Local smoothing and Strichartz estimates for manifolds with degenerate hyperbolic trapping

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1 Local smoothing and Strichartz estimates for manifolds with degenerate hyperbolic trapping H. Christianson partly joint work with J. Wunsch (Northwestern) Department of Mathematics University of North Carolina - Chapel Hill April 23, 2012

2 Outline 1 Introduction: Dispersive estimates and trapping

3 Outline 1 Introduction: Dispersive estimates and trapping 2 Main results

4 Outline 1 Introduction: Dispersive estimates and trapping 2 Main results 3 Sketch of proofs

5 What is dispersion? In R n, solutions to Schrödinger equation: { (D t )u = 0, u(0, x) = u 0,

6 What is dispersion? In R n, solutions to Schrödinger equation: { (D t )u = 0, u(0, x) = u 0, look like e ixξ it ξ 2 û 0 (ξ)dξ

7 What is dispersion? In R n, solutions to Schrödinger equation: { (D t )u = 0, u(0, x) = u 0, look like e ixξ it ξ 2 û 0 (ξ)dξ If û 0 is localized at a frequency ξ 0, stationary phase says x = 2tξ 2tξ 0,

8 What is dispersion? In R n, solutions to Schrödinger equation: { (D t )u = 0, u(0, x) = u 0, look like e ixξ it ξ 2 û 0 (ξ)dξ If û 0 is localized at a frequency ξ 0, stationary phase says x = 2tξ 2tξ 0, so speed depends on frequency.

9 What is dispersion? In R n, solutions to Schrödinger equation: { (D t )u = 0, u(0, x) = u 0, look like e ixξ it ξ 2 û 0 (ξ)dξ If û 0 is localized at a frequency ξ 0, stationary phase says x = 2tξ 2tξ 0, so speed depends on frequency. Practical consequences: on average in time solution is nicer than initial conditions (local smoothing/strichartz estimates).

10 Moral of talk Trapping: if some geodesics do not go to infinity, this doesn t work!

11 Moral of talk Trapping: if some geodesics do not go to infinity, this doesn t work! We will look at some examples which suggest the following:

12 Moral of talk Trapping: if some geodesics do not go to infinity, this doesn t work! We will look at some examples which suggest the following: Conjecture Local smoothing estimates depend quantitatively on the nature of the trapping. Strichartz estimates depend on the dimension of the trapping.

13 What is local smoothing? In R n, Schrödinger propagator e it is unitary on H s spaces (energy conservation).

14 What is local smoothing? In R n, Schrödinger propagator e it is unitary on H s spaces (energy conservation). However, on average in time, and locally in space, propagator is smoothing:

15 What is local smoothing? In R n, Schrödinger propagator e it is unitary on H s spaces (energy conservation). However, on average in time, and locally in space, propagator is smoothing: T 0 x 1/2 ǫ e it 2 u 0 dt C u 0 2 H 1/2 L. 2

16 What is local smoothing? In R n, Schrödinger propagator e it is unitary on H s spaces (energy conservation). However, on average in time, and locally in space, propagator is smoothing: T 0 x 1/2 ǫ e it u 0 2 H 1/2 dt C u 0 2 L 2. First local smoothing result for KdV by Kato ( 83)

17 What is local smoothing? In R n, Schrödinger propagator e it is unitary on H s spaces (energy conservation). However, on average in time, and locally in space, propagator is smoothing: T 0 x 1/2 ǫ e it u 0 2 H 1/2 dt C u 0 2 L 2. First local smoothing result for KdV by Kato ( 83) Generalized for dispersive equations by Sjölin ( 87) and Constantin-Saut ( 88)

18 What is local smoothing? In R n, Schrödinger propagator e it is unitary on H s spaces (energy conservation). However, on average in time, and locally in space, propagator is smoothing: T 0 x 1/2 ǫ e it u 0 2 H 1/2 dt C u 0 2 L 2. First local smoothing result for KdV by Kato ( 83) Generalized for dispersive equations by Sjölin ( 87) and Constantin-Saut ( 88) What about different geometries?

19 What is local smoothing? In R n, Schrödinger propagator e it is unitary on H s spaces (energy conservation). However, on average in time, and locally in space, propagator is smoothing: T 0 x 1/2 ǫ e it u 0 2 H 1/2 dt C u 0 2 L 2. First local smoothing result for KdV by Kato ( 83) Generalized for dispersive equations by Sjölin ( 87) and Constantin-Saut ( 88) What about different geometries? Doi ( 96): H 1/2 smoothing effect manifold is non-trapping (all geodesics escape to infinity)

20 What are Strichartz estimates? Strichartz estimates express a smoothing effect in terms of integrability rather than honest derivatives.

21 What are Strichartz estimates? Strichartz estimates express a smoothing effect in terms of integrability rather than honest derivatives. In R n, the scale-invariant estimate holds: e it u 0 L p ([0,T])L q (R n ) C T u 0 L 2 (R n ), for 2 p + n q = n 2.

