intuition. For λ = 1, the Schwartz kernel of ϕ ( − ∆) e it ∆ is the oscillatory integral (2 π ) n = i d ξ � � 2 | ξ | 2 · ∂ ξ e − it | ξ | 2 � ξ d ξ � e i ( x − y ) · ξ − it | ξ | 2 ϕ ( | ξ | 2 ) e i ( x − y ) · ξ ϕ ( | ξ | 2 ) (2 π ) n 2 t which leads to � � � � x � − k ϕ ( − ∆) e it ∆ � x � − k � � L 2 → L 2 � � t � − k . � � � � � � �

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (continued) ◮ Propagation / time decay estimates: given a (spectral) cutoff ϕ ∈ C ∞ 0 (0 , + ∞ ), understand the time decay of ϕ ( − ∆ /λ ) e it ∆ as t → ∞ , in term of the parameter λ > 0. Intuition. For λ = 1, the Schwartz kernel of ϕ ( − ∆) e it ∆ is the oscillatory integral (2 π ) n = i d ξ � � 2 | ξ | 2 · ∂ ξ e − it | ξ | 2 � ξ d ξ � e i ( x − y ) · ξ − it | ξ | 2 ϕ ( | ξ | 2 ) e i ( x − y ) · ξ ϕ ( | ξ | 2 ) (2 π ) n 2 t which leads to � � � � x � − k ϕ ( − ∆) e it ∆ � x � − k � � L 2 → L 2 � � t � − k . � � � � � � � By scaling � � 1 2 x � − k � 1 � 2 x � − k ϕ ( − ∆ /λ ) e it ∆ � λ L 2 → L 2 � � λ t � − k � � λ � � � � � � �

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (end) ◮ Integrated decay/ smoothing estimates:

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (end) ◮ Integrated decay/ smoothing estimates: Integrated space-time decay estimates are of the form � 1 �� 2 ||� x � − ν ϕ ( − ∆ /λ ) e it ∆ u 0 || 2 L 2 dt � λ || u 0 || L 2 , R with ν > 1 / 2.

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (end) ◮ Integrated decay/ smoothing estimates: Integrated space-time decay estimates are of the form � 1 �� 2 ||� x � − ν ϕ ( − ∆ /λ ) e it ∆ u 0 || 2 L 2 dt � λ || u 0 || L 2 , R with ν > 1 / 2. By tracking the dependence on λ , one may obtain the non spectrally localized estimate ( n ≥ 3) � 1 �� 2 1 ||� x � − 1 � D � 2 e it ∆ u 0 || 2 L 2 dt � || u 0 || L 2 R

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (end) ◮ Integrated decay/ smoothing estimates: Integrated space-time decay estimates are of the form � 1 �� 2 ||� x � − ν ϕ ( − ∆ /λ ) e it ∆ u 0 || 2 L 2 dt � λ || u 0 || L 2 , R with ν > 1 / 2. By tracking the dependence on λ , one may obtain the non spectrally localized estimate ( n ≥ 3) � 1 �� 2 1 ||� x � − 1 � D � 2 e it ∆ u 0 || 2 L 2 dt � || u 0 || L 2 R which is the 1 2 -smoothing effect for the Schr¨ odinger equation.

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (end) ◮ Integrated decay/ smoothing estimates: Integrated space-time decay estimates are of the form � 1 �� 2 ||� x � − ν ϕ ( − ∆ /λ ) e it ∆ u 0 || 2 L 2 dt � λ || u 0 || L 2 , R with ν > 1 / 2. By tracking the dependence on λ , one may obtain the non spectrally localized estimate ( n ≥ 3) � 1 �� 2 1 ||� x � − 1 � D � 2 e it ∆ u 0 || 2 L 2 dt � || u 0 || L 2 R which is the 1 2 -smoothing effect for the Schr¨ odinger equation. Note that even locally in time (i.e. with R replaced by [ − T , T ]) this is non trivial.

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (end) ◮ Integrated decay/ smoothing estimates: Integrated space-time decay estimates are of the form � 1 �� 2 ||� x � − ν ϕ ( − ∆ /λ ) e it ∆ u 0 || 2 L 2 dt � λ || u 0 || L 2 , R with ν > 1 / 2. By tracking the dependence on λ , one may obtain the non spectrally localized estimate ( n ≥ 3) � 1 �� 2 1 ||� x � − 1 � D � 2 e it ∆ u 0 || 2 L 2 dt � || u 0 || L 2 R which is the 1 2 -smoothing effect for the Schr¨ odinger equation. Note that even locally in time (i.e. with R replaced by [ − T , T ]) this is non trivial. Intuition. More on the next slides.

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (end) ◮ Integrated decay/ smoothing estimates: Integrated space-time decay estimates are of the form � 1 �� 2 ||� x � − ν ϕ ( − ∆ /λ ) e it ∆ u 0 || 2 L 2 dt � λ || u 0 || L 2 , R with ν > 1 / 2. By tracking the dependence on λ , one may obtain the non spectrally localized estimate ( n ≥ 3) � 1 �� 2 1 ||� x � − 1 � D � 2 e it ∆ u 0 || 2 L 2 dt � || u 0 || L 2 R which is the 1 2 -smoothing effect for the Schr¨ odinger equation. Note that even locally in time (i.e. with R replaced by [ − T , T ]) this is non trivial. Intuition. More on the next slides. Technically, they follow from resolvent estimates via a Parseval argument, using that e it ∆ is the Fourier transform ( λ → t ) of the spectral measure R 0 ( λ + i 0) − R 0 ( λ − i 0) .

