Wave packets for semiclassical Schr¨ odinger operators From now on, we use a semiclassical normalization � i � h ζ · ( x − z ) − | x − z | 2 z ,ζ ( x ) = ( π h ) − n ψ h 4 exp 2 h ⇒ Localization around z on a scale h 1 / 2 = odinger operator on R n Consider a semiclassical Schr¨ H ( h ) = − h 2 ∆ p ( x , ξ ) = | ξ | 2 + V ( x ) , + V ( x ) , 2 2 with V ∈ C ∞ ( R n , R ). Denote � A t � B t ( z t , ζ t ) = Φ t := D Φ t p ( z , ζ ) , p ( z , ζ ) C t D t and � t S t = z s · ζ s − p ( z s , ζ s ) ds ˙ 0 Proposition [action of the symplectic group on the Siegel half space] A t + i B t is invertible and Γ t := ( C t + i D t )( A t + i B t ) − 1 is symmetric complex, with positive definite imaginary part
Wave packets for semiclassical Schr¨ odinger operators Theorem (Hagedorn-Joye, Combescure-Robert) In the limit h → 0 , and under general conditions on V , e − i t h H ( h ) ψ h z ,ζ ( x ) is well approximated by � � S t + ζ t · ( x − z t ) + Γ t t ( x ) exp i ( π h ) − n 4 γ t A h 2 ( x − z t ) · ( x − z t ) h
Wave packets for semiclassical Schr¨ odinger operators Theorem (Hagedorn-Joye, Combescure-Robert) In the limit h → 0 , and under general conditions on V , e − i t h H ( h ) ψ h z ,ζ ( x ) is well approximated by � � S t + ζ t · ( x − z t ) + Γ t t ( x ) exp i ( π h ) − n 4 γ t A h 2 ( x − z t ) · ( x − z t ) h for times | t | ≤ C 0 | ln h | (C 0 dynamical constant). Here γ t = det ( A t + i B t ) − 1 / 2 .
Wave packets for semiclassical Schr¨ odinger operators Theorem (Hagedorn-Joye, Combescure-Robert) In the limit h → 0 , and under general conditions on V , e − i t h H ( h ) ψ h z ,ζ ( x ) is well approximated by � � S t + ζ t · ( x − z t ) + Γ t t ( x ) exp i ( π h ) − n 4 γ t A h 2 ( x − z t ) · ( x − z t ) h for times | t | ≤ C 0 | ln h | (C 0 dynamical constant). Here γ t = det ( A t + i B t ) − 1 / 2 . The amplitude is of the form � � z , ζ, t , x − z t � j A h 2 A j t ( x ) ∼ 1 + h 1 h 2 j ≥ 1 with A j ( z , ζ, t , X ) polynomial of degree ≤ 3 j in X, with coeff. depending on the classical trajectory t �→ ( z t , ζ t ) and the Taylor expansion of V at z t
Wave packets for semiclassical Schr¨ odinger operators Theorem (Hagedorn-Joye, Combescure-Robert) In the limit h → 0 , and under general conditions on V , e − i t h H ( h ) ψ h z ,ζ ( x ) is well approximated by � � S t + ζ t · ( x − z t ) + Γ t t ( x ) exp i ( π h ) − n 4 γ t A h 2 ( x − z t ) · ( x − z t ) h for times | t | ≤ C 0 | ln h | (C 0 dynamical constant). Here γ t = det ( A t + i B t ) − 1 / 2 . The amplitude is of the form � � z , ζ, t , x − z t � j A h 2 A j t ( x ) ∼ 1 + h 1 h 2 j ≥ 1 with A j ( z , ζ, t , X ) polynomial of degree ≤ 3 j in X, with coeff. depending on the classical trajectory t �→ ( z t , ζ t ) and the Taylor expansion of V at z t 1 Rem. The polynomial growth of the amplitude in ( x − z t ) / h 2 is beaten by the exponential decay of the exponential since Im (Γ t ) is positive definite
Wave packets for semiclassical Schr¨ odinger operators Theorem (Hagedorn-Joye, Combescure-Robert) In the limit h → 0 , and under general conditions on V , e − i t h H ( h ) ψ h z ,ζ ( x ) is well approximated by � � S t + ζ t · ( x − z t ) + Γ t t ( x ) exp i ( π h ) − n 4 γ t A h 2 ( x − z t ) · ( x − z t ) h for times | t | ≤ C 0 | ln h | (C 0 dynamical constant). Here γ t = det ( A t + i B t ) − 1 / 2 . The amplitude is of the form � � z , ζ, t , x − z t � j A h 2 A j t ( x ) ∼ 1 + h 1 h 2 j ≥ 1 with A j ( z , ζ, t , X ) polynomial of degree ≤ 3 j in X, with coeff. depending on the classical trajectory t �→ ( z t , ζ t ) and the Taylor expansion of V at z t 1 Rem. The polynomial growth of the amplitude in ( x − z t ) / h 2 is beaten by the exponential decay of the exponential since Im (Γ t ) is positive definite = ⇒ Concentration near the classical trajectory,
Wave packets for semiclassical Schr¨ odinger operators Theorem (Hagedorn-Joye, Combescure-Robert) In the limit h → 0 , and under general conditions on V , e − i t h H ( h ) ψ h z ,ζ ( x ) is well approximated by � � S t + ζ t · ( x − z t ) + Γ t t ( x ) exp i ( π h ) − n 4 γ t A h 2 ( x − z t ) · ( x − z t ) h for times | t | ≤ C 0 | ln h | (C 0 dynamical constant). Here γ t = det ( A t + i B t ) − 1 / 2 . The amplitude is of the form � � z , ζ, t , x − z t � j A h 2 A j t ( x ) ∼ 1 + h 1 h 2 j ≥ 1 with A j ( z , ζ, t , X ) polynomial of degree ≤ 3 j in X, with coeff. depending on the classical trajectory t �→ ( z t , ζ t ) and the Taylor expansion of V at z t 1 Rem. The polynomial growth of the amplitude in ( x − z t ) / h 2 is beaten by the exponential decay of the exponential since Im (Γ t ) is positive definite ⇒ Concentration near the classical trajectory, at least as long as Im (Γ t ) ≫ h =
Wave packets in semiclassical analysis Sketch of proof. Lemma The matrix Γ t satisfies the Ricatti equation Γ t = − V (2) ( z t ) − (Γ t ) 2 , Γ 0 = i I n , ˙
Wave packets in semiclassical analysis Sketch of proof. Lemma The matrix Γ t satisfies the Ricatti equation Γ t = − V (2) ( z t ) − (Γ t ) 2 , Γ 0 = i I n , ˙ and the function γ t satisfies γ t = − tr (Γ t ) γ t . ˙ 2
Wave packets in semiclassical analysis Sketch of proof. Lemma The matrix Γ t satisfies the Ricatti equation Γ t = − V (2) ( z t ) − (Γ t ) 2 , Γ 0 = i I n , ˙ and the function γ t satisfies γ t = − tr (Γ t ) γ t . ˙ 2 Set ϕ := S t + ζ t · ( x − z t ) + Γ t 2 ( x − z t ) · ( x − z t ) .
