Fractal Weyl Laws and Wave Decay for General Trapping

Transcription

1 Fractal Weyl Laws and Wave Decay for General Trapping Jeffrey Galkowski McGill University July 26, 2017 Joint w/ Semyon Dyatlov

2 The Plan The setting and a brief review of scattering resonances Heuristic analysis Fractal Weyl laws and probabilistic wave decay Sketch of the proofs Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

3 The setting Let (M, g) be a manifold with Euclidean ends and g the (negative) Laplacian. Consider ( 2 t g )u = 0, u t=0 = u 0, t u t=0 = u 1. Question What is the long time (t ) behavior of u on compact sets for u 0, u 1 compactly supported? Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

4 Conversion to a stationary problem Taking the Fourier transform û = F t λ u = 0 e iλt u(t, x)dt. Gives for Im λ 1, ( g λ 2 )û = F (λ). So, we study the meromorphic continuation of ( g λ 2 ) 1 : L 2 comp L 2 loc from Im λ > 0. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

5 Scattering resonances Poles of this continuation are called scattering resonances. Characterized by the existence of u 0 0 satisfying ( g λ 2 )u 0 = 0, u 0 is λ outgoing. The solution to the wave equation can often be expanded (at least in odd dimensions) roughly in the form u λ Λ e itλ u λ where Λ denotes the set of scattering resonances. Remark Re λ and Im λ denote respectively the frequency and decay rate of e itλ u λ. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

6 Scattering resonances Question What do the resonances say about the decay of the wave equation in compact sets? Where are they located? Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

7 The Geometry of Trapping The geodesic flow is given by: ϕ t = exp(th p ) : T M \ 0 T M \ 0, p(x, ξ) = ξ g(x), The outgoing/incoming sets are given by: Γ ± := {(x, ξ) T M \ 0 : ϕ t (x, ξ) as t }, The trapped set is given by K := Γ + Γ. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

8 Some examples 1 the trivial example (M, g) is a compact Riemannian manifold without boundary eigenvalues on the real axis. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

9 Some examples 1 the trivial example (M, g) is a compact Riemannian manifold without boundary eigenvalues on the real axis. 2 K has an elliptic closed orbit resonances with Re λ j, Im λ j C N Re λ j N. (Babich Lazutkin, Stefanov, Tang Zworski) Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

10 Some examples 1 the trivial example (M, g) is a compact Riemannian manifold without boundary eigenvalues on the real axis. 2 K has an elliptic closed orbit resonances with Re λ j, Im λ j C N Re λ j N. (Babich Lazutkin, Stefanov, Tang Zworski) 3 K is a hyperbolic closed orbit resonances with Re λ j, Im λ j β (Ikawa, Gerard). under various conditions, none with Re λ C, Im λ c (Nonnenmacher Zworski, Naud, Dyatlov Zahl...) Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

11 Some examples 1 the trivial example (M, g) is a compact Riemannian manifold without boundary eigenvalues on the real axis. 2 K has an elliptic closed orbit resonances with Re λ j, Im λ j C N Re λ j N. (Babich Lazutkin, Stefanov, Tang Zworski) 3 K is a hyperbolic closed orbit resonances with Re λ j, Im λ j β (Ikawa, Gerard). under various conditions, none with Re λ C, Im λ c (Nonnenmacher Zworski, Naud, Dyatlov Zahl...) 4 K = no resonances with Re λ C, Im λ M log Re λ (Vainberg, Tang Zworski) Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

12 What happens when there may not be a gap? Question How many resonances are there close to the real axis? How is this related to wave decay and the trapped set? Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

13 Heuristics wave decay Consider ( ) u(t) = exp it g u 0 =: U(t)u 0. Question: What is the long time behavior of a solution? Energy of u 0 propagates along ϕ t, so escapes in t for (x, ξ) / Γ. on compact sets then, only see mass of u 0 near Γ. for random data, most mass will not be near Γ and so escapes forward in time at some rate. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

14 Heuristics fractal Weyl law Consider a solution (with β Im λ < 0) ( g λ 2 )u 0 = 0, u 0 is λ outgoing. Question: How many can there be? Note U(t)u 0 = e itλ u 0, so u 0 does not escape in finite time (forward or backward). Therefore, u 0 has mass near K. How localized must u 0 be? The number of independent functions concentrated in a set in phase space is controlled by the volume. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

15 The Weyl law on a compact manifold Theorem (Duistermaat Guillemin 75, Hörmander 68) Let (M, g) be a compact manifold without boundary. Then #{λ [0, R]: λ is an eigenvalue } = C M R d + O(R d 1 ). If the Liouville volume of closed trajectories is 0 then #{λ [0, R]: λ is an eigenvalue } = C M R d + o(r d 1 ). Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