Strichartz and scattering estimates on the Euclidean space Scattering inequalities (end) ◮ Integrated decay/ smoothing estimates: Integrated space-time decay estimates are of the form � 1 �� 2 ||� x � − ν ϕ ( − ∆ /λ ) e it ∆ u 0 || 2 L 2 dt � λ || u 0 || L 2 , R with ν > 1 / 2. By tracking the dependence on λ , one may obtain the non spectrally localized estimate ( n ≥ 3) � 1 �� 2 1 ||� x � − 1 � D � 2 e it ∆ u 0 || 2 L 2 dt � || u 0 || L 2 R which is the 1 2 -smoothing effect for the Schr¨ odinger equation. Note that even locally in time (i.e. with R replaced by [ − T , T ]) this is non trivial. Intuition. More on the next slides. Technically, they follow from resolvent estimates via a Parseval argument, using that e it ∆ is the Fourier transform ( λ → t ) of the spectral measure R 0 ( λ + i 0) − R 0 ( λ − i 0) . Rem. This correspondence λ → t also allows to convert resolvent estimates into time decay/propagation estimates (smoothness of R 0 ( λ ± i 0) ↔ decay of e itP )

Strichartz inequalities vs smoothing effect for a wave packet Strichartz inequalities

Strichartz inequalities vs smoothing effect for a wave packet Strichartz inequalities Consider the L 2 normalized semiclassical wave packet � i h ζ · ( x − z ) − | x − z | 2 � G z ,ζ, h ( x ) = ( π h ) − n 4 exp . 2 h

Strichartz inequalities vs smoothing effect for a wave packet Strichartz inequalities Consider the L 2 normalized semiclassical wave packet � i h ζ · ( x − z ) − | x − z | 2 � G z ,ζ, h ( x ) = ( π h ) − n 4 exp . 2 h Then, � e i t 2 ∆ G z ,ζ, h ( x ) � � �

Strichartz inequalities vs smoothing effect for a wave packet Strichartz inequalities Consider the L 2 normalized semiclassical wave packet � i h ζ · ( x − z ) − | x − z | 2 � G z ,ζ, h ( x ) = ( π h ) − n 4 exp . 2 h Then, π − n � 2 � � � � � x − z − ( t / h ) ζ 4 � e i t 2 ∆ G z ,ζ, h ( x ) � = � � exp − h � t / h � 2 � n 2 h � t / h � 2 � 4 1 with � τ � = (1 + τ 2 ) 2 .

Strichartz inequalities vs smoothing effect for a wave packet Strichartz inequalities Consider the L 2 normalized semiclassical wave packet � i h ζ · ( x − z ) − | x − z | 2 � G z ,ζ, h ( x ) = ( π h ) − n 4 exp . 2 h Then, π − n � 2 � � � � � x − z − ( t / h ) ζ 4 � e i t 2 ∆ G z ,ζ, h ( x ) � = � � exp − h � t / h � 2 � n 2 h � t / h � 2 � 4 1 with � τ � = (1 + τ 2 ) 2 . This implies easily � � � n 2 − 1 1 � 1 n 2 q � e i t 2 ∆ G z ,ζ, h � �� �� � L q = (2 / q ) 2 q � π h � t / h � 2

Strichartz inequalities vs smoothing effect for a wave packet Strichartz inequalities Consider the L 2 normalized semiclassical wave packet � i h ζ · ( x − z ) − | x − z | 2 � G z ,ζ, h ( x ) = ( π h ) − n 4 exp . 2 h Then, π − n � 2 � � � � � x − z − ( t / h ) ζ 4 � e i t 2 ∆ G z ,ζ, h ( x ) � = � � exp − h � t / h � 2 � n 2 h � t / h � 2 � 4 1 with � τ � = (1 + τ 2 ) 2 . This implies easily � � � n 2 − 1 1 � 1 n 2 q � e i t 2 ∆ G z ,ζ, h �� � �� � L q = (2 / q ) 2 q � π h � t / h � 2 Remark. The translation by ( t / h ) ζ is not used. Only the spreading/dilation factor � t / h � plays a role.

Strichartz inequalities vs smoothing effect for a wave packet Strichartz inequalities Consider the L 2 normalized semiclassical wave packet � i h ζ · ( x − z ) − | x − z | 2 � G z ,ζ, h ( x ) = ( π h ) − n 4 exp . 2 h Then, π − n � 2 � � � � � x − z − ( t / h ) ζ 4 � e i t 2 ∆ G z ,ζ, h ( x ) � = � � exp − h � t / h � 2 � n 2 h � t / h � 2 � 4 1 with � τ � = (1 + τ 2 ) 2 . This implies easily � � � n 2 − 1 1 � 1 n 2 q � e i t 2 ∆ G z ,ζ, h � �� �� � L q = (2 / q ) 2 q � π h � t / h � 2 Remark. The translation by ( t / h ) ζ is not used. Only the spreading/dilation factor � t / h � plays a role. In particular, for q = 2 ∗ = 2 n / ( n − 2), � T � e i t � 2 2 ∆ G z ,ζ, h � �� � �� L 2 ∗ dt − T

Strichartz inequalities vs smoothing effect for a wave packet Strichartz inequalities Consider the L 2 normalized semiclassical wave packet � i h ζ · ( x − z ) − | x − z | 2 � G z ,ζ, h ( x ) = ( π h ) − n 4 exp . 2 h Then, π − n � 2 � � � � � x − z − ( t / h ) ζ 4 � e i t 2 ∆ G z ,ζ, h ( x ) � = � � exp − h � t / h � 2 � n 2 h � t / h � 2 � 4 1 with � τ � = (1 + τ 2 ) 2 . This implies easily � � � n 1 2 − 1 � 1 n 2 q � e i t 2 ∆ G z ,ζ, h �� � �� � L q = (2 / q ) 2 q � π h � t / h � 2 Remark. The translation by ( t / h ) ζ is not used. Only the spreading/dilation factor � t / h � plays a role. In particular, for q = 2 ∗ = 2 n / ( n − 2), � T � T 1 dt � e i t � 2 2 ∆ G z ,ζ, h � �� �� � L 2 ∗ dt = c n � t / h � 2 h − T − T

