The Schr odinger propagator for scattering metrics Andrew Hassell - PDF document

The Schr¨ odinger propagator for scattering metrics Andrew Hassell (Australian National University) joint work with Jared Wunsch (Northwestern) MSRI, May 5-9, 2003 http://arXiv.org/math.AP/0301341 1

2 Schr¨ odinger equation for free particle: � � i − 1 ∂ t + 1 � 2∆ ψ = 0 ψ � t =0 = ψ 0 . On curved space: ∆ = ∆ g = ∇ ∗ g ∇ � 0 . For example, on flat R n , the fundamental solu- tion is U ( t, w, x ) = (2 πit ) − n/ 2 e i | x − w | 2 / 2 t . • infinite speed of propagation • lack of decay (unitarity) Fix t > 0 , x . Then u ( t, x, w ) is smooth in w , while u (0 , x, w ) = δ ( w − x ) : singularity disappears instantly. Initial data ψ 0 = e i ( − λx 2 / 2+ ξ · x ) : ψ 0 ∈ C ∞ but ψ ( t, w ) | t = λ − 1 = Cδ ( w + ξ/λ ) . Singularity appears!

3 What happens on curved space? Kapitanski-Safarov (’96): If no trapped geodesics, ψ 0 ∈ E ′ = ⇒ ψ ( t ) ∈ C ∞ for all t > 0. (’98): Parametrix modulo C ∞ ( R n ), but no con- trol at ∞ . Craig-Kappeler-Strauss (’95): Regularity of the solution at all times, and in certain directions, under assumption of regular- ity of ψ 0 along all geodesics lying inside a given spatial cone near infinity. Wunsch (’98): Regularity along certain nontrapped geodesics at particular t > 0 described by “quadratic- scattering wavefront set” of ψ 0 : specifies Directions and times of singularities (location still mysterious).

4 Robbiano-Zuily (’02): Analogue of Wunsch’s result in analytic cate- gory. Burq-G´ erard-Tzvetkov (’01), Staffilani-Tataru (’02): Strichartz estimates. Flat R n with a potential perturbation: vari- ous parametrix constructions: Fujiwara (1980), Zelditch (1983), Treves (1995), Yajima (1996),. . .

5 Contrast with well-developed theory for wave equation . Consider √ ( ∂ t − i ∆) u = 0 , u (0) = u 0 on a compact, boundaryless Riemannian mani- √ fold M. Solution is u ( t ) = e it ∆ u 0 . Let Φ t be geodesic flow on S ∗ X at time t . Theorem (H¨ ormander) √ ∆ is a Fourier integral operator which 1. e it quantizes the contact transformation Φ t . 2. ( x, ˆ ξ ) ∈ WF u 0 iff Φ t ( x, ˆ ξ ) ∈ WF u ( t ) . • Φ t is a contact transformation of the contact manifold S ∗ X with contact one-form ˆ ξ dx . • Singularities travel with unit speed along geodesics, and are neither created nor de- stroyed (time reversibility). • Statement 2 follows immediately from state- ment 1.

6 Back to Schr¨ odinger Goal: construct parametrix, describe regularity of ψ ( t ) . Specific questions: (1) When and where can singularities appear in ψ ( t )? (Describe in terms of initial data.) (2) Where do singularities of initial data in E ′ disappear to? (3) What is the structure of the fundamental solution with initial pole at x ? Questions (1) and (2) are dual. A strong enough answer to (3) will address both.

7 General geometric setup: ( X, g ) a Rie- mannian manifold with ends that look asymp- totically like the large ends of cones (1 , ∞ ) × Y : (= ‘manifold with scattering metric’ as defined by Melrose): g = dr 2 + r 2 h ( r − 1 , y, dy ) h ∈ C ∞ , h 0 ≡ h (0 , y, dy ) a metric on Y. Key example: X = asymptotically Euclid- ian space; r = | x | , y = θ = x/ | x | ∈ S n − 1 : g = (1+2 m r ) dr 2 + r 2 dθ 2 + O ( r − 2 )( dr, rdθ ) , r → ∞ ( We stick with this example for remainder of talk .) Crucial geometric assumption: no trapped geodesics (Or, stay microlocally away from trapping re- gion.)

8 Hamiltonian: H ≡ 1 2∆ g + V ( x ) with V ( x ) ∈ C ∞ ( R n ; R ) having asymptotic ex- pansion: V ( x ) ∼ c r + r − 2 V − 2 ( θ ) + r − 3 V − 3 ( θ ) + . . . ∈ r − 1 C ∞ ( r − 1 , θ ) , for r � r 0 > 0 . • Newtonian gravity (the 1 /r term in the po- tential) and Einsteinian gravity (the dr/r term in the metric) are both OK. Schr¨ odinger equation now reads ( i − 1 ∂ t + H ) ψ = 0 . and we are interested in the kernel of the fun- damental solution, e − itH .

9 Re-examine Euclidean fundamental solution (with V = 0): Let r = | w | , θ = w/ | w | . Then e − itH δ x = (2 πit ) − n/ 2 e i | x − w | 2 / 2 t = e ir 2 / 2 t � ae − i ( rx · θ −| x | 2 / 2) /t � with a = (2 πt ) − n/ 2 . We start at t = 0 with δ x . Study later behavior of solution in r, θ variables ( x, t > 0 , fixed): • The e ir 2 / 2 t term is independent of x : loses all information about location of initial sin- gularity. • The e − i ( rx · θ −| x | 2 / 2) /t term retains informa- tion about location of initial pole in its os- cillation as r → ∞ . • The time t appears in the phases in a very simple way.

