Resolvent estimates with mild trapping

Transcription

1 Resolvent estimates with mild trapping Jared Wunsch Northwestern University (joint work with: Dean Baskin, Hans Christianson, Emmanuel Schenck, András Vasy, Maciej Zworski) Michael Taylor Birthday Conference July 17, 2012

2 The resolvent Let = be (positive!) Laplacian on R 3. Resolvent is the family of operators 3 2 x 2 j=1 j R(λ) = ( λ 2 ) 1 defined (say) as operators on L 2 for Im λ > 0 (hence λ 2 / [0, ) = spec( )). In this situation, can compute exactly by Fourier transform: Schwartz kernel is R(λ, x, y) = eiλ x y 4π x y.

3 The resolvent Let = be (positive!) Laplacian on R 3. Resolvent is the family of operators 3 2 x 2 j=1 j R(λ) = ( λ 2 ) 1 defined (say) as operators on L 2 for Im λ > 0 (hence λ 2 / [0, ) = spec( )). In this situation, can compute exactly by Fourier transform: Schwartz kernel is R(λ, x, y) = eiλ x y 4π x y.

4 The resolvent Let = be (positive!) Laplacian on R 3. Resolvent is the family of operators 3 2 x 2 j=1 j R(λ) = ( λ 2 ) 1 defined (say) as operators on L 2 for Im λ > 0 (hence λ 2 / [0, ) = spec( )). In this situation, can compute exactly by Fourier transform: Schwartz kernel is R(λ, x, y) = eiλ x y 4π x y.

5 Analytic continuation, perturbation If we now formally take Im λ < 0, we get exponential growth in x y, but we still have an integral kernel for an operator Thus, analytically continues to all of C. L 2 c(r 3 ) L 2 loc (R3 ). R(λ) : L 2 c(r 3 ) L 2 loc (R3 ) Now add some geometry, e.g., Riemannian manifold (X, g), with X = R 3 outside a compact set. Let denote Laplace-Beltrami operator. Then analytic Fredholm theory:

6 Analytic continuation, perturbation If we now formally take Im λ < 0, we get exponential growth in x y, but we still have an integral kernel for an operator Thus, analytically continues to all of C. L 2 c(r 3 ) L 2 loc (R3 ). R(λ) : L 2 c(r 3 ) L 2 loc (R3 ) Now add some geometry, e.g., Riemannian manifold (X, g), with X = R 3 outside a compact set. Let denote Laplace-Beltrami operator. Then analytic Fredholm theory:

7 Analytic continuation, perturbation If we now formally take Im λ < 0, we get exponential growth in x y, but we still have an integral kernel for an operator Thus, analytically continues to all of C. L 2 c(r 3 ) L 2 loc (R3 ). R(λ) : L 2 c(r 3 ) L 2 loc (R3 ) Now add some geometry, e.g., Riemannian manifold (X, g), with X = R 3 outside a compact set. Let denote Laplace-Beltrami operator. Then analytic Fredholm theory:

8 Analytic continuation in general R(λ) = ( λ 2 ) 1 analytically continues from the upper half plane to a meromorphic operator family on C of operators L 2 c(x ) L 2 loc (X ) The poles of this operator in lower half plane are the resonances. In particular, if χ C c (X ) the cutoff resolvent family χr(λ)χ is meromorphic, and we will mainly study this family.

9 Analytic continuation in general R(λ) = ( λ 2 ) 1 analytically continues from the upper half plane to a meromorphic operator family on C of operators L 2 c(x ) L 2 loc (X ) The poles of this operator in lower half plane are the resonances. In particular, if χ C c (X ) the cutoff resolvent family χr(λ)χ is meromorphic, and we will mainly study this family.

10 Applications to wave equation Understanding this operator family has applications to evolution equations: Let U(t) = sin t be wave propagator. Then χu(t)χ = 1 χ(µ ) 1 χ sin t µ dµ 2πi Γ µ = 1 ( χr(λ)χe itλ dλ + χr(λ)χe itλ dλ ) πi R R Γ For t 0 : Can always slide contour in first integral to R + iν (ν > 0) and get exponential decay, O(e νt ). The second integral is trickier!

11 Applications to wave equation, continued Suppose we have: χr(µ + iν)χ analytic for ν 0 ν 0. χr(µ + iν)χ L 2 µ N in this strip. Im λ= ν 0 Then can shift second contour, at cost of some powers of µ or equivalently, powers of and get O(e ( ν 0+ɛ)t ) estimate for the operator χ K U(t)χ, i.e., Exponential decay for local energy (with derivative loss). In flat space, N = 1, and have no derivative loss in the estimates.