Strichartz inequalities vs smoothing effect for a wave packet Strichartz inequalities Consider the L 2 normalized semiclassical wave packet � i h ζ · ( x − z ) − | x − z | 2 � G z ,ζ, h ( x ) = ( π h ) − n 4 exp . 2 h Then, π − n � 2 � � � � � x − z − ( t / h ) ζ 4 � e i t 2 ∆ G z ,ζ, h ( x ) � = � � exp − h � t / h � 2 � n 2 h � t / h � 2 � 4 1 with � τ � = (1 + τ 2 ) 2 . This implies easily � � � n 1 2 − 1 � 1 n 2 q � e i t 2 ∆ G z ,ζ, h �� � �� � L q = (2 / q ) 2 q � π h � t / h � 2 Remark. The translation by ( t / h ) ζ is not used. Only the spreading/dilation factor � t / h � plays a role. In particular, for q = 2 ∗ = 2 n / ( n − 2), � T � T � T / h 1 dt 1 � e i t � 2 2 ∆ G z ,ζ, h � �� � �� L 2 ∗ dt = c n h = c n 1 + τ 2 d τ ≤ C � t / h � 2 − T − T − T / h

Strichartz inequalities vs smoothing effect for a wave packet Strichartz inequalities Consider the L 2 normalized semiclassical wave packet � i h ζ · ( x − z ) − | x − z | 2 � G z ,ζ, h ( x ) = ( π h ) − n 4 exp . 2 h Then, π − n � 2 � � � � � x − z − ( t / h ) ζ 4 � e i t 2 ∆ G z ,ζ, h ( x ) � = � � exp − h � t / h � 2 � n 2 h � t / h � 2 � 4 1 with � τ � = (1 + τ 2 ) 2 . This implies easily � � � n 1 2 − 1 � 1 n 2 q � e i t 2 ∆ G z ,ζ, h �� � �� � L q = (2 / q ) 2 q � π h � t / h � 2 Remark. The translation by ( t / h ) ζ is not used. Only the spreading/dilation factor � t / h � plays a role. In particular, for q = 2 ∗ = 2 n / ( n − 2), � T � T � T / h 1 dt 1 � e i t � 2 2 ∆ G z ,ζ, h � �� � �� L 2 ∗ dt = c n h = c n 1 + τ 2 d τ ≤ C � t / h � 2 − T − T − T / h for all h ∈ (0 , 1]

Strichartz inequalities vs smoothing effect for a wave packet Strichartz inequalities Consider the L 2 normalized semiclassical wave packet � i h ζ · ( x − z ) − | x − z | 2 � G z ,ζ, h ( x ) = ( π h ) − n 4 exp . 2 h Then, π − n � 2 � � � � � x − z − ( t / h ) ζ 4 � e i t 2 ∆ G z ,ζ, h ( x ) � = � � exp − h � t / h � 2 � n 2 h � t / h � 2 � 4 1 with � τ � = (1 + τ 2 ) 2 . This implies easily � � � n 1 2 − 1 � 1 n 2 q � e i t 2 ∆ G z ,ζ, h �� � �� � L q = (2 / q ) 2 q � π h � t / h � 2 Remark. The translation by ( t / h ) ζ is not used. Only the spreading/dilation factor � t / h � plays a role. In particular, for q = 2 ∗ = 2 n / ( n − 2), � T � T � T / h 1 dt 1 � e i t � 2 2 ∆ G z ,ζ, h � �� � �� L 2 ∗ dt = c n h = c n 1 + τ 2 d τ ≤ C � t / h � 2 − T − T − T / h for all h ∈ (0 , 1] and z ∈ R n .

Strichartz inequalities vs smoothing effect for a wave packet Smoothing effect (local in time) � � D � s e i t 2 ∆ G z ,ζ, h ( x ) � � �

Strichartz inequalities vs smoothing effect for a wave packet Smoothing effect (local in time) π − n � � 2 � � � � x − z − ( t / h ) ζ 4 � � D � s e i t 2 ∆ G z ,ζ, h ( x ) � � � ζ/ h � s ∼ exp − h → 0 , � h � t / h � 2 � n 2 h � t / h � 2 � 4

Strichartz inequalities vs smoothing effect for a wave packet Smoothing effect (local in time) π − n � � 2 � � � � x − z − ( t / h ) ζ 4 � � D � s e i t 2 ∆ G z ,ζ, h ( x ) � � � ζ/ h � s ∼ exp − h → 0 , � h � t / h � 2 � n 2 h � t / h � 2 � 4 � ζ/ h � s G t = z ,ζ, h ( x ) .

Strichartz inequalities vs smoothing effect for a wave packet Smoothing effect (local in time) π − n � � 2 � � � � x − z − ( t / h ) ζ 4 � � D � s e i t 2 ∆ G z ,ζ, h ( x ) � � � ζ/ h � s ∼ exp − h → 0 , � h � t / h � 2 � n 2 h � t / h � 2 � 4 � ζ/ h � s G t = z ,ζ, h ( x ) . We assume that ζ � = 0,

Strichartz inequalities vs smoothing effect for a wave packet Smoothing effect (local in time) π − n � � 2 � � � � x − z − ( t / h ) ζ 4 � � D � s e i t 2 ∆ G z ,ζ, h ( x ) � � � ζ/ h � s ∼ exp − h → 0 , � h � t / h � 2 � n 2 h � t / h � 2 � 4 � ζ/ h � s G t = z ,ζ, h ( x ) . We assume that ζ � = 0, say | ζ | = 1

Strichartz inequalities vs smoothing effect for a wave packet Smoothing effect (local in time) π − n � � 2 � � � � x − z − ( t / h ) ζ 4 � � D � s e i t 2 ∆ G z ,ζ, h ( x ) � � � ζ/ h � s ∼ exp − h → 0 , � h � t / h � 2 � n 2 h � t / h � 2 � 4 � ζ/ h � s G t = z ,ζ, h ( x ) . We assume that ζ � = 0, say | ζ | = 1 and then, by possibly rotating the axis, that ζ = (1 , 0 , . . . , 0).