Wave packets in semiclassical analysis Sketch of proof. Lemma The matrix Γ t satisfies the Ricatti equation Γ t = − V (2) ( z t ) − (Γ t ) 2 , Γ 0 = i I n , ˙ and the function γ t satisfies γ t = − tr (Γ t ) γ t . ˙ 2 Set ϕ := S t + ζ t · ( x − z t ) + Γ t 2 ( x − z t ) · ( x − z t ) . Then i H ( h ) γ t e h ϕ
Wave packets in semiclassical analysis Sketch of proof. Lemma The matrix Γ t satisfies the Ricatti equation Γ t = − V (2) ( z t ) − (Γ t ) 2 , Γ 0 = i I n , ˙ and the function γ t satisfies γ t = − tr (Γ t ) γ t . ˙ 2 Set ϕ := S t + ζ t · ( x − z t ) + Γ t 2 ( x − z t ) · ( x − z t ) . Then � ˙ �� �� � γ t ϕ + ∇ x ϕ · ∇ x ϕ γ t + ∆ ϕ i i H ( h ) γ t e h ϕ γ t e h ϕ = ˙ + V ( x ) − i h 2 2
Wave packets in semiclassical analysis Sketch of proof. Lemma The matrix Γ t satisfies the Ricatti equation Γ t = − V (2) ( z t ) − (Γ t ) 2 , Γ 0 = i I n , ˙ and the function γ t satisfies γ t = − tr (Γ t ) γ t . ˙ 2 Set ϕ := S t + ζ t · ( x − z t ) + Γ t 2 ( x − z t ) · ( x − z t ) . Then � ˙ �� �� � γ t ϕ + ∇ x ϕ · ∇ x ϕ γ t + ∆ ϕ i i H ( h ) γ t e h ϕ γ t e h ϕ = ˙ + V ( x ) − i h 2 2 � � V ( x ) − V ( z t ) − V (1) ( z t ) · ( x − z t ) − V (2) ( z t ) i ( x − z t ) · ( x − z t ) γ t e h ϕ = 2
Wave packets in semiclassical analysis Sketch of proof. Lemma The matrix Γ t satisfies the Ricatti equation Γ t = − V (2) ( z t ) − (Γ t ) 2 , Γ 0 = i I n , ˙ and the function γ t satisfies γ t = − tr (Γ t ) γ t . ˙ 2 Set ϕ := S t + ζ t · ( x − z t ) + Γ t 2 ( x − z t ) · ( x − z t ) . Then � ˙ �� �� � γ t ϕ + ∇ x ϕ · ∇ x ϕ γ t + ∆ ϕ i i H ( h ) γ t e h ϕ γ t e h ϕ = ˙ + V ( x ) − i h 2 2 � � V ( x ) − V ( z t ) − V (1) ( z t ) · ( x − z t ) − V (2) ( z t ) i ( x − z t ) · ( x − z t ) γ t e h ϕ = 2 � | x − z t | 3 � i γ t e h ϕ = O
Wave packets in semiclassical analysis Sketch of proof. Lemma The matrix Γ t satisfies the Ricatti equation Γ t = − V (2) ( z t ) − (Γ t ) 2 , Γ 0 = i I n , ˙ and the function γ t satisfies γ t = − tr (Γ t ) γ t . ˙ 2 Set ϕ := S t + ζ t · ( x − z t ) + Γ t 2 ( x − z t ) · ( x − z t ) . Then � ˙ �� �� � γ t ϕ + ∇ x ϕ · ∇ x ϕ γ t + ∆ ϕ i i H ( h ) γ t e h ϕ γ t e h ϕ = ˙ + V ( x ) − i h 2 2 � � V ( x ) − V ( z t ) − V (1) ( z t ) · ( x − z t ) − V (2) ( z t ) i ( x − z t ) · ( x − z t ) γ t e h ϕ = 2 � | x − z t | 3 � i γ t e h ϕ = O � | x − z t | 3 � i h ϕ h 3 / 2 O γ t e = h 3 / 2
Wave packets on Riemannian manifolds
Wave packets on Riemannian manifolds Goal: to emulate the construction on R n
Wave packets on Riemannian manifolds Goal: to emulate the construction on R n Previous related works: ◮ Construction of quasimodes: by propagating a single wave packet along a closed geodesic (Babich-Lazutkin, Ralston, Paul-Uribe, Nonnenmacher-Eswarathasan...).