22 What are Strichartz estimates? Strichartz estimates express a smoothing effect in terms of integrability rather than honest derivatives. In R n, the scale-invariant estimate holds: for e it u 0 L p ([0,T])L q (R n ) C T u 0 L 2 (R n ), 2 p + n q = n 2. Picture for trapping geometry is much more complicated: you may lose derivatives, but possibly not.

23 Trapping: local smoothing What about trapping geometries? Previously only thin fractal trapped sets were understood:

24 Trapping: local smoothing What about trapping geometries? Previously only thin fractal trapped sets were understood: Doi ( 96): sharp H 1/2 smoothing effect non-trapping

25 Trapping: local smoothing What about trapping geometries? Previously only thin fractal trapped sets were understood: Doi ( 96): sharp H 1/2 smoothing effect non-trapping Burq ( 04): Convex bodies removed from Euclidean space (Ikawa s example), ǫ > 0 loss.

26 Trapping: local smoothing What about trapping geometries? Previously only thin fractal trapped sets were understood: Doi ( 96): sharp H 1/2 smoothing effect non-trapping Burq ( 04): Convex bodies removed from Euclidean space (Ikawa s example), ǫ > 0 loss. C ( 08) One periodic geodesic, boundary value problems, thin fractal sets, ǫ > 0 loss.

27 Trapping: local smoothing What about trapping geometries? Previously only thin fractal trapped sets were understood: Doi ( 96): sharp H 1/2 smoothing effect non-trapping Burq ( 04): Convex bodies removed from Euclidean space (Ikawa s example), ǫ > 0 loss. C ( 08) One periodic geodesic, boundary value problems, thin fractal sets, ǫ > 0 loss. Datchev ( 09): Thin hyperbolic trapped set (different infinite ends ), ǫ > 0 loss.

28 Trapping: local smoothing What about trapping geometries? Previously only thin fractal trapped sets were understood: Doi ( 96): sharp H 1/2 smoothing effect non-trapping Burq ( 04): Convex bodies removed from Euclidean space (Ikawa s example), ǫ > 0 loss. C ( 08) One periodic geodesic, boundary value problems, thin fractal sets, ǫ > 0 loss. Datchev ( 09): Thin hyperbolic trapped set (different infinite ends ), ǫ > 0 loss. Compact manifold, or resonances converging to real axis at exponential rate imply no local smoothing, 1/2 derivative loss.

29 Examples γ Figure: A piece of the catenoid. Figure: R n with two convex bodies removed. γ

30 Trapping: Strichartz estimates Can use local smoothing to glue together Strichartz estimates on semiclassical timescales. This works well in non-trapping situations. In trapping situations:

31 Trapping: Strichartz estimates Can use local smoothing to glue together Strichartz estimates on semiclassical timescales. This works well in non-trapping situations. In trapping situations: Burq-Gérard-Tzvetkov ( 04) and Burq ( 04): Convex bodies removed from Euclidean space (Ikawa s example), 1/p +ǫ > 0 loss, 1/p due to boundary.

32 Trapping: Strichartz estimates Can use local smoothing to glue together Strichartz estimates on semiclassical timescales. This works well in non-trapping situations. In trapping situations: Burq-Gérard-Tzvetkov ( 04) and Burq ( 04): Convex bodies removed from Euclidean space (Ikawa s example), 1/p +ǫ > 0 loss, 1/p due to boundary. C ( 08): One periodic geodesic or thin fractal set without boundary ǫ > 0 loss.

33 Trapping: Strichartz estimates Can use local smoothing to glue together Strichartz estimates on semiclassical timescales. This works well in non-trapping situations. In trapping situations: Burq-Gérard-Tzvetkov ( 04) and Burq ( 04): Convex bodies removed from Euclidean space (Ikawa s example), 1/p +ǫ > 0 loss, 1/p due to boundary. C ( 08): One periodic geodesic or thin fractal set without boundary ǫ > 0 loss. Burq-Guillarmou-Hassell ( 10): using work of Anantharaman, extend parametrix logarithmically leading to lossless estimates.

34 Trapping: Strichartz estimates Can use local smoothing to glue together Strichartz estimates on semiclassical timescales. This works well in non-trapping situations. In trapping situations: Burq-Gérard-Tzvetkov ( 04) and Burq ( 04): Convex bodies removed from Euclidean space (Ikawa s example), 1/p +ǫ > 0 loss, 1/p due to boundary. C ( 08): One periodic geodesic or thin fractal set without boundary ǫ > 0 loss. Burq-Guillarmou-Hassell ( 10): using work of Anantharaman, extend parametrix logarithmically leading to lossless estimates. Burq-Gérard-Tzvetkov ( 04): Compact manifold 1/p loss is sharp.

35 Obvious questions Question 1: Is there something in between ǫ > 0 loss and total (1/2 derivatve) loss?

36 Obvious questions Question 1: Is there something in between ǫ > 0 loss and total (1/2 derivatve) loss? Answer: Yes...

37 Obvious questions Question 1: Is there something in between ǫ > 0 loss and total (1/2 derivatve) loss? Answer: Yes... Can lose 1/2 1/(m+1).