Strichartz inequalities vs smoothing effect for a wave packet Smoothing effect (local in time) π − n � � 2 � � � � x − z − ( t / h ) ζ 4 � � D � s e i t 2 ∆ G z ,ζ, h ( x ) � � � ζ/ h � s ∼ exp − h → 0 , � h � t / h � 2 � n 2 h � t / h � 2 � 4 � ζ/ h � s G t = z ,ζ, h ( x ) . We assume that ζ � = 0, say | ζ | = 1 and then, by possibly rotating the axis, that ζ = (1 , 0 , . . . , 0). Then � 2 � � x � − ν � ζ/ h � s G t � �� � �� z ,ζ, h L 2 x

Strichartz inequalities vs smoothing effect for a wave packet Smoothing effect (local in time) π − n � � 2 � � � � x − z − ( t / h ) ζ 4 � � D � s e i t 2 ∆ G z ,ζ, h ( x ) � � � ζ/ h � s ∼ exp − h → 0 , � h � t / h � 2 � n 2 h � t / h � 2 � 4 � ζ/ h � s G t = z ,ζ, h ( x ) . We assume that ζ � = 0, say | ζ | = 1 and then, by possibly rotating the axis, that ζ = (1 , 0 , . . . , 0). Then y 2 � � � � 2 2 y + z + t ζ/ h � − 2 ν exp 1 � � x � − ν � ζ/ h � s G t x = c n � ζ/ h � 2 s � t / h � − n � �� �� � � h − dy z ,ζ, h L 2 � t / h � 2

Strichartz inequalities vs smoothing effect for a wave packet Smoothing effect (local in time) π − n � � 2 � � � � x − z − ( t / h ) ζ 4 � � D � s e i t 2 ∆ G z ,ζ, h ( x ) � � � ζ/ h � s ∼ exp − h → 0 , � h � t / h � 2 � n 2 h � t / h � 2 � 4 � ζ/ h � s G t = z ,ζ, h ( x ) . We assume that ζ � = 0, say | ζ | = 1 and then, by possibly rotating the axis, that ζ = (1 , 0 , . . . , 0). Then y 2 � � � � 2 2 y + z + t ζ/ h � − 2 ν exp 1 � � x � − ν � ζ/ h � s G t x = c n � ζ/ h � 2 s � t / h � − n � �� �� � � h − dy z ,ζ, h L 2 � t / h � 2 If we further integrate in time on [ − T , T ] t ,

Strichartz inequalities vs smoothing effect for a wave packet Smoothing effect (local in time) π − n � � 2 � � � � x − z − ( t / h ) ζ 4 � � D � s e i t 2 ∆ G z ,ζ, h ( x ) � � � ζ/ h � s ∼ exp − h → 0 , � h � t / h � 2 � n 2 h � t / h � 2 � 4 � ζ/ h � s G t = z ,ζ, h ( x ) . We assume that ζ � = 0, say | ζ | = 1 and then, by possibly rotating the axis, that ζ = (1 , 0 , . . . , 0). Then y 2 � � � � 2 2 y + z + t ζ/ h � − 2 ν exp 1 � � x � − ν � ζ/ h � s G t x = c n � ζ/ h � 2 s � t / h � − n �� � �� � � h − dy z ,ζ, h L 2 � t / h � 2 If we further integrate in time on [ − T , T ] t , � T / h − y 2 � � � 1 2 y + z + τζ � − 2 ν exp c n h � ζ/ h � 2 s � τ � − n � h dyd τ � τ � 2 − T / h

Strichartz inequalities vs smoothing effect for a wave packet Smoothing effect (local in time) π − n � � 2 � � � � x − z − ( t / h ) ζ 4 � � D � s e i t 2 ∆ G z ,ζ, h ( x ) � � � ζ/ h � s ∼ exp − h → 0 , � h � t / h � 2 � n 2 h � t / h � 2 � 4 � ζ/ h � s G t = z ,ζ, h ( x ) . We assume that ζ � = 0, say | ζ | = 1 and then, by possibly rotating the axis, that ζ = (1 , 0 , . . . , 0). Then y 2 � � � � 2 2 y + z + t ζ/ h � − 2 ν exp 1 � � x � − ν � ζ/ h � s G t x = c n � ζ/ h � 2 s � t / h � − n �� � �� � � h − dy z ,ζ, h L 2 � t / h � 2 If we further integrate in time on [ − T , T ] t , � T / h − y 2 � � � 1 2 y + z + τζ � − 2 ν exp c n h � ζ/ h � 2 s � τ � − n � h dyd τ � τ � 2 − T / h which is bounded by � T / h � 1 2 Y 1 � τ � + z 1 + τ � − 2 ν exp c n h � 1 / h � 2 s − Y 2 � � � h dYd τ − T / h

Strichartz inequalities vs smoothing effect for a wave packet Smoothing effect (local in time) π − n � � 2 � � � � x − z − ( t / h ) ζ 4 � � D � s e i t 2 ∆ G z ,ζ, h ( x ) � � � ζ/ h � s ∼ exp − h → 0 , � h � t / h � 2 � n 2 h � t / h � 2 � 4 � ζ/ h � s G t = z ,ζ, h ( x ) . We assume that ζ � = 0, say | ζ | = 1 and then, by possibly rotating the axis, that ζ = (1 , 0 , . . . , 0). Then y 2 � � � � 2 2 y + z + t ζ/ h � − 2 ν exp 1 � � x � − ν � ζ/ h � s G t x = c n � ζ/ h � 2 s � t / h � − n � �� �� � � h − dy z ,ζ, h L 2 � t / h � 2 If we further integrate in time on [ − T , T ] t , � T / h − y 2 � � � 1 2 y + z + τζ � − 2 ν exp c n h � ζ/ h � 2 s � τ � − n � h dyd τ � τ � 2 − T / h which is bounded by � T / h � 1 2 Y 1 � τ � + z 1 + τ � − 2 ν exp c n h � 1 / h � 2 s − Y 2 � � � h dYd τ − T / h Remark. Up to the term Y 1 � τ � , there is no more contribution of the spreading � τ � .