12 Applications to wave equation, continued Suppose we have: χr(µ + iν)χ analytic for ν 0 ν 0. χr(µ + iν)χ L 2 µ N in this strip. Im λ= ν 0 Then can shift second contour, at cost of some powers of µ or equivalently, powers of and get O(e ( ν 0+ɛ)t ) estimate for the operator χ K U(t)χ, i.e., Exponential decay for local energy (with derivative loss). In flat space, N = 1, and have no derivative loss in the estimates.

13 Applications to wave equation, continued Suppose we have: χr(µ + iν)χ analytic for ν 0 ν 0. χr(µ + iν)χ L 2 µ N in this strip. Im λ= ν 0 Then can shift second contour, at cost of some powers of µ or equivalently, powers of and get O(e ( ν 0+ɛ)t ) estimate for the operator χ K U(t)χ, i.e., Exponential decay for local energy (with derivative loss). In flat space, N = 1, and have no derivative loss in the estimates.

14 An application to the Schrödinger propagator Suppose we have the estimate for µ R, as on flat R 3. IVP for Schrödinger equation: χr(µ)χ L 2 µ 1 ( i 1 t + )u = 0, u t=0 = u 0. We then have local smoothing i.e. 1 0 χu 2 H 1/2 u 0 2 L 2, e it : L 2 L 2 ([0, 1]; H 1/2 loc (X )). This is of essential importance in dealing with Strichartz estimates, which are used in study of NLS.

15 An application to the Schrödinger propagator Suppose we have the estimate for µ R, as on flat R 3. IVP for Schrödinger equation: χr(µ)χ L 2 µ 1 ( i 1 t + )u = 0, u t=0 = u 0. We then have local smoothing i.e. 1 0 χu 2 H 1/2 u 0 2 L 2, e it : L 2 L 2 ([0, 1]; H 1/2 loc (X )). This is of essential importance in dealing with Strichartz estimates, which are used in study of NLS.

16 Sketch of proof Fourier transform arguments show that if ( i 1 t + )v = χf, with zero initial data, then χu L 2 t H 1/2 x χf L 2 t Hx 1/2, and the original claim follows by a TT argument. (Cf. Constantin-Saut ( 87), Sjölin ( 87), Vega ( 88). On asymptotically Euclidean spaces: Craig-Kappeler-Strauss ( 95) by commutator arguments.)

17 Semiclassical rescaling Since we know meromorphic continuation, the existence of a resonance free strip, and polynomial estimates reduces to question of high energy behavior. Semiclassical rescaling: Set study the operator family λ 2 = z h 2 ; P h (z) = h 2 z and its inverse R h (z) = (h 2 z) 1.

18 Semiclassical rescaling Since we know meromorphic continuation, the existence of a resonance free strip, and polynomial estimates reduces to question of high energy behavior. Semiclassical rescaling: Set study the operator family λ 2 = z h 2 ; P h (z) = h 2 z and its inverse R h (z) = (h 2 z) 1.

19 Then resonance free strip in λ is equivalent to pole-free region in z for χr h (z)χ of form [1 δ, 1 + δ] + i[ ν 0 h, 0] and the free resolvent estimate on the real axis is χr h (z)χ 1 h, z [1 δ, 1 + δ]. 1 δ 1+δ 1 δ iν 0 h 1+δ iν 0 h

20 When do these estimates hold? Let H p denote Hamilton vector field (generator of geodesic flow). Let K S X be the trapped set defined by ρ K C > 0, exp(th p )(ρ) < C t. We say the metric is non-trapping if K =. The classic result: Theorem If the metric is nontrapping then there is a resonance free region Im z ν 0 h log h (hence a resonance free strip in λ-plane) and the free resolvent estimate holds: χr h (z)χ h 1.

21 When do these estimates hold? Let H p denote Hamilton vector field (generator of geodesic flow). Let K S X be the trapped set defined by ρ K C > 0, exp(th p )(ρ) < C t. We say the metric is non-trapping if K =. The classic result: Theorem If the metric is nontrapping then there is a resonance free region Im z ν 0 h log h (hence a resonance free strip in λ-plane) and the free resolvent estimate holds: χr h (z)χ h 1.

22 Further nice results in this simplest case: Get resonance wave expansion (Lax-Phillips, Vainberg). Any region Im z Ch log h is known to contain finitely many resonances. Cf. Morawetz ( 61) Lax-Phillips ( 67), Morawetz-Ralston-Strauss ( 77), Vainberg ( 73); also work of Andersson, Melrose, Sjöstrand, Taylor for necessary propagation of singularities results in boundary case. Partial converse: The estimate χr h (z)χ h 1 fails if K (Ralston ( 69)).