16 The Weyl law on a compact manifold Corollary #{λ [R, R + 1]: λ is an eigenvalue } = O(R d 1 ). If the Liouville volume of closed trajectories is 0. #{λ [R, R + 1]: λ is an eigenvalue } cr d 1. Remark Note that Γ + = Γ = T M \ 0, so µ L (K S M) = µ L (S M). Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

17 A weak fractal Weyl law in Strips Define the number of resonances in a strip by: N (R, β) := #{λ [R, R + 1] + i[ β, 0]: λ is a resonance}, β 0. Theorem (Dyatlov G 17) For all β > 0 we have N (R, β) = O(R d 1 ), R. Moreover, if the trapped set K T M \ 0 has Liouville volume zero, N (R, β) = o(r d 1 ), R. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

18 Dynamical neighborhoods of the trapped set Let B M with π(k) B where π : T M \ 0 M is the natural projection. We define the time t neighborhood of Γ by T (t) := π 1 (B) ϕ t (π 1 (B)). The Liouville volume of T (t) is denoted V(t) := µ L (S M T (t)). Remark Note that while these definitions depend on the choice of B, their long time behavior does not. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

19 The Ehrenfest time Maximal expansion rate: smallest Λ max such that dϕ t (x, ξ) = O(e (Λmax+ε) t ) for all ε > 0. Define the Ehrenfest time t e (R) := log R 2Λ max Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

20 Fractal Weyl law Theorem (Dyatlov G 17) For each β 0, 0 < ρ < 1, there exists a constant C > 0 such that [ N (R, β) CR d 1 min V ( ρt e (R) ), e 2βte(R) V ( 2ρt e (R) )]. Remark Flowing up to once the Ehrenfest time comes at no loss, but flowing to twice the Ehrenfest time requires estimating backwards in time which leads to exponential growth when Im λ < 0. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

21 Comparison with previous results Previous work requires the trapped set to be hyperbolic (Sjöstrand 90, Zworski 99, Guillopé Lin Zworski 04, Sjöstrand Zworski 07, Datchev Dyatlov 13, Nonnenmacher Sjöstrand Zworski 14.) N (R, β) = O(R δ+ ), dim box (K S M) = 2δ + 1 Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

22 Comparison with previous results Previous work requires the trapped set to be hyperbolic (Sjöstrand 90, Zworski 99, Guillopé Lin Zworski 04, Sjöstrand Zworski 07, Datchev Dyatlov 13, Nonnenmacher Sjöstrand Zworski 14.) N (R, β) = O(R δ+ ), dim box (K S M) = 2δ + 1 On the other hand R d 1 min [V ( ρt e (R) ), e 2βte(R) V ( 2ρt e (R) )] c min ( R d 1+δ+ 2, R δ+β/λmax+). Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

23 Comparison with previous results Previous work requires the trapped set to be hyperbolic (Sjöstrand 90, Zworski 99, Guillopé Lin Zworski 04, Sjöstrand Zworski 07, Datchev Dyatlov 13, Nonnenmacher Sjöstrand Zworski 14.) N (R, β) = O(R δ+ ), dim box (K S M) = 2δ + 1 Improvements are possible if (M, g) is a manifold of constant negative curvature (Dyatlov 15) corresponding to flowing 2t e (R) forward and 2t e (R) backward (see also Naud 14, Jakobson Naud 16). On the other hand R d 1 min [V ( ρt e (R) ), e 2βte(R) V ( 2ρt e (R) )] c min ( R d 1+δ+ 2, R δ+β/λmax+). Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

24 Trapping with positive escape rate Define the classical escape rate Theorem 3 gives γ := lim sup t 1 log V(t) 0. t N (R, β) = O(R m(β,γ)+ ) d 1 γ β, 0 β γ Λ m(β, γ) := max 2 ; d 1 γ, β γ 2Λ max 2. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

25 Trapping with positive escape rate d 1 m d 1 γ 2Λ max m(β, γ) d 1 γ Λ max γ/2 β Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

26 Examples g = dr 2 + α(r) 2 dθ 2 Figure: α(r) 1 on [ 1, 1], V(t) Ct 1 Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

27 Examples g = dr 2 + α(r) 2 dθ 2 Figure: α(r) 1 + Cr 2n near r = 0, n > 1, V(t) Ct n+1 n 1 Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

28 Examples g = dr 2 + α(r) 2 dθ 2 Figure: α(r) 1 + Cr 2 near r = 0, V(t) e ct Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

29 Randomizing the Initial Data Let B M with smooth boundary, B be the Dirichlet Laplacian on B with respect to the metric g. Let {(e k, λ k )} k=1 an orthonormal basis of eigenfunctions of B L 2 (B) with ( B λ 2 k)e k = 0. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

30 Randomizing the Initial Data For R > 0 consider { } E R := a k e k (x), a k C, I R := {k λ k [R, 2R]} k I R Let u R S R := {u E R : u L 2 = 1} be chosen at random with respect to the standard measure on the sphere. Note: dim E R = cr d + O(R d 1 ). Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