Strichartz inequalities vs smoothing effect for a wave packet Smoothing effect (local in time) π − n � � 2 � � � � x − z − ( t / h ) ζ 4 � � D � s e i t 2 ∆ G z ,ζ, h ( x ) � � � ζ/ h � s ∼ exp − h → 0 , � h � t / h � 2 � n 2 h � t / h � 2 � 4 � ζ/ h � s G t = z ,ζ, h ( x ) . We assume that ζ � = 0, say | ζ | = 1 and then, by possibly rotating the axis, that ζ = (1 , 0 , . . . , 0). Then y 2 � � � � 2 2 y + z + t ζ/ h � − 2 ν exp 1 � � x � − ν � ζ/ h � s G t x = c n � ζ/ h � 2 s � t / h � − n � �� �� � � h − dy z ,ζ, h L 2 � t / h � 2 If we further integrate in time on [ − T , T ] t , � T / h − y 2 � � � 1 2 y + z + τζ � − 2 ν exp c n h � ζ/ h � 2 s � τ � − n � h dyd τ � τ � 2 − T / h which is bounded by � T / h � 1 2 Y 1 � τ � + z 1 + τ � − 2 ν exp c n h � 1 / h � 2 s − Y 2 � � � h dYd τ − T / h Remark. Up to the term Y 1 � τ � , there is no more contribution of the spreading � τ � . Here, the main role will be played the translation by ( t / h ) ζ = τζ .

Strichartz inequalities vs smoothing effect for a wave packet Recall we are estimating � T / h � 2 Y 1 � τ � + z 1 + τ � − 2 ν exp 1 � 2 � �� � � x � − ν � ζ/ h � s G t � �� t , x � h 1 − 2 s − Y 2 � � h � dYd τ. z ,ζ, h L 2 − T / h

Strichartz inequalities vs smoothing effect for a wave packet Recall we are estimating � T / h � 2 Y 1 � τ � + z 1 + τ � − 2 ν exp 1 � 2 � �� � � x � − ν � ζ/ h � s G t � �� t , x � h 1 − 2 s − Y 2 � � h � dYd τ. z ,ζ, h L 2 − T / h ◮ In the region | h 1 / 2 Y 1 | ≤ ǫ ( ǫ ≪ 1 fixed),

Strichartz inequalities vs smoothing effect for a wave packet Recall we are estimating � T / h � 1 2 Y 1 � τ � + z 1 + τ � − 2 ν exp � 2 �� � � � x � − ν � ζ/ h � s G t � �� t , x � h 1 − 2 s − Y 2 � � h � dYd τ. z ,ζ, h L 2 − T / h ◮ In the region | h 1 / 2 Y 1 | ≤ ǫ ( ǫ ≪ 1 fixed), we integrate in time by using the variable 1 2 Y 1 � τ � ˜ τ = τ + h (Jacobian = 1 + O ( ǫ ))

Strichartz inequalities vs smoothing effect for a wave packet Recall we are estimating � T / h � 2 Y 1 � τ � + z 1 + τ � − 2 ν exp 1 � 2 � �� � � x � − ν � ζ/ h � s G t � �� t , x � h 1 − 2 s − Y 2 � � h � dYd τ. z ,ζ, h L 2 − T / h ◮ In the region | h 1 / 2 Y 1 | ≤ ǫ ( ǫ ≪ 1 fixed), we integrate in time by using the variable 1 2 Y 1 � τ � ˜ τ = τ + h (Jacobian = 1 + O ( ǫ )) so we bound the integral by � �� CT / h � τ � − 2 ν exp h 1 − 2 s − Y 2 � � � z 1 + ˜ d ˜ τ dY (1) − CT / h

Strichartz inequalities vs smoothing effect for a wave packet Recall we are estimating � T / h � 2 Y 1 � τ � + z 1 + τ � − 2 ν exp 1 � 2 � �� � � x � − ν � ζ/ h � s G t � �� t , x � h 1 − 2 s − Y 2 � � h � dYd τ. z ,ζ, h L 2 − T / h ◮ In the region | h 1 / 2 Y 1 | ≤ ǫ ( ǫ ≪ 1 fixed), we integrate in time by using the variable 1 2 Y 1 � τ � ˜ τ = τ + h (Jacobian = 1 + O ( ǫ )) so we bound the integral by � �� CT / h � τ � − 2 ν exp h 1 − 2 s − Y 2 � � � z 1 + ˜ d ˜ τ dY (1) − CT / h ◮ If | h 1 / 2 Y 1 | ≥ ǫ

Strichartz inequalities vs smoothing effect for a wave packet Recall we are estimating � T / h � 1 2 Y 1 � τ � + z 1 + τ � − 2 ν exp � 2 �� � � � x � − ν � ζ/ h � s G t � �� t , x � h 1 − 2 s − Y 2 � � h � dYd τ. z ,ζ, h L 2 − T / h ◮ In the region | h 1 / 2 Y 1 | ≤ ǫ ( ǫ ≪ 1 fixed), we integrate in time by using the variable 1 2 Y 1 � τ � ˜ τ = τ + h (Jacobian = 1 + O ( ǫ )) so we bound the integral by � �� CT / h � τ � − 2 ν exp h 1 − 2 s − Y 2 � � � z 1 + ˜ d ˜ τ dY (1) − CT / h ◮ If | h 1 / 2 Y 1 | ≥ ǫ , then | Y 1 | � h − 1 2 and 1 2 Y 1 � τ � + z 1 + τ � − 2 ν exp � � z 1 + τ � − 2 ν exp − Y 2 � − Y 2 / 2 O ( h ∞ ) � h � � � ⇒ Integral ≤ (1) × O ( h ∞ )