Wave packets on Riemannian manifolds Goal: to emulate the construction on R n Previous related works: ◮ Construction of quasimodes: by propagating a single wave packet along a closed geodesic (Babich-Lazutkin, Ralston, Paul-Uribe, Nonnenmacher-Eswarathasan...). Allows to use Fermi coordinates.
Wave packets on Riemannian manifolds Goal: to emulate the construction on R n Previous related works: ◮ Construction of quasimodes: by propagating a single wave packet along a closed geodesic (Babich-Lazutkin, Ralston, Paul-Uribe, Nonnenmacher-Eswarathasan...). Allows to use Fermi coordinates. ◮ More general propagation results: Paul-Uribe, Guillemin-Uribe-Wang: qualitative description of wave packets and their evolutions (for Hamiltonians with non homogeneous symbols).
Wave packets on Riemannian manifolds Goal: to emulate the construction on R n Previous related works: ◮ Construction of quasimodes: by propagating a single wave packet along a closed geodesic (Babich-Lazutkin, Ralston, Paul-Uribe, Nonnenmacher-Eswarathasan...). Allows to use Fermi coordinates. ◮ More general propagation results: Paul-Uribe, Guillemin-Uribe-Wang: qualitative description of wave packets and their evolutions (for Hamiltonians with non homogeneous symbols). General but not so explicit,
Wave packets on Riemannian manifolds Goal: to emulate the construction on R n Previous related works: ◮ Construction of quasimodes: by propagating a single wave packet along a closed geodesic (Babich-Lazutkin, Ralston, Paul-Uribe, Nonnenmacher-Eswarathasan...). Allows to use Fermi coordinates. ◮ More general propagation results: Paul-Uribe, Guillemin-Uribe-Wang: qualitative description of wave packets and their evolutions (for Hamiltonians with non homogeneous symbols). General but not so explicit, using local coordinates
Wave packets on Riemannian manifolds Goal: to emulate the construction on R n Previous related works: ◮ Construction of quasimodes: by propagating a single wave packet along a closed geodesic (Babich-Lazutkin, Ralston, Paul-Uribe, Nonnenmacher-Eswarathasan...). Allows to use Fermi coordinates. ◮ More general propagation results: Paul-Uribe, Guillemin-Uribe-Wang: qualitative description of wave packets and their evolutions (for Hamiltonians with non homogeneous symbols). General but not so explicit, using local coordinates and given for finite times
Wave packets on Riemannian manifolds Goal: to emulate the construction on R n Previous related works: ◮ Construction of quasimodes: by propagating a single wave packet along a closed geodesic (Babich-Lazutkin, Ralston, Paul-Uribe, Nonnenmacher-Eswarathasan...). Allows to use Fermi coordinates. ◮ More general propagation results: Paul-Uribe, Guillemin-Uribe-Wang: qualitative description of wave packets and their evolutions (for Hamiltonians with non homogeneous symbols). General but not so explicit, using local coordinates and given for finite times Motivations and interests: 1. Consider more than the propagation along a single trajectory ⇒ vary ( z , ζ )
Wave packets on Riemannian manifolds Goal: to emulate the construction on R n Previous related works: ◮ Construction of quasimodes: by propagating a single wave packet along a closed geodesic (Babich-Lazutkin, Ralston, Paul-Uribe, Nonnenmacher-Eswarathasan...). Allows to use Fermi coordinates. ◮ More general propagation results: Paul-Uribe, Guillemin-Uribe-Wang: qualitative description of wave packets and their evolutions (for Hamiltonians with non homogeneous symbols). General but not so explicit, using local coordinates and given for finite times Motivations and interests: 1. Consider more than the propagation along a single trajectory ⇒ vary ( z , ζ ) 2. Get an (at most as possible) intrinsinc description of wave packets propagation
Wave packets on Riemannian manifolds Goal: to emulate the construction on R n Previous related works: ◮ Construction of quasimodes: by propagating a single wave packet along a closed geodesic (Babich-Lazutkin, Ralston, Paul-Uribe, Nonnenmacher-Eswarathasan...). Allows to use Fermi coordinates. ◮ More general propagation results: Paul-Uribe, Guillemin-Uribe-Wang: qualitative description of wave packets and their evolutions (for Hamiltonians with non homogeneous symbols). General but not so explicit, using local coordinates and given for finite times Motivations and interests: 1. Consider more than the propagation along a single trajectory ⇒ vary ( z , ζ ) 2. Get an (at most as possible) intrinsinc description of wave packets propagation 3. Get (relatively) explicit approximation of e i tH ( h ) / h as a single integral