31 Heuristics Relation of fractal Weyl law and decay There are R d eigenvalues in [R, 2R]. There are R d resonances states in [R, 2R] i[0, β]. Most of the energy of u R should miss these resonant states. The larger portion of energy decays faster than e βt. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

32 Probabilistic Wave Decay Theorem (Dyatlov G 17) Suppose that K and ψ Cc (B ). Fix C 0, α > 0, 0 < ρ < 1 Then there exists C > 0 such that for all m C 2, ψu(t)ψu R 2 L 2 mv ( ρ min(t, 2t e (R)) ) for all t [α log R, C 0 R] with probability 1 Ce m/c. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

33 Schematic of the wave decay ψu(t)ψu R 1 2t e (R) t Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

34 Sketch of proof - fractal Weyl law χ C c (T M \ 0; [0, 1]), χ 1 on K S M. χ + t = χ(χ ϕ t ), χ t = χ(χ ϕ t ). Goal: write (with norm estimates on Z, J ) Then Id = Z(ω)( h 2 g ω 2 ) + A(ω) A(ω) = J (ω) Op(χ t e ) Op(χ + t e )χ E ( h 2 g ω 2 h ) + O(h ). ( h 2 g ω 2 ) 1 = (Id A(ω) 2 ) 1 (Id +A(ω))Z(ω). Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

35 Sketch of proof - fractal Weyl law Then ( h 2 g ω 2 ) 1 = (Id A(ω) 2 ) 1 (Id +A(ω))Z(ω). So, need to estimate zeros of det(id A(ω) 2 ). Apply Jensen s inequality together with det(id A(ω) 2 ) exp( A(ω) 2 tr ) exp( A(ω) 2 HS) and similarly for det(id A(ω) 2 ) 1. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

36 Sketch of proof - Parametrix construction If a avoids Γ + S M, then Op(a) = Z(ω)( h 2 g ω 2 ) + O(h ) (1) If b controls a at time t, i.e. ϕ t (supp a) supp(1 b) = Then Op(a) = Z(ω)( h 2 g ω 2 ) + e iωt/h Op(a)U(t) Op(b) + O(h ). (2) Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

37 Sketch of proof - Parametrix construction First goal: Op(χ 2 ) = Z(ω)( h 2 g ω 2 ) + J (ω) Op(χ s ) + O(h ), J (ω) L 2 L 2 C exp(βt). Write χ 2 = χ 2 1 UΓ+ + χ 2 1 UΓ c and estimate χ 2 1 UΓ c by (1). + + Estimate χ 2 1 UΓ+ using (2) with ϕ s (supp χ 2 Γ + ) supp(1 χ s ) =. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

38 Sketch of proof - Parametrix construction Next goal- Estimate χ (and hence χ s ) using χ + t e, i.e. Op(χ) = Z(ω)( h 2 g ω 2 ) + Op(χ + t e ) + O(h ). Apply above argument to χ χ + t (flowing backwards in time) so that Op(χ χ + t e ) = Z(ω)( h 2 g ω 2 ) + J (ω) Op(χ χ + t 0 ) + O(h ). and use that χ χ + t 0 = χ(1 χ ϕ t0 ) is supported away from Γ + to estimate the right hand side. Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

39 Sketch of proof - Parametrix construction Together, Op(χ 2 ) = Z(ω)( h 2 g ω 2 ) + J (ω) Op(χ s ) Op(χ + t e ) + O(h ). Insert energy cutoff Op(χ + t e ) = Z(ω)( h 2 g ω 2 )+Op(χ + t e )χ E ( h 2 g ω 2 h ) +O(h ). Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

40 Sketch of proof - wave decay Note for v R N chosen uniformly at random from the sphere of radius 1 with respect to the L 2 norm, and A : L 2 (R N ) L 2 (R N ), E( Av L 2) = N 1 2 A HS. So, estimate ψu(t)ψχ E ( h 2 g ) HS for ψ Cc (M), ψ 1 on B (we write h = R 1 ). Idea: Insert cutoffs as above ψu(2t)ψχ E ( h 2 g ) = ψu(2t) χe ( h 2 g )ψχ E ( h 2 g ) Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37

41 Sketch of proof - wave decay Note that if (x, ξ) B and ϕ 2t (x, ξ) B, then ϕ t (x, ξ) T (t) T ( t). Insert a cutoff near K ψu(2t)ψχ E ( h 2 g ) = ψχu(t)χ 3 U(t)χ χ E ψχ E Insert U(t)U( t), and apply Egorov s theorm So we have = ψu(t)u( t)χu(t)χ 2 U(t)χU( t)u(t) χ E ψχ E = ψu(t)χ t χ + t U(t) χ E ψχ E + O(h ) ψu(2t)ψχ E ( h 2 g ) HS C χ + t χ t χ E HS Ch d 2 V(2t) 1 2 Jeffrey Galkowski (McGill) Fractal Weyl Laws, Wave Decay July / 37