Strichartz inequalities vs smoothing effect for a wave packet Recall we are estimating � T / h � 1 2 Y 1 � τ � + z 1 + τ � − 2 ν exp � 2 � �� � � x � − ν � ζ/ h � s G t � �� t , x � h 1 − 2 s − Y 2 � � h � dYd τ. z ,ζ, h L 2 − T / h ◮ In the region | h 1 / 2 Y 1 | ≤ ǫ ( ǫ ≪ 1 fixed), we integrate in time by using the variable 1 2 Y 1 � τ � ˜ τ = τ + h (Jacobian = 1 + O ( ǫ )) so we bound the integral by � �� CT / h � τ � − 2 ν exp h 1 − 2 s − Y 2 � � � z 1 + ˜ d ˜ τ dY (1) − CT / h ◮ If | h 1 / 2 Y 1 | ≥ ǫ , then | Y 1 | � h − 1 2 and 1 2 Y 1 � τ � + z 1 + τ � − 2 ν exp � � z 1 + τ � − 2 ν exp − Y 2 � − Y 2 / 2 O ( h ∞ ) � h � � � ⇒ Integral ≤ (1) × O ( h ∞ ) Conclusion: If s = 1 2 and ν > 1 2 1 � �� � � x � − ν � ζ/ h � 2 G t � �� L 2 ([ − T , T ] × R n ) ≤ C z ,ζ, h �

Strichartz inequalities vs smoothing effect for a wave packet Recall we are estimating � T / h � 1 2 Y 1 � τ � + z 1 + τ � − 2 ν exp � 2 � �� � � x � − ν � ζ/ h � s G t �� � t , x � h 1 − 2 s − Y 2 � � h � dYd τ. z ,ζ, h L 2 − T / h ◮ In the region | h 1 / 2 Y 1 | ≤ ǫ ( ǫ ≪ 1 fixed), we integrate in time by using the variable 1 2 Y 1 � τ � τ = τ + h ˜ (Jacobian = 1 + O ( ǫ )) so we bound the integral by � �� CT / h � τ � − 2 ν exp h 1 − 2 s − Y 2 � � � z 1 + ˜ d ˜ τ dY (1) − CT / h ◮ If | h 1 / 2 Y 1 | ≥ ǫ , then | Y 1 | � h − 1 2 and 1 2 Y 1 � τ � + z 1 + τ � − 2 ν exp � � z 1 + τ � − 2 ν exp − Y 2 � − Y 2 / 2 O ( h ∞ ) � h � � � ⇒ Integral ≤ (1) × O ( h ∞ ) Conclusion: If s = 1 2 and ν > 1 2 1 � �� � � x � − ν � ζ/ h � 2 G t � �� L 2 ([ − T , T ] × R n ) ≤ C uniformly in h ∈ (0 , 1] z ,ζ, h �

Strichartz inequalities vs smoothing effect for a wave packet Recall we are estimating � T / h � 1 2 Y 1 � τ � + z 1 + τ � − 2 ν exp � 2 � �� � � x � − ν � ζ/ h � s G t �� � t , x � h 1 − 2 s − Y 2 � � h � dYd τ. z ,ζ, h L 2 − T / h ◮ In the region | h 1 / 2 Y 1 | ≤ ǫ ( ǫ ≪ 1 fixed), we integrate in time by using the variable 1 2 Y 1 � τ � τ = τ + h ˜ (Jacobian = 1 + O ( ǫ )) so we bound the integral by � �� CT / h � τ � − 2 ν exp h 1 − 2 s − Y 2 � � � z 1 + ˜ d ˜ τ dY (1) − CT / h ◮ If | h 1 / 2 Y 1 | ≥ ǫ , then | Y 1 | � h − 1 2 and 1 2 Y 1 � τ � + z 1 + τ � − 2 ν exp � � z 1 + τ � − 2 ν exp − Y 2 � − Y 2 / 2 O ( h ∞ ) � h � � � ⇒ Integral ≤ (1) × O ( h ∞ ) Conclusion: If s = 1 2 and ν > 1 2 1 � �� � � x � − ν � ζ/ h � 2 G t � �� uniformly in h ∈ (0 , 1] and in z ∈ R n . L 2 ([ − T , T ] × R n ) ≤ C z ,ζ, h �

Global Strichartz inequalities on asymptotically flat manifolds General problem: Extend Strichartz estimates to asymptotically flat manifolds

Global Strichartz inequalities on asymptotically flat manifolds General problem: Extend Strichartz estimates to asymptotically flat manifolds 1. see which properties persist or can be lost

Global Strichartz inequalities on asymptotically flat manifolds General problem: Extend Strichartz estimates to asymptotically flat manifolds 1. see which properties persist or can be lost 2. more specifically, try to decouple what happens near infinity

Global Strichartz inequalities on asymptotically flat manifolds General problem: Extend Strichartz estimates to asymptotically flat manifolds 1. see which properties persist or can be lost 2. more specifically, try to decouple what happens near infinity (where one expects the same behavior as on R n )

Global Strichartz inequalities on asymptotically flat manifolds General problem: Extend Strichartz estimates to asymptotically flat manifolds 1. see which properties persist or can be lost 2. more specifically, try to decouple what happens near infinity (where one expects the same behavior as on R n ) from what happens inside a compact set

Global Strichartz inequalities on asymptotically flat manifolds General problem: Extend Strichartz estimates to asymptotically flat manifolds 1. see which properties persist or can be lost 2. more specifically, try to decouple what happens near infinity (where one expects the same behavior as on R n ) from what happens inside a compact set (where the geometry/geodesic flow may be arbitrary and complicated)