Wave packets on Riemannian manifolds Goal: to emulate the construction on R n Previous related works: ◮ Construction of quasimodes: by propagating a single wave packet along a closed geodesic (Babich-Lazutkin, Ralston, Paul-Uribe, Nonnenmacher-Eswarathasan...). Allows to use Fermi coordinates. ◮ More general propagation results: Paul-Uribe, Guillemin-Uribe-Wang: qualitative description of wave packets and their evolutions (for Hamiltonians with non homogeneous symbols). General but not so explicit, using local coordinates and given for finite times Motivations and interests: 1. Consider more than the propagation along a single trajectory ⇒ vary ( z , ζ ) 2. Get an (at most as possible) intrinsinc description of wave packets propagation 3. Get (relatively) explicit approximation of e i tH ( h ) / h as a single integral, without need to go to the universal cover
Wave packets on Riemannian manifolds Goal: to emulate the construction on R n Previous related works: ◮ Construction of quasimodes: by propagating a single wave packet along a closed geodesic (Babich-Lazutkin, Ralston, Paul-Uribe, Nonnenmacher-Eswarathasan...). Allows to use Fermi coordinates. ◮ More general propagation results: Paul-Uribe, Guillemin-Uribe-Wang: qualitative description of wave packets and their evolutions (for Hamiltonians with non homogeneous symbols). General but not so explicit, using local coordinates and given for finite times Motivations and interests: 1. Consider more than the propagation along a single trajectory ⇒ vary ( z , ζ ) 2. Get an (at most as possible) intrinsinc description of wave packets propagation 3. Get (relatively) explicit approximation of e i tH ( h ) / h as a single integral, without need to go to the universal cover, up to | t | ≤ C 0 | log h |
Wave packets on Riemannian manifolds Goal: to emulate the construction on R n Previous related works: ◮ Construction of quasimodes: by propagating a single wave packet along a closed geodesic (Babich-Lazutkin, Ralston, Paul-Uribe, Nonnenmacher-Eswarathasan...). Allows to use Fermi coordinates. ◮ More general propagation results: Paul-Uribe, Guillemin-Uribe-Wang: qualitative description of wave packets and their evolutions (for Hamiltonians with non homogeneous symbols). General but not so explicit, using local coordinates and given for finite times Motivations and interests: 1. Consider more than the propagation along a single trajectory ⇒ vary ( z , ζ ) 2. Get an (at most as possible) intrinsinc description of wave packets propagation 3. Get (relatively) explicit approximation of e i tH ( h ) / h as a single integral, without need to go to the universal cover, up to | t | ≤ C 0 | log h | 4. See e.g. quite explicitly the effect of (negative) curvature
Wave packets on Riemannian manifolds Goal: to emulate the construction on R n Previous related works: ◮ Construction of quasimodes: by propagating a single wave packet along a closed geodesic (Babich-Lazutkin, Ralston, Paul-Uribe, Nonnenmacher-Eswarathasan...). Allows to use Fermi coordinates. ◮ More general propagation results: Paul-Uribe, Guillemin-Uribe-Wang: qualitative description of wave packets and their evolutions (for Hamiltonians with non homogeneous symbols). General but not so explicit, using local coordinates and given for finite times Motivations and interests: 1. Consider more than the propagation along a single trajectory ⇒ vary ( z , ζ ) 2. Get an (at most as possible) intrinsinc description of wave packets propagation 3. Get (relatively) explicit approximation of e i tH ( h ) / h as a single integral, without need to go to the universal cover, up to | t | ≤ C 0 | log h | 4. See e.g. quite explicitly the effect of (negative) curvature 5. ...
Wave packets on Riemannian manifolds Let ( M n , g ) be a Riemannian manifold with bounded geometry
Wave packets on Riemannian manifolds Let ( M n , g ) be a Riemannian manifold with bounded geometry i.e. 1. injectivity radius bounded from below by r 0 > 0
Wave packets on Riemannian manifolds Let ( M n , g ) be a Riemannian manifold with bounded geometry i.e. 1. injectivity radius bounded from below by r 0 > 0 2. all covariant derivatives of the Riemann curvature tensor bounded on M
Wave packets on Riemannian manifolds Let ( M n , g ) be a Riemannian manifold with bounded geometry i.e. 1. injectivity radius bounded from below by r 0 > 0 2. all covariant derivatives of the Riemann curvature tensor bounded on M 3. complete (for simplicity)