23 Further nice results in this simplest case: Get resonance wave expansion (Lax-Phillips, Vainberg). Any region Im z Ch log h is known to contain finitely many resonances. Cf. Morawetz ( 61) Lax-Phillips ( 67), Morawetz-Ralston-Strauss ( 77), Vainberg ( 73); also work of Andersson, Melrose, Sjöstrand, Taylor for necessary propagation of singularities results in boundary case. Partial converse: The estimate χr h (z)χ h 1 fails if K (Ralston ( 69)).

24 Open questions Main questions: How badly do these estimates fail in presence of trapping? What is rate of energy decay? What kind of Schrödinger local smoothing might we get? If trapping is stable, e.g. elliptic closed trajectory, get very bad failure associated to existence of associated quasimodes. In general, get resonance free region only of form Im z Ce ɛ/h (Burq ( 98)) hence local energy decay as (log t) k with k derivatives of loss.

25 Open questions Main questions: How badly do these estimates fail in presence of trapping? What is rate of energy decay? What kind of Schrödinger local smoothing might we get? If trapping is stable, e.g. elliptic closed trajectory, get very bad failure associated to existence of associated quasimodes. In general, get resonance free region only of form Im z Ce ɛ/h (Burq ( 98)) hence local energy decay as (log t) k with k derivatives of loss.

26 Weak trapping On the other hand, if the trapped set is unstable we get back estimates almost as strong as free ones: Famous example of Ikawa ( 82) shows that unstable (hyperbolic!) trapping between two strictly convex bodies gives resonance-free strip of width ch and resolvent bound O(h N ), hence exponential energy decay for wave equation. Refinement of Tang-Zworski, Burq: R h (λ) log h, λ [1 δ, 1 + δ], h hence local smoothing for Schrödinger holds with ɛ loss ɛ > 0: e it : L 2 L 2 ([0, 1]; H 1/2 ɛ loc ).

27 Weak trapping On the other hand, if the trapped set is unstable we get back estimates almost as strong as free ones: Famous example of Ikawa ( 82) shows that unstable (hyperbolic!) trapping between two strictly convex bodies gives resonance-free strip of width ch and resolvent bound O(h N ), hence exponential energy decay for wave equation. Refinement of Tang-Zworski, Burq: R h (λ) log h, λ [1 δ, 1 + δ], h hence local smoothing for Schrödinger holds with ɛ loss ɛ > 0: e it : L 2 L 2 ([0, 1]; H 1/2 ɛ loc ).

28 Weak trapping On the other hand, if the trapped set is unstable we get back estimates almost as strong as free ones: Famous example of Ikawa ( 82) shows that unstable (hyperbolic!) trapping between two strictly convex bodies gives resonance-free strip of width ch and resolvent bound O(h N ), hence exponential energy decay for wave equation. Refinement of Tang-Zworski, Burq: R h (λ) log h, λ [1 δ, 1 + δ], h hence local smoothing for Schrödinger holds with ɛ loss ɛ > 0: e it : L 2 L 2 ([0, 1]; H 1/2 ɛ loc ).

29 Weak trapping On the other hand, if the trapped set is unstable we get back estimates almost as strong as free ones: Famous example of Ikawa ( 82) shows that unstable (hyperbolic!) trapping between two strictly convex bodies gives resonance-free strip of width ch and resolvent bound O(h N ), hence exponential energy decay for wave equation. Refinement of Tang-Zworski, Burq: R h (λ) log h, λ [1 δ, 1 + δ], h hence local smoothing for Schrödinger holds with ɛ loss ɛ > 0: e it : L 2 L 2 ([0, 1]; H 1/2 ɛ loc ).

30 Similar setting in boundaryless case: hyperbolic cylinder. More generally: K given by single hyperbolic closed geodesic (Colin de Verdière-Parisse ( 94), Christianson ( 07)). A somewhat broader class of interesting geometric examples: normally hyperbolic trapped sets. In this setting, K is a smooth manifold, and dynamics in normal directions to K are extremely unstable. We make the following dynamical assumptions:

31 Similar setting in boundaryless case: hyperbolic cylinder. More generally: K given by single hyperbolic closed geodesic (Colin de Verdière-Parisse ( 94), Christianson ( 07)). A somewhat broader class of interesting geometric examples: normally hyperbolic trapped sets. In this setting, K is a smooth manifold, and dynamics in normal directions to K are extremely unstable. We make the following dynamical assumptions:

32 Define forward- and backward-trapped sets ( stable/unstable manifolds ): Γ ± = { ρ S X : C > 0, exp(th p )(ρ) < C. } Hence K = Γ + Γ, (Figure: Arnold Fiedler, Stefan Liebscher, James C. Alexander.)