Global Strichartz inequalities on asymptotically flat manifolds General problem: Extend Strichartz estimates to asymptotically flat manifolds 1. see which properties persist or can be lost 2. more specifically, try to decouple what happens near infinity (where one expects the same behavior as on R n ) from what happens inside a compact set (where the geometry/geodesic flow may be arbitrary and complicated) 3. see the influence of the geometry on nonlinear equations

Global Strichartz inequalities on asymptotically flat manifolds General problem: Extend Strichartz estimates to asymptotically flat manifolds 1. see which properties persist or can be lost 2. more specifically, try to decouple what happens near infinity (where one expects the same behavior as on R n ) from what happens inside a compact set (where the geometry/geodesic flow may be arbitrary and complicated) 3. see the influence of the geometry on nonlinear equations 4. the Schr¨ odinger equation can be replaced by other dispersive PDE (wave, Klein-Gordon) which are relevant on asymptotically flat manifolds

Global Strichartz inequalities on asymptotically flat manifolds General problem: Extend Strichartz estimates to asymptotically flat manifolds 1. see which properties persist or can be lost 2. more specifically, try to decouple what happens near infinity (where one expects the same behavior as on R n ) from what happens inside a compact set (where the geometry/geodesic flow may be arbitrary and complicated) 3. see the influence of the geometry on nonlinear equations 4. the Schr¨ odinger equation can be replaced by other dispersive PDE (wave, Klein-Gordon) which are relevant on asymptotically flat manifolds 5. good motivation / test to understand which scattering properties are robust and relevant (in particular in the low energy analysis)

Global Strichartz inequalities on asymptotically flat manifolds General problem: Extend Strichartz estimates to asymptotically flat manifolds 1. see which properties persist or can be lost 2. more specifically, try to decouple what happens near infinity (where one expects the same behavior as on R n ) from what happens inside a compact set (where the geometry/geodesic flow may be arbitrary and complicated) 3. see the influence of the geometry on nonlinear equations 4. the Schr¨ odinger equation can be replaced by other dispersive PDE (wave, Klein-Gordon) which are relevant on asymptotically flat manifolds 5. good motivation / test to understand which scattering properties are robust and relevant (in particular in the low energy analysis) Scattering inequalities turn out to play a crucial role in this problem.

Asymptotically flat manifolds ◮ The model: R n , equipped with the flat metric, G 0 = dx 2 1 + · · · + dx 2 � n = G jk dx j dx k , G 0 := ( G jk ) = I . j , k

Asymptotically flat manifolds ◮ The model: R n , equipped with the flat metric, G 0 = dx 2 1 + · · · + dx 2 � n = G jk dx j dx k , G 0 := ( G jk ) = I . j , k The geodesic flow φ t : R n × R n

Asymptotically flat manifolds ◮ The model: R n , equipped with the flat metric, G 0 = dx 2 1 + · · · + dx 2 � n = G jk dx j dx k , G 0 := ( G jk ) = I . j , k The geodesic flow φ t : R n × R n (= T ∗ R n )

Asymptotically flat manifolds ◮ The model: R n , equipped with the flat metric, G 0 = dx 2 1 + · · · + dx 2 � n = G jk dx j dx k , G 0 := ( G jk ) = I . j , k The geodesic flow φ t : R n × R n (= T ∗ R n ) → R n × R n is given by φ t ( x , ξ ) = ( x + 2 t ξ, ξ )

Asymptotically flat manifolds ◮ The model: R n , equipped with the flat metric, G 0 = dx 2 1 + · · · + dx 2 � n = G jk dx j dx k , G 0 := ( G jk ) = I . j , k The geodesic flow φ t : R n × R n (= T ∗ R n ) → R n × R n is given by φ t ( x , ξ ) = ( x + 2 t ξ, ξ ) =: ( x t , ξ t ) ,

Asymptotically flat manifolds ◮ The model: R n , equipped with the flat metric, G 0 = dx 2 1 + · · · + dx 2 � n = G jk dx j dx k , G 0 := ( G jk ) = I . j , k The geodesic flow φ t : R n × R n (= T ∗ R n ) → R n × R n is given by φ t ( x , ξ ) = ( x + 2 t ξ, ξ ) =: ( x t , ξ t ) , it solves the Hamilton equations

Asymptotically flat manifolds ◮ The model: R n , equipped with the flat metric, G 0 = dx 2 1 + · · · + dx 2 � n = G jk dx j dx k , G 0 := ( G jk ) = I . j , k The geodesic flow φ t : R n × R n (= T ∗ R n ) → R n × R n is given by φ t ( x , ξ ) = ( x + 2 t ξ, ξ ) =: ( x t , ξ t ) , it solves the Hamilton equations x t = ( ∂ ξ p )( x t , ξ t ) , ξ t = − ( ∂ x p )( x t , ξ t ) ˙ ˙ where p ( x , ξ ) = | ξ | 2 = ξ · G − 1 ξ 0

Asymptotically flat manifolds ◮ The model: R n , equipped with the flat metric, G 0 = dx 2 1 + · · · + dx 2 � n = G jk dx j dx k , G 0 := ( G jk ) = I . j , k The geodesic flow φ t : R n × R n (= T ∗ R n ) → R n × R n is given by φ t ( x , ξ ) = ( x + 2 t ξ, ξ ) =: ( x t , ξ t ) , it solves the Hamilton equations x t = ( ∂ ξ p )( x t , ξ t ) , ξ t = − ( ∂ x p )( x t , ξ t ) ˙ ˙ where p ( x , ξ ) = | ξ | 2 = ξ · G − 1 ξ 0 n with D j = 1 ∂ is the (principal) symbol of − ∆ = D 2 1 + · · · + D 2 i ∂ x j