Wave packets on Riemannian manifolds Let ( M n , g ) be a Riemannian manifold with bounded geometry i.e. 1. injectivity radius bounded from below by r 0 > 0 2. all covariant derivatives of the Riemann curvature tensor bounded on M 3. complete (for simplicity) Example. Any closed Riemannian manifold
Wave packets on Riemannian manifolds Let ( M n , g ) be a Riemannian manifold with bounded geometry i.e. 1. injectivity radius bounded from below by r 0 > 0 2. all covariant derivatives of the Riemann curvature tensor bounded on M 3. complete (for simplicity) Example. Any closed Riemannian manifold Lemma [Inverse exponential map close to the diagonal of M × M ] If d g ( z , m ) < r 0 , there is a unique W m ∈ T z M such that z � � W m m = exp z . z
Wave packets on Riemannian manifolds Let ( M n , g ) be a Riemannian manifold with bounded geometry i.e. 1. injectivity radius bounded from below by r 0 > 0 2. all covariant derivatives of the Riemann curvature tensor bounded on M 3. complete (for simplicity) Example. Any closed Riemannian manifold Lemma [Inverse exponential map close to the diagonal of M × M ] If d g ( z , m ) < r 0 , there is a unique W m ∈ T z M such that z � � W m m = exp z . z For fixed m, z �→ W m is a vector field z
Wave packets on Riemannian manifolds Let ( M n , g ) be a Riemannian manifold with bounded geometry i.e. 1. injectivity radius bounded from below by r 0 > 0 2. all covariant derivatives of the Riemann curvature tensor bounded on M 3. complete (for simplicity) Example. Any closed Riemannian manifold Lemma [Inverse exponential map close to the diagonal of M × M ] If d g ( z , m ) < r 0 , there is a unique W m ∈ T z M such that z � � W m m = exp z . z For fixed m, z �→ W m is a vector field and one can expand its covariant derivative z ∼ − I + 1 z + 1 ∇ W m 3 R z ( ., W m z ) W m 12 ( ∇ R ) z ( W m z ; ., W m z ) W m z + · · · z
Wave packets on Riemannian manifolds Let ( M n , g ) be a Riemannian manifold with bounded geometry i.e. 1. injectivity radius bounded from below by r 0 > 0 2. all covariant derivatives of the Riemann curvature tensor bounded on M 3. complete (for simplicity) Example. Any closed Riemannian manifold Lemma [Inverse exponential map close to the diagonal of M × M ] If d g ( z , m ) < r 0 , there is a unique W m ∈ T z M such that z � � W m m = exp z . z For fixed m, z �→ W m is a vector field and one can expand its covariant derivative z ∼ − I + 1 z + 1 ∇ W m 3 R z ( ., W m z ) W m 12 ( ∇ R ) z ( W m z ; ., W m z ) W m z + · · · z All tensors in this expansion are bounded
Wave packets on Riemannian manifolds Let ( M n , g ) be a Riemannian manifold with bounded geometry i.e. 1. injectivity radius bounded from below by r 0 > 0 2. all covariant derivatives of the Riemann curvature tensor bounded on M 3. complete (for simplicity) Example. Any closed Riemannian manifold Lemma [Inverse exponential map close to the diagonal of M × M ] If d g ( z , m ) < r 0 , there is a unique W m ∈ T z M such that z � � W m m = exp z . z For fixed m, z �→ W m is a vector field and one can expand its covariant derivative z ∼ − I + 1 z + 1 ∇ W m 3 R z ( ., W m z ) W m 12 ( ∇ R ) z ( W m z ; ., W m z ) W m z + · · · z All tensors in this expansion are bounded (similar result for higher covariant derivatives)
Wave packets on Riemannian manifolds Let ( M n , g ) be a Riemannian manifold with bounded geometry i.e. 1. injectivity radius bounded from below by r 0 > 0 2. all covariant derivatives of the Riemann curvature tensor bounded on M 3. complete (for simplicity) Example. Any closed Riemannian manifold Lemma [Inverse exponential map close to the diagonal of M × M ] If d g ( z , m ) < r 0 , there is a unique W m ∈ T z M such that z � � W m m = exp z . z For fixed m, z �→ W m is a vector field and one can expand its covariant derivative z ∼ − I + 1 z + 1 ∇ W m 3 R z ( ., W m z ) W m 12 ( ∇ R ) z ( W m z ; ., W m z ) W m z + · · · z All tensors in this expansion are bounded (similar result for higher covariant derivatives) Rem: on R n , W m = m − z . z
Wave packets on Riemannian manifolds Consider V ∈ C ∞ ( M , R ) and H ( h ) := − h 2 ∆ g + V 2
Wave packets on Riemannian manifolds Consider V ∈ C ∞ ( M , R ) and H ( h ) := − h 2 ∆ g + V 2 Hamiltonian flow of | ξ | 2 ( z t , ζ t ) = Φ t ( z , ζ ) , m + V ( m ) 2
Wave packets on Riemannian manifolds Consider V ∈ C ∞ ( M , R ) and H ( h ) := − h 2 ∆ g + V 2 Hamiltonian flow of | ξ | 2 ( z t , ζ t ) = Φ t ( z , ζ ) , m + V ( m ) 2 Proposition. Let U be a coordinate patch, with coordinates y 1 , . . . , y n .