33 Assume Γ ± are codimension-one smooth manifolds intersecting transversely at K There exist E ± T K (Γ ± ) such that T K Γ ± = TK E ±, where d exp(th p ) : E ± E ± and there exists θ > 0 such that for all λ < δ, d(exp(th p )(v) Ce θ t v for all v E, ±t 0.

34 These hypotheses are not structurally stable, but do follow (at least up to loss of derivatives) from stronger hypothesis that dynamics be r-normally-hyperbolic for every r in the sense of Hirsch-Pugh-Shub. (The implication and the stability follow from a deep theorem of Hirsch-Pugh-Shub ( 77), Fenichel ( 72).) This geometric setup arises, e.g., in situation of single hyperbolic trapped geodesic (dim K = 1); more interestingly, also when K is photon sphere of trapped orbits in exterior of rotating stationary black hole ( Kerr metric ) (cf. Wunsch-Zworski ( 10)). K

35 These hypotheses are not structurally stable, but do follow (at least up to loss of derivatives) from stronger hypothesis that dynamics be r-normally-hyperbolic for every r in the sense of Hirsch-Pugh-Shub. (The implication and the stability follow from a deep theorem of Hirsch-Pugh-Shub ( 77), Fenichel ( 72).) This geometric setup arises, e.g., in situation of single hyperbolic trapped geodesic (dim K = 1); more interestingly, also when K is photon sphere of trapped orbits in exterior of rotating stationary black hole ( Kerr metric ) (cf. Wunsch-Zworski ( 10)). K

36 Theorem (W.-Zworski ( 10)) In the case when the dynamics of K are normally hyperbolic, we have a resonance-free region [1 δ, 1 + δ] + i[ ν 0 h, 0] in which polynomial resolvent bounds hold; the resolvent estimate on the real axis is log h χr h (z)χ. h (Cf. Gérard-Sjöstrand ( 88) in analytic setting but without polynomial bounds, also G.-S., Nonnenmacher-Zworski ( 12?) for sharp estimate on width of strip.) Consequently, we again have exponential energy decay for the wave equation, and Schrödinger local smoothing with epsilon loss. Proof is by positive commutator estimates using degenerate escape function (in somewhat singular pseudodifferential calculus). See end of talk. 1 1 Time permitting!

37 Fractal trapped set Similar estimates in cases when trapped set is fractal, but sufficiently filamentary, with condition on topological pressure (Nonnenmacher-Zworski ( 09); cf. Anantharaman ( 08), Anantharaman-Nonnenmacher ( 07)). Here we again get exponential energy decay, epsilon loss in resolvent estimate on the real axis with corresponding wave decay and local smoothing (Christianson ( 09)).

38 Fractal trapped set Similar estimates in cases when trapped set is fractal, but sufficiently filamentary, with condition on topological pressure (Nonnenmacher-Zworski ( 09); cf. Anantharaman ( 08), Anantharaman-Nonnenmacher ( 07)). Here we again get exponential energy decay, epsilon loss in resolvent estimate on the real axis with corresponding wave decay and local smoothing (Christianson ( 09)).

39 Really weak trapping An example where trapping is even weaker than hyperbolic is that of diffraction by cone points. Let X have conic singularities. Geodesics circulating among the cone points are in some sense admissible geodesics: they do arise in propagation of singularities for the wave equation but in some sense carry weaker singularities (cf. Cheeger-Taylor ( 82)). Assume: Any geometric geodesic escapes a compact set in time T 0. (I.e., more or less: any geodesic missing the cone points escapes in uniform time.) No three cone points are collinear. No two cone points are conjugate to one another.

40 Really weak trapping An example where trapping is even weaker than hyperbolic is that of diffraction by cone points. Let X have conic singularities. Geodesics circulating among the cone points are in some sense admissible geodesics: they do arise in propagation of singularities for the wave equation but in some sense carry weaker singularities (cf. Cheeger-Taylor ( 82)). Assume: Any geometric geodesic escapes a compact set in time T 0. (I.e., more or less: any geodesic missing the cone points escapes in uniform time.) No three cone points are collinear. No two cone points are conjugate to one another.

41 Really weak trapping An example where trapping is even weaker than hyperbolic is that of diffraction by cone points. Let X have conic singularities. Geodesics circulating among the cone points are in some sense admissible geodesics: they do arise in propagation of singularities for the wave equation but in some sense carry weaker singularities (cf. Cheeger-Taylor ( 82)). Assume: Any geometric geodesic escapes a compact set in time T 0. (I.e., more or less: any geodesic missing the cone points escapes in uniform time.) No three cone points are collinear. No two cone points are conjugate to one another.