Asymptotically flat manifolds ◮ The model: R n , equipped with the flat metric, G 0 = dx 2 1 + · · · + dx 2 � n = G jk dx j dx k , G 0 := ( G jk ) = I . j , k The geodesic flow φ t : R n × R n (= T ∗ R n ) → R n × R n is given by φ t ( x , ξ ) = ( x + 2 t ξ, ξ ) =: ( x t , ξ t ) , it solves the Hamilton equations x t = ( ∂ ξ p )( x t , ξ t ) , ξ t = − ( ∂ x p )( x t , ξ t ) ˙ ˙ where p ( x , ξ ) = | ξ | 2 = ξ · G − 1 ξ 0 n with D j = 1 ∂ is the (principal) symbol of − ∆ = D 2 1 + · · · + D 2 i ∂ x j ◮ Pertubed model: R n , equipped with a metric � j , k G jk ( x ) dx j dx k such that � � G ( x ) − I → 0 as x → ∞ , G ( x ) := G jk ( x )

Asymptotically flat manifolds ◮ The model: R n , equipped with the flat metric, G 0 = dx 2 1 + · · · + dx 2 � n = G jk dx j dx k , G 0 := ( G jk ) = I . j , k The geodesic flow φ t : R n × R n (= T ∗ R n ) → R n × R n is given by φ t ( x , ξ ) = ( x + 2 t ξ, ξ ) =: ( x t , ξ t ) , it solves the Hamilton equations x t = ( ∂ ξ p )( x t , ξ t ) , ξ t = − ( ∂ x p )( x t , ξ t ) ˙ ˙ where p ( x , ξ ) = | ξ | 2 = ξ · G − 1 ξ 0 n with D j = 1 ∂ is the (principal) symbol of − ∆ = D 2 1 + · · · + D 2 i ∂ x j ◮ Pertubed model: R n , equipped with a metric � j , k G jk ( x ) dx j dx k such that � � G ( x ) − I → 0 as x → ∞ , G ( x ) := G jk ( x ) more precisely, ∂ α ( G jk ( x ) − δ jk ) = O ( � x � − µ −| α | ) for some µ > 0.

Asymptotically flat manifolds ◮ The model: R n , equipped with the flat metric, G 0 = dx 2 1 + · · · + dx 2 � n = G jk dx j dx k , G 0 := ( G jk ) = I . j , k The geodesic flow φ t : R n × R n (= T ∗ R n ) → R n × R n is given by φ t ( x , ξ ) = ( x + 2 t ξ, ξ ) =: ( x t , ξ t ) , it solves the Hamilton equations x t = ( ∂ ξ p )( x t , ξ t ) , ξ t = − ( ∂ x p )( x t , ξ t ) ˙ ˙ where p ( x , ξ ) = | ξ | 2 = ξ · G − 1 ξ 0 n with D j = 1 ∂ is the (principal) symbol of − ∆ = D 2 1 + · · · + D 2 i ∂ x j ◮ Pertubed model: R n , equipped with a metric � j , k G jk ( x ) dx j dx k such that � � G ( x ) − I → 0 as x → ∞ , G ( x ) := G jk ( x ) more precisely, ∂ α ( G jk ( x ) − δ jk ) = O ( � x � − µ −| α | ) for some µ > 0. The geodesic flow is defined analogously with p ( x , ξ ) = ξ · G ( x ) − 1 ξ

Asymptotically flat manifolds ◮ The model: R n , equipped with the flat metric, G 0 = dx 2 1 + · · · + dx 2 � n = G jk dx j dx k , G 0 := ( G jk ) = I . j , k The geodesic flow φ t : R n × R n (= T ∗ R n ) → R n × R n is given by φ t ( x , ξ ) = ( x + 2 t ξ, ξ ) =: ( x t , ξ t ) , it solves the Hamilton equations x t = ( ∂ ξ p )( x t , ξ t ) , ξ t = − ( ∂ x p )( x t , ξ t ) ˙ ˙ where p ( x , ξ ) = | ξ | 2 = ξ · G − 1 ξ 0 n with D j = 1 ∂ is the (principal) symbol of − ∆ = D 2 1 + · · · + D 2 i ∂ x j ◮ Pertubed model: R n , equipped with a metric � j , k G jk ( x ) dx j dx k such that � � G ( x ) − I → 0 as x → ∞ , G ( x ) := G jk ( x ) more precisely, ∂ α ( G jk ( x ) − δ jk ) = O ( � x � − µ −| α | ) for some µ > 0. The geodesic flow is defined analogously with p ( x , ξ ) = ξ · G ( x ) − 1 ξ = � G jk ( x ) ξ j ξ k j , k

Asymptotically flat manifolds ◮ The model: R n , equipped with the flat metric, G 0 = dx 2 1 + · · · + dx 2 � n = G jk dx j dx k , G 0 := ( G jk ) = I . j , k The geodesic flow φ t : R n × R n (= T ∗ R n ) → R n × R n is given by φ t ( x , ξ ) = ( x + 2 t ξ, ξ ) =: ( x t , ξ t ) , it solves the Hamilton equations x t = ( ∂ ξ p )( x t , ξ t ) , ξ t = − ( ∂ x p )( x t , ξ t ) ˙ ˙ where p ( x , ξ ) = | ξ | 2 = ξ · G − 1 ξ 0 n with D j = 1 ∂ is the (principal) symbol of − ∆ = D 2 1 + · · · + D 2 i ∂ x j ◮ Pertubed model: R n , equipped with a metric � j , k G jk ( x ) dx j dx k such that � � G ( x ) − I → 0 as x → ∞ , G ( x ) := G jk ( x ) more precisely, ∂ α ( G jk ( x ) − δ jk ) = O ( � x � − µ −| α | ) for some µ > 0. The geodesic flow is defined analogously with p ( x , ξ ) = ξ · G ( x ) − 1 ξ = � G jk ( x ) ξ j ξ k j , k the (principal) symbol of the Laplace-Beltrami operator � G jk ( x ) ∂ x j ∂ x k + � G jk ( x )Γ ℓ − ∆ G = − jk ( x ) ∂ x ℓ j , k j , k ,ℓ