Wave packets on Riemannian manifolds Consider V ∈ C ∞ ( M , R ) and H ( h ) := − h 2 ∆ g + V 2 Hamiltonian flow of | ξ | 2 ( z t , ζ t ) = Φ t ( z , ζ ) , m + V ( m ) 2 Proposition. Let U be a coordinate patch, with coordinates y 1 , . . . , y n . Along each trajectory starting at ( z , ζ ) ∈ T ∗ U, one can define intrinsincally Γ t : T z t M C → T z t M C , where T z t M C = T z t M ⊗ C
Wave packets on Riemannian manifolds Consider V ∈ C ∞ ( M , R ) and H ( h ) := − h 2 ∆ g + V 2 Hamiltonian flow of | ξ | 2 ( z t , ζ t ) = Φ t ( z , ζ ) , m + V ( m ) 2 Proposition. Let U be a coordinate patch, with coordinates y 1 , . . . , y n . Along each trajectory starting at ( z , ζ ) ∈ T ∗ U, one can define intrinsincally Γ t : T z t M C → T z t M C , where T z t M C = T z t M ⊗ C (i.e. Γ t is a complex tensor along the curve t �→ z t )
Wave packets on Riemannian manifolds Consider V ∈ C ∞ ( M , R ) and H ( h ) := − h 2 ∆ g + V 2 Hamiltonian flow of | ξ | 2 ( z t , ζ t ) = Φ t ( z , ζ ) , m + V ( m ) 2 Proposition. Let U be a coordinate patch, with coordinates y 1 , . . . , y n . Along each trajectory starting at ( z , ζ ) ∈ T ∗ U, one can define intrinsincally Γ t : T z t M C → T z t M C , where T z t M C = T z t M ⊗ C (i.e. Γ t is a complex tensor along the curve t �→ z t ) which is
Wave packets on Riemannian manifolds Consider V ∈ C ∞ ( M , R ) and H ( h ) := − h 2 ∆ g + V 2 Hamiltonian flow of | ξ | 2 ( z t , ζ t ) = Φ t ( z , ζ ) , m + V ( m ) 2 Proposition. Let U be a coordinate patch, with coordinates y 1 , . . . , y n . Along each trajectory starting at ( z , ζ ) ∈ T ∗ U, one can define intrinsincally Γ t : T z t M C → T z t M C , where T z t M C = T z t M ⊗ C (i.e. Γ t is a complex tensor along the curve t �→ z t ) which is symmetric � � � � Γ t X , Y X , Γ t Y z t = z t , X , Y ∈ T z t M
Wave packets on Riemannian manifolds Consider V ∈ C ∞ ( M , R ) and H ( h ) := − h 2 ∆ g + V 2 Hamiltonian flow of | ξ | 2 ( z t , ζ t ) = Φ t ( z , ζ ) , m + V ( m ) 2 Proposition. Let U be a coordinate patch, with coordinates y 1 , . . . , y n . Along each trajectory starting at ( z , ζ ) ∈ T ∗ U, one can define intrinsincally Γ t : T z t M C → T z t M C , where T z t M C = T z t M ⊗ C (i.e. Γ t is a complex tensor along the curve t �→ z t ) which is symmetric � � � � Γ t X , Y X , Γ t Y z t = z t , X , Y ∈ T z t M has positive definite imaginary part � � Γ t X , X Im z t > 0 , X � = 0 , X ∈ T z t M
Wave packets on Riemannian manifolds Consider V ∈ C ∞ ( M , R ) and H ( h ) := − h 2 ∆ g + V 2 Hamiltonian flow of | ξ | 2 ( z t , ζ t ) = Φ t ( z , ζ ) , m + V ( m ) 2 Proposition. Let U be a coordinate patch, with coordinates y 1 , . . . , y n . Along each trajectory starting at ( z , ζ ) ∈ T ∗ U, one can define intrinsincally Γ t : T z t M C → T z t M C , where T z t M C = T z t M ⊗ C (i.e. Γ t is a complex tensor along the curve t �→ z t ) which is symmetric � � � � Γ t X , Y X , Γ t Y z t = z t , X , Y ∈ T z t M has positive definite imaginary part � � Γ t X , X Im z t > 0 , X � = 0 , X ∈ T z t M and satisfies the Ricatti equation z t Γ t = − Hess ( V ) z t − R z t � z t � z t − � Γ t � 2 ∇ ˙ ., ˙ ˙ where R z t is the Riemann tensor at z t
Wave packets on Riemannian manifolds Proof. To construct Γ t on R n , we have used the natural identifications T ( z ,ζ ) ( T ∗ R n ) = R n ⊕ R n , T ( z t ,ζ t ) ( T ∗ R n ) = R n ⊕ R n
Wave packets on Riemannian manifolds Proof. To construct Γ t on R n , we have used the natural identifications T ( z ,ζ ) ( T ∗ R n ) = R n ⊕ R n , T ( z t ,ζ t ) ( T ∗ R n ) = R n ⊕ R n How to proceed on a manifold ?
Wave packets on Riemannian manifolds Proof. To construct Γ t on R n , we have used the natural identifications T ( z ,ζ ) ( T ∗ R n ) = R n ⊕ R n , T ( z t ,ζ t ) ( T ∗ R n ) = R n ⊕ R n How to proceed on a manifold ? 1. At starting points ( z , ζ ) with z ∈ U , we split T ( z ,ζ ) ( T ∗ M ) ≈ R n y ⊕ R n η using the (symplectic) coordinates ( y 1 , . . . , y n , η 1 , . . . , η n ) on T ∗ U
Wave packets on Riemannian manifolds Proof. To construct Γ t on R n , we have used the natural identifications T ( z ,ζ ) ( T ∗ R n ) = R n ⊕ R n , T ( z t ,ζ t ) ( T ∗ R n ) = R n ⊕ R n How to proceed on a manifold ? 1. At starting points ( z , ζ ) with z ∈ U , we split T ( z ,ζ ) ( T ∗ M ) ≈ R n y ⊕ R n η using the (symplectic) coordinates ( y 1 , . . . , y n , η 1 , . . . , η n ) on T ∗ U 2. At points ( z t , ζ t ), we use the (global) identification I g : T ∗ M → TM