42 Really weak trapping An example where trapping is even weaker than hyperbolic is that of diffraction by cone points. Let X have conic singularities. Geodesics circulating among the cone points are in some sense admissible geodesics: they do arise in propagation of singularities for the wave equation but in some sense carry weaker singularities (cf. Cheeger-Taylor ( 82)). Assume: Any geometric geodesic escapes a compact set in time T 0. (I.e., more or less: any geodesic missing the cone points escapes in uniform time.) No three cone points are collinear. No two cone points are conjugate to one another.

43 Theorem (Baskin-W. ( 12)) Here we have the same resolvent estimate as in the non-trapping case: χr h (z)χ 1 h with a resonance free region of width h log h and exponential energy decay, resonance wave expansion. Applies, e.g., to (doubled) exterior of polygon(s) in R 2. Cf. Duyckaerts ( 04) for inverse-square potentials and Burq ( 97) for difraction in R 2 between analytic corner and smooth obstacle.

44 Theorem (Baskin-W. ( 12)) Here we have the same resolvent estimate as in the non-trapping case: χr h (z)χ 1 h with a resonance free region of width h log h and exponential energy decay, resonance wave expansion. Applies, e.g., to (doubled) exterior of polygon(s) in R 2. Cf. Duyckaerts ( 04) for inverse-square potentials and Burq ( 97) for difraction in R 2 between analytic corner and smooth obstacle.

45 Propagation of singularities in conic geometry The proof has two main elements: Weak escape of singularities, a kind of ultra-weak Huygens principle, says that for any s there exists T s such that for t > T s, U(t) : L 2 H s. Idea is that diffracted wave is conormal, and more regular than incident wave if incident wave is nonfocused. Cf. Cheeger-Taylor ( 82), Melrose-Wunsch ( 04). Microlocalizing along trapped trajectory, we must diffract through a cone point at least once every T 0 seconds. First diffraction gives a conormal wave and second diffraction gives increase in regularity (by nonconjugacy hypothesis)... Second part of proof is modification of argument of Vainberg ( 73) (cf. also Lax-Phillips ( 64)) that uses escape of singularities to obtain a logarithmic region free of resonances.

46 Propagation of singularities in conic geometry The proof has two main elements: Weak escape of singularities, a kind of ultra-weak Huygens principle, says that for any s there exists T s such that for t > T s, U(t) : L 2 H s. Idea is that diffracted wave is conormal, and more regular than incident wave if incident wave is nonfocused. Cf. Cheeger-Taylor ( 82), Melrose-Wunsch ( 04). Microlocalizing along trapped trajectory, we must diffract through a cone point at least once every T 0 seconds. First diffraction gives a conormal wave and second diffraction gives increase in regularity (by nonconjugacy hypothesis)... Second part of proof is modification of argument of Vainberg ( 73) (cf. also Lax-Phillips ( 64)) that uses escape of singularities to obtain a logarithmic region free of resonances.

47 Propagation of singularities in conic geometry The proof has two main elements: Weak escape of singularities, a kind of ultra-weak Huygens principle, says that for any s there exists T s such that for t > T s, U(t) : L 2 H s. Idea is that diffracted wave is conormal, and more regular than incident wave if incident wave is nonfocused. Cf. Cheeger-Taylor ( 82), Melrose-Wunsch ( 04). Microlocalizing along trapped trajectory, we must diffract through a cone point at least once every T 0 seconds. First diffraction gives a conormal wave and second diffraction gives increase in regularity (by nonconjugacy hypothesis)... Second part of proof is modification of argument of Vainberg ( 73) (cf. also Lax-Phillips ( 64)) that uses escape of singularities to obtain a logarithmic region free of resonances.

48 Propagation of singularities in conic geometry The proof has two main elements: Weak escape of singularities, a kind of ultra-weak Huygens principle, says that for any s there exists T s such that for t > T s, U(t) : L 2 H s. Idea is that diffracted wave is conormal, and more regular than incident wave if incident wave is nonfocused. Cf. Cheeger-Taylor ( 82), Melrose-Wunsch ( 04). Microlocalizing along trapped trajectory, we must diffract through a cone point at least once every T 0 seconds. First diffraction gives a conormal wave and second diffraction gives increase in regularity (by nonconjugacy hypothesis)... Second part of proof is modification of argument of Vainberg ( 73) (cf. also Lax-Phillips ( 64)) that uses escape of singularities to obtain a logarithmic region free of resonances.