38 Obvious questions Question 1: Is there something in between ǫ > 0 loss and total (1/2 derivatve) loss? Answer: Yes... Can lose 1/2 1/(m+1). Question 2: Is there something between no loss and 1/p loss Strichartz estimates?

39 Obvious questions Question 1: Is there something in between ǫ > 0 loss and total (1/2 derivatve) loss? Answer: Yes... Can lose 1/2 1/(m+1). Question 2: Is there something between no loss and 1/p loss Strichartz estimates? Answer: Yes...

40 Obvious questions Question 1: Is there something in between ǫ > 0 loss and total (1/2 derivatve) loss? Answer: Yes... Can lose 1/2 1/(m+1). Question 2: Is there something between no loss and 1/p loss Strichartz estimates? Answer: Yes... Can lose (n 2)/2n.

41 Degenerate hyperbolic orbits Let X = R x R θ /2πZ, with g = dx 2 +(1+x 2m ) 1/m dθ 2, m Z +. X is asymptotically Euclidean: (1+x 2m ) 1/m x 2 as x

42 Degenerate hyperbolic orbits Let X = R x R θ /2πZ, with g = dx 2 +(1+x 2m ) 1/m dθ 2, m Z +. X is asymptotically Euclidean: (1+x 2m ) 1/m x 2 as x X has one periodic geodesic at min of (1+x 2m ) 1/m, and no other trapping

43 Degenerate hyperbolic orbits Let X = R x R θ /2πZ, with g = dx 2 +(1+x 2m ) 1/m dθ 2, m Z +. X is asymptotically Euclidean: (1+x 2m ) 1/m x 2 as x X has one periodic geodesic at min of (1+x 2m ) 1/m, and no other trapping Min is of order x 2m at x = 0, which is degenerate for m > 1. The Gaussian curvature K 0, K 0 as x ±, K = 0 to order 2m 2 at x = 0.

44 Examples γ Figure: A piece of the manifold X and the periodic geodesic γ. The Gaussian curvature is K = A /A = (2m 1)x 2m 2 (1+x 2m ) 2.

45 The Schrödinger equation on X Theorem (Local Smoothing) Suppose X is as above for m 2, and assume u solves { (D t )u = 0 in R X, u t=0 = u 0 H s for some s m/(m+1). Then for any T <, there exists a constant C > 0 such that T 0 x 3/2 u 2 H 1 (X) dt C( D θ m/(m+1) u 0 2 L 2 + D x 1/2 u 0 2 L 2 ).

46 The Schrödinger equation on X Theorem (Local Smoothing) Suppose X is as above for m 2, and assume u solves { (D t )u = 0 in R X, u t=0 = u 0 H s for some s m/(m+1). Then for any T <, there exists a constant C > 0 such that T 0 x 3/2 u 2 H 1 (X) dt C( D θ m/(m+1) u 0 2 L 2 + D x 1/2 u 0 2 L 2 ). Theorem (Sharpness) Theorem 2 is sharp, and the estimate can be saturated on a (weak) semiclassical time scale.

47 Remarks The case m = 1 was already known in great detail, and the loss there is actually logarithmic.

48 Remarks The case m = 1 was already known in great detail, and the loss there is actually logarithmic. Shows a relationship between the curvature near the trapped set and the local smoothing effect.

49 Remarks The case m = 1 was already known in great detail, and the loss there is actually logarithmic. Shows a relationship between the curvature near the trapped set and the local smoothing effect. The weak semiclassical timescale is a frequency dependent timescale, which also depends on m.

50 Remarks The case m = 1 was already known in great detail, and the loss there is actually logarithmic. Shows a relationship between the curvature near the trapped set and the local smoothing effect. The weak semiclassical timescale is a frequency dependent timescale, which also depends on m. The choice of infinite ends is somewhat irrelevant

51 Remarks The case m = 1 was already known in great detail, and the loss there is actually logarithmic. Shows a relationship between the curvature near the trapped set and the local smoothing effect. The weak semiclassical timescale is a frequency dependent timescale, which also depends on m. The choice of infinite ends is somewhat irrelevant As a byproduct of the proof, we obtain a sharp resolvent estimate with a polynomial loss (next slide).

52 A sharp resolvent estimate Theorem Fix m 2, and assume X = R 2 for x C. For any χ C c (X), there exists a constant C = C m,χ > 0 such that for λ 0, χ( g (λ i0) 2 ) 1 χ L 2 L 2 Cλ 2/(m+1). Moreover, this estimate is sharp, in the sense that no better polynomial rate of decay holds true.

53 n-dimensional extension Smoothing theorem relies on separation of variables to produce 1-dim l semiclassical Schrödinger operator with potential.