Wave packets on Riemannian manifolds Proof. To construct Γ t on R n , we have used the natural identifications T ( z ,ζ ) ( T ∗ R n ) = R n ⊕ R n , T ( z t ,ζ t ) ( T ∗ R n ) = R n ⊕ R n How to proceed on a manifold ? 1. At starting points ( z , ζ ) with z ∈ U , we split T ( z ,ζ ) ( T ∗ M ) ≈ R n y ⊕ R n η using the (symplectic) coordinates ( y 1 , . . . , y n , η 1 , . . . , η n ) on T ∗ U 2. At points ( z t , ζ t ), we use the (global) identification I g : T ∗ M → TM I g ( z t , ζ t ) = ( z t , ˙ z t )
Wave packets on Riemannian manifolds Proof. To construct Γ t on R n , we have used the natural identifications T ( z ,ζ ) ( T ∗ R n ) = R n ⊕ R n , T ( z t ,ζ t ) ( T ∗ R n ) = R n ⊕ R n How to proceed on a manifold ? 1. At starting points ( z , ζ ) with z ∈ U , we split T ( z ,ζ ) ( T ∗ M ) ≈ R n y ⊕ R n η using the (symplectic) coordinates ( y 1 , . . . , y n , η 1 , . . . , η n ) on T ∗ U 2. At points ( z t , ζ t ), we use the (global) identification I g : T ∗ M → TM I g ( z t , ζ t ) = ( z t , ˙ z t ) and split along horizontal and vertical spaces z t ) ( I g T ∗ M ) = H ( z t , ˙ T ( z t , ˙ z t ) ⊕ V ( z t , ˙ z t )
Wave packets on Riemannian manifolds Proof. To construct Γ t on R n , we have used the natural identifications T ( z ,ζ ) ( T ∗ R n ) = R n ⊕ R n , T ( z t ,ζ t ) ( T ∗ R n ) = R n ⊕ R n How to proceed on a manifold ? 1. At starting points ( z , ζ ) with z ∈ U , we split T ( z ,ζ ) ( T ∗ M ) ≈ R n y ⊕ R n η using the (symplectic) coordinates ( y 1 , . . . , y n , η 1 , . . . , η n ) on T ∗ U 2. At points ( z t , ζ t ), we use the (global) identification I g : T ∗ M → TM I g ( z t , ζ t ) = ( z t , ˙ z t ) and split along horizontal and vertical spaces z t ) ( I g T ∗ M ) = H ( z t , ˙ T ( z t , ˙ z t ) ⊕ V ( z t , ˙ z t ) This gives a natural block decomposition � L A � L B � I g ◦ Φ t � : R n y ⊕ R n d = η → H ( z t , ˙ z t ) ⊕ V ( z t , ˙ z t ) L C L D
Wave packets on Riemannian manifolds Proof (continued). One can then define � − 1 : H C � �� z t ) → V C L C + i L D L A + i L B ( z t , ˙ ( z t , ˙ z t )
Wave packets on Riemannian manifolds Proof (continued). One can then define � − 1 : H C � �� z t ) → V C L C + i L D L A + i L B ( z t , ˙ ( z t , ˙ z t ) and then define Γ t by composition with the natural isomorphisms T z t M C → H C V C z t ) → T z t M C z t ) , ( z t , ˙ ( z t , ˙
Wave packets on Riemannian manifolds Proof (continued). One can then define � − 1 : H C � �� z t ) → V C L C + i L D L A + i L B ( z t , ˙ ( z t , ˙ z t ) and then define Γ t by composition with the natural isomorphisms T z t M C → H C V C z t ) → T z t M C z t ) , ( z t , ˙ ( z t , ˙ More concretely, using local coordinates ( x 1 , . . . , x n ) near z t , the matrix of Γ t reads
Wave packets on Riemannian manifolds Proof (continued). One can then define � − 1 : H C � �� z t ) → V C L C + i L D L A + i L B ( z t , ˙ ( z t , ˙ z t ) and then define Γ t by composition with the natural isomorphisms T z t M C → H C V C z t ) → T z t M C z t ) , ( z t , ˙ ( z t , ˙ More concretely, using local coordinates ( x 1 , . . . , x n ) near z t , the matrix of Γ t reads G − 1 ( C t + i D t )( A t + i B t ) − 1 − G − 1 Σ t
Wave packets on Riemannian manifolds Proof (continued). One can then define � − 1 : H C � �� z t ) → V C L C + i L D L A + i L B ( z t , ˙ ( z t , ˙ z t ) and then define Γ t by composition with the natural isomorphisms T z t M C → H C V C z t ) → T z t M C z t ) , ( z t , ˙ ( z t , ˙ More concretely, using local coordinates ( x 1 , . . . , x n ) near z t , the matrix of Γ t reads G − 1 ( C t + i D t )( A t + i B t ) − 1 − G − 1 Σ t with G − 1 = ( g ij ( x t )) ,
Wave packets on Riemannian manifolds Proof (continued). One can then define � − 1 : H C � �� z t ) → V C L C + i L D L A + i L B ( z t , ˙ ( z t , ˙ z t ) and then define Γ t by composition with the natural isomorphisms T z t M C → H C V C z t ) → T z t M C z t ) , ( z t , ˙ ( z t , ˙ More concretely, using local coordinates ( x 1 , . . . , x n ) near z t , the matrix of Γ t reads G − 1 ( C t + i D t )( A t + i B t ) − 1 − G − 1 Σ t with � G − 1 = ( g ij ( x t )) , x t = x ( z t ) Σ t g kl ( x t )Γ l ij ( x t )˙ x t ij = k , k , l
Wave packets on Riemannian manifolds Proof (continued). One can then define � − 1 : H C � �� z t ) → V C L C + i L D L A + i L B ( z t , ˙ ( z t , ˙ z t ) and then define Γ t by composition with the natural isomorphisms T z t M C → H C V C z t ) → T z t M C z t ) , ( z t , ˙ ( z t , ˙ More concretely, using local coordinates ( x 1 , . . . , x n ) near z t , the matrix of Γ t reads G − 1 ( C t + i D t )( A t + i B t ) − 1 − G − 1 Σ t with � G − 1 = ( g ij ( x t )) , x t = x ( z t ) Σ t g kl ( x t )Γ l ij ( x t )˙ x t ij = k , k , l and � A t � � ∂ x t /∂ y � B t ∂ x t /∂η = C t D t ∂ξ t /∂ y ∂ξ t /∂η