49 Bigger losses in resolvent estimates Does there exist a situation where loss in estimate in on χr h (z)χ is polynomial (unlike in case of stable closed orbits) but greater than logarithmic? Yes: occurs in case of degenerate hyperbolic trapping, e.g. X is locally X = ( 1, 1) x Sθ 1, equipped with the metric ds 2 = dx 2 + (1 + x 2m )dθ 2 with m 2. Theorem (Christianson-W. ( 10)) In this situation, χr h (λ)χ h 2m/(m+1), λ [1 δ, 1 + δ] and this estimate is sharp. Thus local smoothing holds as follows: e it : L 2 L 2 ([0, 1]; H 1/(m+1) loc ) (and polynomial energy decay for wave equation).

50 Bigger losses in resolvent estimates Does there exist a situation where loss in estimate in on χr h (z)χ is polynomial (unlike in case of stable closed orbits) but greater than logarithmic? Yes: occurs in case of degenerate hyperbolic trapping, e.g. X is locally X = ( 1, 1) x Sθ 1, equipped with the metric ds 2 = dx 2 + (1 + x 2m )dθ 2 with m 2. Theorem (Christianson-W. ( 10)) In this situation, χr h (λ)χ h 2m/(m+1), λ [1 δ, 1 + δ] and this estimate is sharp. Thus local smoothing holds as follows: e it : L 2 L 2 ([0, 1]; H 1/(m+1) loc ) (and polynomial energy decay for wave equation).

51 The damped wave equation We now describe a problem that involves resolvent estimates of a nominally different type than those considered here: the damped wave equation. Let (X, g) be a compact, connected Riemannian manifold and a 0 a smooth function. Damped wave equation: { ( 2 t + + a(x) t ) u(x, t) = 0, u(x, 0) = u 0 H 1 (X ), t u(x, 0) = u 1 H 0 (X ). Lebeau ( 93): norm of solution decays to 0 (logarithmically!) if a > 0 somewhere. Let N = {ρ S X : t R, a exp(th p )(ρ) = 0} ( undamped set analogous to trapped set K above). If N = ( geometric control ) get exponential decay of solutions (Rauch-Taylor ( 74), Bardos-Lebeau-Rauch ( 92)).

52 The damped wave equation We now describe a problem that involves resolvent estimates of a nominally different type than those considered here: the damped wave equation. Let (X, g) be a compact, connected Riemannian manifold and a 0 a smooth function. Damped wave equation: { ( 2 t + + a(x) t ) u(x, t) = 0, u(x, 0) = u 0 H 1 (X ), t u(x, 0) = u 1 H 0 (X ). Lebeau ( 93): norm of solution decays to 0 (logarithmically!) if a > 0 somewhere. Let N = {ρ S X : t R, a exp(th p )(ρ) = 0} ( undamped set analogous to trapped set K above). If N = ( geometric control ) get exponential decay of solutions (Rauch-Taylor ( 74), Bardos-Lebeau-Rauch ( 92)).

53 The damped wave equation We now describe a problem that involves resolvent estimates of a nominally different type than those considered here: the damped wave equation. Let (X, g) be a compact, connected Riemannian manifold and a 0 a smooth function. Damped wave equation: { ( 2 t + + a(x) t ) u(x, t) = 0, u(x, 0) = u 0 H 1 (X ), t u(x, 0) = u 1 H 0 (X ). Lebeau ( 93): norm of solution decays to 0 (logarithmically!) if a > 0 somewhere. Let N = {ρ S X : t R, a exp(th p )(ρ) = 0} ( undamped set analogous to trapped set K above). If N = ( geometric control ) get exponential decay of solutions (Rauch-Taylor ( 74), Bardos-Lebeau-Rauch ( 92)).

54 The damped wave equation We now describe a problem that involves resolvent estimates of a nominally different type than those considered here: the damped wave equation. Let (X, g) be a compact, connected Riemannian manifold and a 0 a smooth function. Damped wave equation: { ( 2 t + + a(x) t ) u(x, t) = 0, u(x, 0) = u 0 H 1 (X ), t u(x, 0) = u 1 H 0 (X ). Lebeau ( 93): norm of solution decays to 0 (logarithmically!) if a > 0 somewhere. Let N = {ρ S X : t R, a exp(th p )(ρ) = 0} ( undamped set analogous to trapped set K above). If N = ( geometric control ) get exponential decay of solutions (Rauch-Taylor ( 74), Bardos-Lebeau-Rauch ( 92)).