54 n-dimensional extension Smoothing theorem relies on separation of variables to produce 1-dim l semiclassical Schrödinger operator with potential. If X = R x S n 1, with g = dx 2 +(1+x 2m ) 1/m g S n 1, m Z +, (or any other compact (n 1)-dim l manifold),

55 n-dimensional extension Smoothing theorem relies on separation of variables to produce 1-dim l semiclassical Schrödinger operator with potential. If X = R x S n 1, with g = dx 2 +(1+x 2m ) 1/m g S n 1, m Z +, (or any other compact (n 1)-dim l manifold), (n 1)-dim l trapped sphere,

56 n-dimensional extension Smoothing theorem relies on separation of variables to produce 1-dim l semiclassical Schrödinger operator with potential. If X = R x S n 1, with g = dx 2 +(1+x 2m ) 1/m g S n 1, m Z +, (or any other compact (n 1)-dim l manifold), (n 1)-dim l trapped sphere, but same theorem holds: T 0 x 3/2 u 2 H 1 (X) dt C( D θ m/(m+1) u 0 2 L 2 + D x 1/2 u 0 2 L 2 ).

57 n-dimensional extension Smoothing theorem relies on separation of variables to produce 1-dim l semiclassical Schrödinger operator with potential. If X = R x S n 1, with g = dx 2 +(1+x 2m ) 1/m g S n 1, m Z +, (or any other compact (n 1)-dim l manifold), (n 1)-dim l trapped sphere, but same theorem holds: T 0 x 3/2 u 2 H 1 (X) dt C( D θ m/(m+1) u 0 2 L 2 + D x 1/2 u 0 2 L 2 ). Proof is exactly the same.

58 Strichartz estimates with trapping Theorem (C ( 11)) For our manifold X, if u 0 H k localized along a single harmonic subspace, for any ǫ > 0, e it u 0 L p T Lq C ǫ D θ (n 2)/2n+ǫ u 0 L 2,

59 Strichartz estimates with trapping Theorem (C ( 11)) For our manifold X, if u 0 H k localized along a single harmonic subspace, for any ǫ > 0, e it u 0 L p T Lq C ǫ D θ (n 2)/2n+ǫ u 0 L 2, provided 2 p + n q = n 2.

60 Remarks ǫ > 0 loss is probably an artifact of the method. Fractional loss is sharp.

61 Remarks ǫ > 0 loss is probably an artifact of the method. Fractional loss is sharp. Suggests that Strichartz estimates depend only on dimension of trapped set, but local smoothing depends only on degeneracy.

62 Remarks ǫ > 0 loss is probably an artifact of the method. Fractional loss is sharp. Suggests that Strichartz estimates depend only on dimension of trapped set, but local smoothing depends only on degeneracy. Fractional loss is (n 2)/2n < 1/2 derivatives, which is always better than the (endpoint) Burq-Gérard-Tzvetkov estimate.

63 The Schrödinger equation in polar coordinates I In R 2 with polar coordinates, have { (D t 2 r r 1 r r 2 2 θ )u = 0, u t=0 = u 0.

64 The Schrödinger equation in polar coordinates I In R 2 with polar coordinates, have { (D t r 2 r 1 r r 2 θ 2 )u = 0, u t=0 = u 0. Using equation, for B self-adjoint, 0 = 2i Im = = T 0 T 0 T 0 B(D t )u, u dt B(D t )u, u u, B(D t )u dt [B, ]u, u dt + i Bu, u T 0.

65 The Schrödinger equation in polar coordinates II If B = arctan(r)d r, last term is bounded using energy estimates by Bu, u T C u 0 2. H 1/2 0

66 The Schrödinger equation in polar coordinates II If B = arctan(r)d r, last term is bounded using energy estimates by Bu, u T C u 0 2. H 1/2 Then 0 i[b,( 2 r r 1 r r 2 2 θ )] = 2 r 2 2 r + 2 arctan rr 3 2 θ + l.o.t.,

67 The Schrödinger equation in polar coordinates II If B = arctan(r)d r, last term is bounded using energy estimates by Bu, u T C u 0 2. H 1/2 Then 0 i[b,( r 2 r 1 r r 2 θ 2 )] = 2 r 2 r arctan rr 3 θ 2 + l.o.t., IBP in r and θ, T 0 ( r 1 r u 2 L 2 + r 1/2 r 1 θ u 2 L 2 )dt C u 0 2 H 1/2.

68 Positive commutator again On X, conjugate to flat problem. With Tu(x,θ) = (1+x 2m ) 1/4m u(x,θ), T : L 2 (dvol) L 2 (dxdθ), := T T 1 = 2 x A 2 (x) 2 θ + l.o.t.,

69 Positive commutator again On X, conjugate to flat problem. With Tu(x,θ) = (1+x 2m ) 1/4m u(x,θ), T : L 2 (dvol) L 2 (dxdθ), := T T 1 = 2 x A 2 (x) 2 θ + l.o.t., Use B = arctan(x) x and commutator idea: [, B] = 2 x 2 2 x + 2A A 3 arctan(x) 2 θ + l.o.t.