Wave packets on Riemannian manifolds Proof (continued). One can then define � − 1 : H C � �� z t ) → V C L C + i L D L A + i L B ( z t , ˙ ( z t , ˙ z t ) and then define Γ t by composition with the natural isomorphisms T z t M C → H C V C z t ) → T z t M C z t ) , ( z t , ˙ ( z t , ˙ More concretely, using local coordinates ( x 1 , . . . , x n ) near z t , the matrix of Γ t reads G − 1 ( C t + i D t )( A t + i B t ) − 1 − G − 1 Σ t with � G − 1 = ( g ij ( x t )) , x t = x ( z t ) Σ t g kl ( x t )Γ l ij ( x t )˙ x t ij = k , k , l and � A t � � ∂ x t /∂ y � B t ∂ x t /∂η = C t D t ∂ξ t /∂ y ∂ξ t /∂η ⇒ Symmetry of Γ t , =
Wave packets on Riemannian manifolds Proof (continued). One can then define � − 1 : H C � �� z t ) → V C L C + i L D L A + i L B ( z t , ˙ ( z t , ˙ z t ) and then define Γ t by composition with the natural isomorphisms T z t M C → H C V C z t ) → T z t M C z t ) , ( z t , ˙ ( z t , ˙ More concretely, using local coordinates ( x 1 , . . . , x n ) near z t , the matrix of Γ t reads G − 1 ( C t + i D t )( A t + i B t ) − 1 − G − 1 Σ t with � G − 1 = ( g ij ( x t )) , x t = x ( z t ) Σ t g kl ( x t )Γ l ij ( x t )˙ x t ij = k , k , l and � A t � � ∂ x t /∂ y � B t ∂ x t /∂η = C t D t ∂ξ t /∂ y ∂ξ t /∂η ⇒ Symmetry of Γ t , positivity of Im (Γ t ) =
Wave packets on Riemannian manifolds Proof (continued). One can then define � − 1 : H C � �� z t ) → V C L C + i L D L A + i L B ( z t , ˙ ( z t , ˙ z t ) and then define Γ t by composition with the natural isomorphisms T z t M C → H C V C z t ) → T z t M C z t ) , ( z t , ˙ ( z t , ˙ More concretely, using local coordinates ( x 1 , . . . , x n ) near z t , the matrix of Γ t reads G − 1 ( C t + i D t )( A t + i B t ) − 1 − G − 1 Σ t with � G − 1 = ( g ij ( x t )) , x t = x ( z t ) Σ t g kl ( x t )Γ l ij ( x t )˙ x t ij = k , k , l and � A t � � ∂ x t /∂ y � B t ∂ x t /∂η = C t D t ∂ξ t /∂ y ∂ξ t /∂η ⇒ Symmetry of Γ t , positivity of Im (Γ t ) + Ricatti equation by direct computation # =
Wave packets on Riemannian manifolds Proof (continued). One can then define � − 1 : H C � �� z t ) → V C L C + i L D L A + i L B ( z t , ˙ ( z t , ˙ z t ) and then define Γ t by composition with the natural isomorphisms T z t M C → H C V C z t ) → T z t M C z t ) , ( z t , ˙ ( z t , ˙ More concretely, using local coordinates ( x 1 , . . . , x n ) near z t , the matrix of Γ t reads G − 1 ( C t + i D t )( A t + i B t ) − 1 − G − 1 Σ t with � G − 1 = ( g ij ( x t )) , x t = x ( z t ) Σ t g kl ( x t )Γ l ij ( x t )˙ x t ij = k , k , l and � A t � � ∂ x t /∂ y � B t ∂ x t /∂η = C t D t ∂ξ t /∂ y ∂ξ t /∂η ⇒ Symmetry of Γ t , positivity of Im (Γ t ) + Ricatti equation by direct computation # = y n ) are other coordinates on U , the matrix of Γ t is changed into Rem. If (˜ y 1 , . . . , ˜ � − 1 − G − 1 Σ t , G − 1 � ˜ C t + ˜ �� ˜ A t + ˜ D t Z B t Z
Wave packets on Riemannian manifolds Proof (continued). One can then define � − 1 : H C � �� z t ) → V C L C + i L D L A + i L B ( z t , ˙ ( z t , ˙ z t ) and then define Γ t by composition with the natural isomorphisms T z t M C → H C V C z t ) → T z t M C z t ) , ( z t , ˙ ( z t , ˙ More concretely, using local coordinates ( x 1 , . . . , x n ) near z t , the matrix of Γ t reads G − 1 ( C t + i D t )( A t + i B t ) − 1 − G − 1 Σ t with � G − 1 = ( g ij ( x t )) , x t = x ( z t ) Σ t g kl ( x t )Γ l ij ( x t )˙ x t ij = k , k , l and � A t � � ∂ x t /∂ y � B t ∂ x t /∂η = C t D t ∂ξ t /∂ y ∂ξ t /∂η ⇒ Symmetry of Γ t , positivity of Im (Γ t ) + Ricatti equation by direct computation # = y n ) are other coordinates on U , the matrix of Γ t is changed into Rem. If (˜ y 1 , . . . , ˜ � ∂ ˜ � � ∂ ˜ � − 1 ∂ y + i ∂ ˜ η η ∂ y + i ∂ ˜ y y � − 1 − G − 1 Σ t , G − 1 � ˜ C t + ˜ �� ˜ A t + ˜ D t Z B t Z Z = ∂η ∂η
Wave packets on Riemannian manifolds Definition of gaussian wave packets
Wave packets on Riemannian manifolds Definition of gaussian wave packets Let ρ ∈ C ∞ 0 ( − r 0 , r 0 ) , equal to 1 near 0 .
Wave packets on Riemannian manifolds Definition of gaussian wave packets Let ρ ∈ C ∞ 0 ( − r 0 , r 0 ) , equal to 1 near 0 . � � 4 γ 0 exp i z + 1 z ,ζ ( m ) := ( π h ) − n Ψ h ζ · W m 2 � Γ 0 W m z , W m z � z ρ ( d g ( z , m )) , h for m ∈ M and ( z , ζ ) ∈ T ∗ U (i.e. ζ ∈ T ∗ z U) � − 1 γ 0 = det � g jk ( y ( z )) 4
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