55 Damped resolvent To understand issue of decay rates more generally, employ FT and semiclassical rescaling: need to study the inverse of an operator close to P h (z) = h 2 g + iha z. (Note: a only appears in subprincipal semiclassical symbol.) Again, a pole free region z [1 δ, 1 + δ] + i[ ν 0 h, 0] together with polynomial estimates on the inverse gives exponential decay. Smaller region: various subexponential decay rates possible.

56 Damped resolvent To understand issue of decay rates more generally, employ FT and semiclassical rescaling: need to study the inverse of an operator close to P h (z) = h 2 g + iha z. (Note: a only appears in subprincipal semiclassical symbol.) Again, a pole free region z [1 δ, 1 + δ] + i[ ν 0 h, 0] together with polynomial estimates on the inverse gives exponential decay. Smaller region: various subexponential decay rates possible.

57 Damping resolvent estimates Some known estimates on width of resonance free region and subexponential decay: single hyperbolic orbit (Christianson ( 07)), fractal trapped set with pressure condition (Schenck ( 10)),... Turns out that a certain amount (indeed quite a lot) can be obtained just from the dynamics near N ; if we know a polynomial resolvent estimate for a noncompact manifold X isometric to X in a neighborhood of π(n )\ supp a and with Euclidean ends, with N = K, then we know an estimate on P h (z) 1 sufficiently near the real axis, i.e. have a gluing theorem.

58 Damping resolvent estimates Some known estimates on width of resonance free region and subexponential decay: single hyperbolic orbit (Christianson ( 07)), fractal trapped set with pressure condition (Schenck ( 10)),... Turns out that a certain amount (indeed quite a lot) can be obtained just from the dynamics near N ; if we know a polynomial resolvent estimate for a noncompact manifold X isometric to X in a neighborhood of π(n )\ supp a and with Euclidean ends, with N = K, then we know an estimate on P h (z) 1 sufficiently near the real axis, i.e. have a gluing theorem.

59 a>0 a>0 N The compact manifold X with damping K The noncompact manifold X

60 Gluing theorem Let R h (z) denote the resolvent on X, our noncompact model. Theorem (Christianson-Schenck-Vasy-W. ( 12)) Assume there is α(h) h N such that for z [1 δ, 1 + δ]. Then R h (z) L 2 L 2 α(h) h, (h 2 g + iha z) 1 L 2 L 2 C α(h) h, for z [1 δ, 1 + δ] + i[ c 0, c 0 ]h/α(h). (Cf. Sjöstrand-Zworski ( 91), Burq-Zworski ( 04) for black box set-up, Datchev-Vasy ( 10, 12) for resolvent estimates through trapping and gluing.)

61 Applications Applications in all the examples above (normally hyperbolic trapping, fractal hyperbolic trapping, degenerate hyperbolic trapping) giving various subexponential decay rates. Sharp in some cases (e.g. surface of revolution with single undamped orbit Burq-Christianson ( 12)). Interesting open questions remain concerning hypotheses on the global dynamics which give stronger estimates.

62 Applications Applications in all the examples above (normally hyperbolic trapping, fractal hyperbolic trapping, degenerate hyperbolic trapping) giving various subexponential decay rates. Sharp in some cases (e.g. surface of revolution with single undamped orbit Burq-Christianson ( 12)). Interesting open questions remain concerning hypotheses on the global dynamics which give stronger estimates.

63 How to prove resolvent estimates? Proofs of resolvent estimates above are by commutator methods, i.e. microlocal energy estimates. To see how these work, say we are in the non-trapping setting. First, we switch to a problem elliptic near infinity in one of two ways: Complex scale, or, more simply Add a complex absorbing potential iw vanishing in a large compact set F but equal to i on Euclidean end. Note similarity of absorbing potential to damping term, but no factor of h! (For equivalence, cf. Datchev-Vasy ( 10).)

64 How to prove resolvent estimates? Proofs of resolvent estimates above are by commutator methods, i.e. microlocal energy estimates. To see how these work, say we are in the non-trapping setting. First, we switch to a problem elliptic near infinity in one of two ways: Complex scale, or, more simply Add a complex absorbing potential iw vanishing in a large compact set F but equal to i on Euclidean end. Note similarity of absorbing potential to damping term, but no factor of h! (For equivalence, cf. Datchev-Vasy ( 10).)

65 How to prove resolvent estimates? Proofs of resolvent estimates above are by commutator methods, i.e. microlocal energy estimates. To see how these work, say we are in the non-trapping setting. First, we switch to a problem elliptic near infinity in one of two ways: Complex scale, or, more simply Add a complex absorbing potential iw vanishing in a large compact set F but equal to i on Euclidean end. Note similarity of absorbing potential to damping term, but no factor of h! (For equivalence, cf. Datchev-Vasy ( 10).)