70 Positive commutator again On X, conjugate to flat problem. With Tu(x,θ) = (1+x 2m ) 1/4m u(x,θ), T : L 2 (dvol) L 2 (dxdθ), := T T 1 = 2 x A 2 (x) 2 θ + l.o.t., Use B = arctan(x) x and commutator idea: [, B] = 2 x 2 2 x + 2A A 3 arctan(x) 2 θ + l.o.t. Coefficient A arctan(x) vanishes to order 2m at x = 0.

71 Positive commutator again On X, conjugate to flat problem. With Tu(x,θ) = (1+x 2m ) 1/4m u(x,θ), T : L 2 (dvol) L 2 (dxdθ), := T T 1 = 2 x A 2 (x) 2 θ + l.o.t., Use B = arctan(x) x and commutator idea: [, B] = 2 x 2 2 x + 2A A 3 arctan(x) 2 θ + l.o.t. Coefficient A arctan(x) vanishes to order 2m at x = 0. IBP = T 0 ( x 1 x u 2 L 2 + x m x m 3/2 θ u 2 L 2 )dt C u 0 2 H 1/2. (3.1)

72 Estimate near 0 Separate variables: u(t, x,θ) = k e ikθ u k (t, x), and u 0 (x,θ) = k e ikθ u 0,k (x). Estimate on each mode u k.

73 Estimate near 0 Separate variables: u(t, x,θ) = k e ikθ u k (t, x), and u 0 (x,θ) = k e ikθ u 0,k (x). Estimate on each mode u k. Suffices to show T 0 χ(x)ku k 2 L 2 (R) dt C( k m/(m+1) u 0,k 2 L 2 + u 0,k 2 H 1/2 ) for χ C c (R), χ(x) 1 near x = 0.

74 Estimate near 0 Separate variables: u(t, x,θ) = k e ikθ u k (t, x), and u 0 (x,θ) = k e ikθ u 0,k (x). Estimate on each mode u k. Suffices to show T 0 χ(x)ku k 2 L 2 (R) dt C( k m/(m+1) u 0,k 2 L 2 + u 0,k 2 H 1/2 ) for χ C c (R), χ(x) 1 near x = 0. Modulo l.o.t., we have e ikθ u k = (D 2 x + A 2 (x)k 2 )e ikθ u k =: Q k e ikθ u k.

75 Reductions By TT argument, energy cutoff, and Fourier transform t τ, suffices to show cutoff resolvent estimate:

76 Reductions By TT argument, energy cutoff, and Fourier transform t τ, suffices to show cutoff resolvent estimate: ϕ(x, D/ k )(τ + Q) 1 ϕ(x, D/ k ) L 2 x L 2 x Ck 2/(m+1) with ϕ 1 near (0, 0) with compact support.

77 Reductions By TT argument, energy cutoff, and Fourier transform t τ, suffices to show cutoff resolvent estimate: ϕ(x, D/ k )(τ + Q) 1 ϕ(x, D/ k ) L 2 x L 2 x Ck 2/(m+1) with ϕ 1 near (0, 0) with compact support. Rescale: h = k 1 : Q = (hd x ) 2 + A 2 z + l.o.t., z 1 and theorem follows if we can prove:

78 Reductions By TT argument, energy cutoff, and Fourier transform t τ, suffices to show cutoff resolvent estimate: ϕ(x, D/ k )(τ + Q) 1 ϕ(x, D/ k ) L 2 x L 2 x Ck 2/(m+1) with ϕ 1 near (0, 0) with compact support. Rescale: h = k 1 : Q = (hd x ) 2 + A 2 z + l.o.t., z 1 and theorem follows if we can prove: Proposition Let ϕ Φ 0 have wavefront set sufficiently close to (0, 0), and let m 2. Then for each ǫ > 0 sufficiently small, there exists a constant C > 0 such that ϕ(x, hd) Q 1 ϕ(x, hd) L 2 L 2 Ch 2m/m+1, z [1 ǫ, 1+ǫ].

79 Local dynamics Near (0, 0), symbol q of Q has Taylor expansion q ξ 2 1 m x 2m + 1 z, z 1.

80 Local dynamics Near (0, 0), symbol q of Q has Taylor expansion q ξ 2 1 m x 2m + 1 z, z 1. Idea: Escape function. Cook up a function G which increases along the Hamiltonian flow: H q G > 0.

81 Local dynamics Near (0, 0), symbol q of Q has Taylor expansion q ξ 2 1 m x 2m + 1 z, z 1. Idea: Escape function. Cook up a function G which increases along the Hamiltonian flow: H q G > 0. Nondegenerate case (m = 1): q = ξ 2 x 2. Hamiltonian flow: {ẋ = 2ξ, ξ = 2x,

82 Local dynamics Near (0, 0), symbol q of Q has Taylor expansion q ξ 2 1 m x 2m + 1 z, z 1. Idea: Escape function. Cook up a function G which increases along the Hamiltonian flow: H q G > 0. Nondegenerate case (m = 1): q = ξ 2 x 2. Hamiltonian flow: {ẋ = 2ξ, Invariant manifolds: ξ = 2x, (ξ ± x) = ±2(ξ ± x).