10 Use this form as ansatz in more gen- eral geometric setting: divide by the explicit quadratic oscillatory factor e ir 2 / 2 t and try to construct the resulting kernel, which is hopefully only linearly oscillatory. Theorem. Let χ ∈ C ∞ c ( R n ). The fundamen- tal solution is of the form e − itH χ = e ir 2 / 2 t W t χ, where the kernel of W t is a scattering fibered Legendrian (Melrose-Zworski, H.-Vasy). • Inserting the function χ means that we only consider the asymptotics as | w | → ∞ , keep- ing x in a fixed (but arbitrary) compact set. On R t × R n r,θ × R n x , W is a finite sum of terms of the form (0.1) � t − n 2 − k a ( t, r − 1 , θ, x, v ) e iφ ( r − 1 ,θ,x,v ) r/t dv. 2 U ⋐ R k

11 We can state a slightly weaker version of the the- orem more easily by composing with the Fourier transform F : Let W t = e − ir 2 / 2 t e − itH for fixed t > 0. Then F ◦ W t is a Fourier integral operator. To analyze W t further we recall the definition of the scattering wavefront set, which in R n can be specified in terms of the usual wavefront set and the Fourier transform. Let S n − 1 denote the “sphere at infinity” of our ∞ asymptotically Euclidian space. Can identify S ∗ R n ≡ R n × S n − 1 , ∞ R n ≡ S n − 1 × R n . T ∗ S n − 1 Hence exchanging coordinates ( θ, ζ ) → ( ζ, θ ) gives diffeomorphism between these spaces (and ∞ R n a contact structure). gives T ∗ S n − 1

12 Definition. The scattering wavefront set of ∞ R n de- a distribution u is the subset of T ∗ S n − 1 fined by ( θ, ζ ) ∈ WF sc ( u ) iff ( ζ, θ ) ∈ WF( F u ) . (Definition originates with Melrose in more gen- eral setting (scattering metrics).) WF sc measures linear oscillation near infinity. For example, let u ( x ) = e iα · x . Then sc u = { ( θ, α ) : θ ∈ S n − 1 WF ∞ } . Hence e − ir 2 / 2 t e − itH is a “scattering FIO” in- terchanging scattering wavefront set and ordi- nary wavefront set.

13 Question: what is the canonical relation of W t = e − ir 2 / 2 t e − itH ? Let ( x, ˆ ξ ) ∈ S ∗ R n . Let γ ( t ) be geodesic with γ ′ (0) = ˆ γ (0) = x, ξ. Define Φ : S ∗ R n → T ∗ ∞ R n by S n − 1 Φ( x, ˆ ξ ) = ( θ, λθ + µ ) with µ ⊥ θ given by γ ( t ) | γ ( t ) | ∈ S n − 1 θ = lim ∞ t →∞ λ = lim t →∞ t − | γ ( t ) | � γ ( t ) � µ = lim t →∞ | γ | | γ ( t ) | − θ Thus • θ is asymptotic direction; • λ is “sojourn time” (cf. Guillemin) that a particle spends in finite region before head- ing out to ∞ (finite by assumption); • µ measures angle of approach to S n − 1 ∞ .

14 Proposition. Φ is a contact transformation from S ∗ R n to T ∗ ∞ R n . S n − 1 The canonical relation for W : The canonical relation parametrized by W t = e − ir 2 / 2 t e − itH is t − 1 Φ (scaling acts in fiber variable). Special case: x ∈ R n , θ ∈ S n − 1 ∞ , and there exists a unique geodesic γ ( t ) from x to θ (non- degenerate case). Then locally W = ae irS ( x,θ ) /t (no integral required), where S ( x, θ ) = lim t →∞ t − | γ ( t ) | . “ sojourn time .”

15 Euclidean example once more: e − ir 2 / 2 t e − itH δ x = ae i ( − x · θ + O ( r − 1 )) r/t . Sojourn time in R n for line through x in direc- tion θ : S ( x, θ ) = lim t − | x + tθ | = − x · θ, as appears in phase!

16 Egorov Theorem. Let A be a properly sup- ported, zeroth order pseudo on R n . Then A = W t AW ∗ ˜ t is a zeroth order scattering pseudodifferential operator, and σ sc ( ˜ A )(Φ( q )) = σ ( A )( q ) . Propagation Theorem. Let ψ ( t ) = e − itH ψ 0 . Fix a t � = 0. Then ( x, ˆ ξ ) ∈ WF ψ ( t ) iff − 1 sc ( e ir 2 / 2 t ψ 0 ) . t Φ( x, − ˆ ξ ) ∈ WF Thus, we have a characterization of the sin- gularities at nonzero time t in terms of the as- ymptotic behaviour of the initial data. Previ- ous propagation results are immediate conse- quences.

17 Some words on the proof A parametrix for e − itH is constructed as a Legendrian distribution, starting at the diago- nal near t = 0. Here it takes the form U ( w, x, t ) = e i Ψ( w,x ) /t a ( t, w, x ) , a smooth , with Ψ( w, x ) equal to d ( w, x ) 2 / 2. The func- tion Ψ determines a Legendrian submanifold of T ∗ R n × T ∗ R n × R , namely L = { ( w, ζ, x, ξ, τ ) | ζ = d w Ψ , ξ = d x Ψ , τ = Ψ } which is Legendrian with respect to the contact form ζ · dw + ξ · dx − dτ . This Legendrian becomes non-projectable outside the injectivity radius, meaning ( w, x ) are no longer coordinates on it, but it remains perfectly smooth. It may be defined by ( w, ζ ) = exp sg/ 2 ( x, ξ ) , τ = s 2 / 2 , s ∈ (0 , ∞ ) .