66 Escape function By non-trapping there is an escape function a 0 C c such that H p (a 0 ) = b 2 + e where b = 1 on F, e is supported where iw = i. Set a = a 0 + δ 1 and A = a w (x, hd). For λ R let P = h 2 λ + iw and note that 2 Im Au, Pu = i (P A AP)u, u = i(p P )Au, u + i[p, A]u, u = 2 WAu, u h Bu 2 + h Eu, u + O(h 2 ) u 2.

67 For δ small, the real part of WAu, u dominates hence rearranging gives W u 2 ɛh u 2, h Bu 2 + W u 2 ɛh u 2 + Ch Eu, u + C Pu Au. Since LHS then dominates h u 2, can absorb first term on right as well absorbing E term into W to get u h 1 Pu, for h sufficiently small, as desired.

68 What if there is trapping? In case of trapping, there can be no escape function! As a substitute, we seek a such that H p (a) 0 with vanishing only on K, the trapped set, in a controlled way. Optimal choices tend to be somewhat singular, hence to live in slightly exotic calculi. For instance in case of normally hyperbolic trapping, let ϕ ± be defining functions for Γ ±, the stable/unstable manifolds. Then let a = log ϕ2 (ρ) + h/ h ϕ 2 + (ρ) + h/ h with h a secondary small parameter. The commutator then has transverse harmonic-oscillator type lower bounds. Sharp estimate on width of resonance-free strip relies on method of Bony-Chemin ( 94).

69 What if there is trapping? In case of trapping, there can be no escape function! As a substitute, we seek a such that H p (a) 0 with vanishing only on K, the trapped set, in a controlled way. Optimal choices tend to be somewhat singular, hence to live in slightly exotic calculi. For instance in case of normally hyperbolic trapping, let ϕ ± be defining functions for Γ ±, the stable/unstable manifolds. Then let a = log ϕ2 (ρ) + h/ h ϕ 2 + (ρ) + h/ h with h a secondary small parameter. The commutator then has transverse harmonic-oscillator type lower bounds. Sharp estimate on width of resonance-free strip relies on method of Bony-Chemin ( 94).

70 Similar strategy in degenerate hyperbolic case, following an inhomogeneous semiclassical rescaling of phase space x x h 1/(m+1), ξ ξ h m/(m+1).

71 Damped resolvent In dealing with resolvent with damping, i.e. inverse of P = (h 2 + iha λ), Let P 1 = (h 2 + iw λ), be the complex-scaled noncompact model operator.

72 Let ϕ 0, ϕ 1 be partition of unity with ϕ 0 equal to 1 in a neighborhood of π(n ) and supported where P = P 1. Let ϕ 0 ψ. Then ϕ 0 u 2 = P 1 1 P 1ϕ 0 u 2 = P 1 1 Pϕ 0u 2 = P 1 1 (ϕ 0Pu + [P, ϕ 0 ]u) 2 α(h)2 h 2 Pu 2 + α(h) ψu 2 Presence of α(h) rather than α(h) 2 in second term is an essential new ingredient, derived from commutator estimates of Datchev-Vasy ( 12) which give improved estimates for cutoff resolvents away from the trapped set (cf. Burq ( 02), Cardoso-Vodev ( 02)). (Employs commutant equal to 1 on trapping.)

73 Now propagation of singularities: since ψ is supported away from N, and α(h) ψu 2 α(h) h 2 Pu 2 + α(h) χu 2 + O(h ) u 2. ϕ 1 u 2 1 h 2 Pu 2 + χu 2 + O(h ) u 2, where χ supported on {a δ > 0}. Adding together the estimates for ϕ 0 u and ϕ 1 u we find u 2 α(h)2 h 2 Pu 2 + α(h) χu 2 + O(h ) u 2. Since a δ on χ : χu 2 au, u = h 1 Im Pu, u h 1 Pu u.

74 Now propagation of singularities: since ψ is supported away from N, and α(h) ψu 2 α(h) h 2 Pu 2 + α(h) χu 2 + O(h ) u 2. ϕ 1 u 2 1 h 2 Pu 2 + χu 2 + O(h ) u 2, where χ supported on {a δ > 0}. Adding together the estimates for ϕ 0 u and ϕ 1 u we find u 2 α(h)2 h 2 Pu 2 + α(h) χu 2 + O(h ) u 2. Since a δ on χ : χu 2 au, u = h 1 Im Pu, u h 1 Pu u.

75 Hence u 2 α(h)2 h 2 and Cauchy-Schwarz yields Pu 2 + α(h) h Pu u, u α(h) h Pu.