83 Stable/Unstable Manifolds Λ ± = {ξ = x} is the stable/unstable manifold. ξ ξ = x x ξ = x Figure: The case non-degenerate hyperbolic case m = 1 with the stable/unstable manifolds ξ = x.

84 Stable/Unstable Manifolds II Idea: G ± = dist 2 (,Λ ± ) = (ξ ± x) 2 satisfies H q G ± = const.g ±. Then G = G G + satisfies H q G = 4(ξ 2 + x 2 ) is symbol of quantum harmonic oscillator (bounded below by h).

85 Local dynamics For m 2, Hamiltonian system for q is {ẋ = 2ξ, ξ = 2x 2m 1,

86 Local dynamics For m 2, Hamiltonian system for q is {ẋ = 2ξ, ξ = 2x 2m 1, which has linearization about zero: {ẋ = 2ξ, ξ = 0.

87 Local dynamics For m 2, Hamiltonian system for q is {ẋ = 2ξ, ξ = 2x 2m 1, which has linearization about zero: {ẋ = 2ξ, degenerate critical point. ξ = 0.

88 Local dynamics For m 2, Hamiltonian system for q is {ẋ = 2ξ, ξ = 2x 2m 1, which has linearization about zero: {ẋ = 2ξ, degenerate critical point. Factorizing q: ξ = 0. (ξ ± m 1/2 x m ) = ±2m 1/2 x m 1 (ξ ± m 1/2 x m ),

89 Weakly stable/unstable manifolds ξ ξ = m 1/2 x m x ξ = m 1/2 x m Figure: The weakly stable/unstable manifolds Λ = {ξ = m 1/2 sgn(x) x m }.

90 Escape function for the degenerate case If G = dist 2 (,Λ ) dist 2 (,Λ + ), then

91 Escape function for the degenerate case If G = dist 2 (,Λ ) dist 2 (,Λ + ), then H q G = H q (4m 1/2 ξsgn(x) x m = 8m 1/2 x m 1 (ξ 2 + x 2m ),

92 Escape function for the degenerate case If G = dist 2 (,Λ ) dist 2 (,Λ + ), then H q G = H q (4m 1/2 ξsgn(x) x m = 8m 1/2 x m 1 (ξ 2 + x 2m ), no longer bounded below.

93 Escape function for the degenerate case If G = dist 2 (,Λ ) dist 2 (,Λ + ), then H q G = H q (4m 1/2 ξsgn(x) x m = 8m 1/2 x m 1 (ξ 2 + x 2m ), no longer bounded below. Homogeneity try same escape function as m = 1:

94 Escape function for the degenerate case If G = dist 2 (,Λ ) dist 2 (,Λ + ), then H q G = H q (4m 1/2 ξsgn(x) x m = 8m 1/2 x m 1 (ξ 2 + x 2m ), no longer bounded below. Homogeneity try same escape function as m = 1: H q (ξx) = 2ξ 2 + 2mx 2m

95 Escape function for the degenerate case If G = dist 2 (,Λ ) dist 2 (,Λ + ), then H q G = H q (4m 1/2 ξsgn(x) x m = 8m 1/2 x m 1 (ξ 2 + x 2m ), no longer bounded below. Homogeneity try same escape function as m = 1: H q (ξx) = 2ξ 2 + 2mx 2m quantum anharmonic oscillator (bounded below by h 2m/(m+1) ).

96 Inhomogeneous blowup calculus Idea: We are working in marginal (h 1/(m+1), h m/(m+1) ) calculus.

97 Inhomogeneous blowup calculus Idea: We are working in marginal (h 1/(m+1), h m/(m+1) ) calculus. Rescaling in phase space (x,ξ) (h 1/(m+1) x, h m/(m+1) ξ) is unitary if we switch from h-calculus to 1-calculus.

98 Inhomogeneous blowup calculus Idea: We are working in marginal (h 1/(m+1), h m/(m+1) ) calculus. Rescaling in phase space (x,ξ) (h 1/(m+1) x, h m/(m+1) ξ) is unitary if we switch from h-calculus to 1-calculus. q q 1 = h 2m/(m+1) (ξ 2 1 m x 2m )

99 Inhomogeneous blowup calculus Idea: We are working in marginal (h 1/(m+1), h m/(m+1) ) calculus. Rescaling in phase space (x,ξ) (h 1/(m+1) x, h m/(m+1) ξ) is unitary if we switch from h-calculus to 1-calculus. q q 1 = h 2m/(m+1) (ξ 2 1 m x 2m ) takes h-dependent operator into scale-invariant calculus

100 Inhomogeneous blowup calculus Idea: We are working in marginal (h 1/(m+1), h m/(m+1) ) calculus. Rescaling in phase space (x,ξ) (h 1/(m+1) x, h m/(m+1) ξ) is unitary if we switch from h-calculus to 1-calculus. q q 1 = h 2m/(m+1) (ξ 2 1 m x 2m ) takes h-dependent operator into scale-invariant calculus Escape function G xξ near (0, 0) (better control away) H q1 G = 2h 2m/(m+1) (ξ 2 +x 2m )

101 Inhomogeneous blowup calculus Idea: We are working in marginal (h 1/(m+1), h m/(m+1) ) calculus. Rescaling in phase space (x,ξ) (h 1/(m+1) x, h m/(m+1) ξ) is unitary if we switch from h-calculus to 1-calculus. q q 1 = h 2m/(m+1) (ξ 2 1 m x 2m ) takes h-dependent operator into scale-invariant calculus Escape function G xξ near (0, 0) (better control away) H q1 G = 2h 2m/(m+1) (ξ 2 +x 2m ) = i[h q1, G]u, u Ch 2m/(m+1) u 2

102 Inhomogeneous blowup calculus Idea: We are working in marginal (h 1/(m+1), h m/(m+1) ) calculus. Rescaling in phase space (x,ξ) (h 1/(m+1) x, h m/(m+1) ξ) is unitary if we switch from h-calculus to 1-calculus. q q 1 = h 2m/(m+1) (ξ 2 1 m x 2m ) takes h-dependent operator into scale-invariant calculus Escape function G xξ near (0, 0) (better control away) H q1 G = 2h 2m/(m+1) (ξ 2 +x 2m ) Proves Proposition. = i[h q1, G]u, u Ch 2m/(m+1) u 2

103 Complex scaling Model operator P = h 2 2 x x 2m near x = 0.

104 Complex scaling Model operator P = h 2 2 x x 2m near x = 0. Complex scaling. Let T be defined: Tu(x) = u(e iθ x), θ [0,π/2). T 1 PTT 1 u = E 0 T 1 u.

105 Complex scaling Model operator P = h 2 2 x x 2m near x = 0. Complex scaling. Let T be defined: Tu(x) = u(e iθ x), θ [0,π/2). T 1 PTT 1 u = E 0 T 1 u. ( e 2iθ h 2 2 x e 2imθ x 2m )w = E 0 w, w = T 1 u. If θ = π/(2m+2), get ( h 2 2 x + x 2m )w = e 2iθ E 0 w.

106 Complex scaling Model operator P = h 2 2 x x 2m near x = 0. Complex scaling. Let T be defined: Tu(x) = u(e iθ x), θ [0,π/2). T 1 PTT 1 u = E 0 T 1 u. ( e 2iθ h 2 x 2 e 2imθ x 2m )w = E 0 w, w = T 1 u. If θ = π/(2m+2), get ( h 2 x 2 + x 2m )w = e 2iθ E 0 w. Anharmonic oscillator e 2iθ E 0 = c 0 h 2m/(m+1), so Im E 0 = αh 2m/(m+1) for some α > 0.

107 Complex scaling Model operator P = h 2 2 x x 2m near x = 0. Complex scaling. Let T be defined: Tu(x) = u(e iθ x), θ [0,π/2). T 1 PTT 1 u = E 0 T 1 u. ( e 2iθ h 2 x 2 e 2imθ x 2m )w = E 0 w, w = T 1 u. If θ = π/(2m+2), get ( h 2 x 2 + x 2m )w = e 2iθ E 0 w. Anharmonic oscillator e 2iθ E 0 = c 0 h 2m/(m+1), so Im E 0 = αh 2m/(m+1) for some α > 0. resonances?

108 Quasimodes Complex WKB quasimodes u satisfying: ( h 2 x 2 x 2 )u = E 0 u +O(h 2m/(m+1) u L 2), (3.2) and supp u { x ǫh 1/(m+1) }.

109 Quasimodes Complex WKB quasimodes u satisfying: ( h 2 2 x x 2 )u = E 0 u +O(h 2m/(m+1) u L 2), (3.2) and supp u { x ǫh 1/(m+1) }. Theorem Let ϕ 0 (x,θ) = e ikθ u(x), with u C c (R) as in (3.2), and h 1 = k > 0. Suppose ψ solves { (D t + )ψ = 0, ψ t=0 = ϕ 0. Then C 0 > 0 independent of k such that k 2/(m+1) /C 0 0 D θ ψ 2 L 2 dt C 1 0 D θ m/(m+1) ϕ 0 2 L 2.

110 Idea of Proof of Strichartz estimates Away from trapping, perfect Strichartz estimates hold.

111 Idea of Proof of Strichartz estimates Away from trapping, perfect Strichartz estimates hold. Near trapping, try to construct parametrix and use stationary phase to compute dispersion

112 Idea of Proof of Strichartz estimates Away from trapping, perfect Strichartz estimates hold. Near trapping, try to construct parametrix and use stationary phase to compute dispersion Rescale and again look at semiclassical Schrödinger operator hd t h 2 2 x + V(x).

113 Idea of Proof of Strichartz estimates Away from trapping, perfect Strichartz estimates hold. Near trapping, try to construct parametrix and use stationary phase to compute dispersion Rescale and again look at semiclassical Schrödinger operator hd t h 2 2 x + V(x). dispersion on h-dependent timescales.

114 Idea of Proof of Strichartz estimates Away from trapping, perfect Strichartz estimates hold. Near trapping, try to construct parametrix and use stationary phase to compute dispersion Rescale and again look at semiclassical Schrödinger operator hd t h 2 2 x + V(x). dispersion on h-dependent timescales. Use local smoothing to glue together Strichartz estimates on semiclassical time scales.

115 Idea of Proof of Strichartz estimates Away from trapping, perfect Strichartz estimates hold. Near trapping, try to construct parametrix and use stationary phase to compute dispersion Rescale and again look at semiclassical Schrödinger operator hd t h 2 2 x + V(x). dispersion on h-dependent timescales. Use local smoothing to glue together Strichartz estimates on semiclassical time scales. Local smoothing has a polynomial loss depending on m

116 Idea of Proof of Strichartz estimates Away from trapping, perfect Strichartz estimates hold. Near trapping, try to construct parametrix and use stationary phase to compute dispersion Rescale and again look at semiclassical Schrödinger operator hd t h 2 2 x + V(x). dispersion on h-dependent timescales. Use local smoothing to glue together Strichartz estimates on semiclassical time scales. Local smoothing has a polynomial loss depending on m need dispersion on timescale polynomially beyond Ehrenfest time.

117 Rough picture ξ ξ = 1 V(x) x ξ = 1 V(x) Figure: The total phase plane. The weakly unstable/stable manifolds are given by Λ ± = {ξ sgn(x) 1 V(x) = 0}.

118 Rough picture ξ ξ = 1 V(x) x ξ = 1 V(x) Figure: The total phase plane. The weakly unstable/stable manifolds are given by Λ ± = {ξ sgn(x) 1 V(x) = 0}. Near x = 0, V 1 x 2m.

119 Phase construction Hamiltonian ODEs for phase construction: {ẋ = 2ξ, so that x(t) = x 0 + 2tξ 0 + ξ = 2mx 2m 1, t 0 (t s)2mx 2m 1 (s)ds.

120 Phase construction Hamiltonian ODEs for phase construction: {ẋ = 2ξ, so that x(t) = x 0 + 2tξ 0 + Need to invert x 0 x(t). ξ = 2mx 2m 1, t 0 (t s)2mx 2m 1 (s)ds.

121 Dyadic decomposition Break up { x ǫ} into O(log(1/h)) regions where x 2 j h 1/(m+1).

122 Dyadic decomposition Break up { x ǫ} into O(log(1/h)) regions where x 2 j h 1/(m+1). Good stationary phase estimates up to time h (1 m)/(m+1) dispersion estimate.

123 Dyadic decomposition Break up { x ǫ} into O(log(1/h)) regions where x 2 j h 1/(m+1). Good stationary phase estimates up to time h (1 m)/(m+1) dispersion estimate. Prove wave packets propagate out of {x 2 j h 1/(m+1) } also in time h (1 m)/(m+1).

124 Dyadic decomposition Break up { x ǫ} into O(log(1/h)) regions where x 2 j h 1/(m+1). Good stationary phase estimates up to time h (1 m)/(m+1) dispersion estimate. Prove wave packets propagate out of {x 2 j h 1/(m+1) } also in time h (1 m)/(m+1). O(h ) cross correlations, at least for timescales h (1 m)/(m+1) ǫ/2.

125 Dyadic decomposition Break up { x ǫ} into O(log(1/h)) regions where x 2 j h 1/(m+1). Good stationary phase estimates up to time h (1 m)/(m+1) dispersion estimate. Prove wave packets propagate out of {x 2 j h 1/(m+1) } also in time h (1 m)/(m+1). O(h ) cross correlations, at least for timescales h (1 m)/(m+1) ǫ/2. Prove approximate solution is close to actual solution for timescales h (1 m)/(m+1)+ǫ/2.

126 Dyadic decomposition Break up { x ǫ} into O(log(1/h)) regions where x 2 j h 1/(m+1). Good stationary phase estimates up to time h (1 m)/(m+1) dispersion estimate. Prove wave packets propagate out of {x 2 j h 1/(m+1) } also in time h (1 m)/(m+1). O(h ) cross correlations, at least for timescales h (1 m)/(m+1) ǫ/2. Prove approximate solution is close to actual solution for timescales h (1 m)/(m+1)+ǫ/2. Sum up h ǫ log(1/h) such approximations to estimate actual solution.

127 Dyadic decomposition Break up { x ǫ} into O(log(1/h)) regions where x 2 j h 1/(m+1). Good stationary phase estimates up to time h (1 m)/(m+1) dispersion estimate. Prove wave packets propagate out of {x 2 j h 1/(m+1) } also in time h (1 m)/(m+1). O(h ) cross correlations, at least for timescales h (1 m)/(m+1) ǫ/2. Prove approximate solution is close to actual solution for timescales h (1 m)/(m+1)+ǫ/2. Sum up h ǫ log(1/h) such approximations to estimate actual solution. Loss of (n 2)/2n comes from L q estimates of spherical harmonics.

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