Gluing semiclassical resolvent estimates via propagation of singularities

Transcription

1 Gluing semiclassical resolvent estimates via propagation of singularities Kiril Datchev (MIT) and András Vasy (Stanford) Postdoc Seminar MSRI Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

2 Propagators for the wave equation The Schrödinger equation on a complete Riemannian manifold (X,g) i t u = g u, u(x,0) = u 0 (x), is solved by u(t) = e it g u 0. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

3 Propagators for the wave equation The Schrödinger equation on a complete Riemannian manifold (X,g) i t u = g u, u(x,0) = u 0 (x), is solved by The wave equation u(t) = e it g u 0. 2 t u = g u, u(x,0) = u 0 (x), t u(x,0) = u 1 (x), by u(t) = cos(t g )u 0 + sin( t g ) g u 1. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

4 Propagators in terms of resolvents Such propagators can be written in terms of resolvents using Stone s formula. e it g = 1 e itλ [ ( g λ i0) 1 ( g λ + i0) 1] dλ 2πi 0 Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

5 Propagators in terms of resolvents Such propagators can be written in terms of resolvents using Stone s formula. e it g = 1 e itλ [ ( g λ i0) 1 ( g λ + i0) 1] dλ 2πi 0 If H = g +V obeys H > α 2 Id in the sense of operators, then sint H +iα = e iλt (H λ 2 ) 1 dλ. H +iα Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

6 Propagators in terms of resolvents Such propagators can be written in terms of resolvents using Stone s formula. e it g = 1 e itλ [ ( g λ i0) 1 ( g λ + i0) 1] dλ 2πi 0 If H = g +V obeys H > α 2 Id in the sense of operators, then sint H +iα = e iλt (H λ 2 ) 1 dλ. H +iα These formulas can be used to deduce propagator estimates from resolvent estimates. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

7 Resolvent estimates and propagators: local smoothing Uniform high energy resolvent estimates up to the spectrum imply local smoothing. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

8 Resolvent estimates and propagators: local smoothing Uniform high energy resolvent estimates up to the spectrum imply local smoothing. uniformly estimate up to here λ 0 spectrum χ( g λ iε) 1 χ L 2 L 2 C λ uniformly for λ λ 0, ε 0 Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

9 Resolvent estimates and propagators: local smoothing Uniform high energy resolvent estimates up to the spectrum imply local smoothing. uniformly estimate up to here λ 0 spectrum χ( g λ iε) 1 χ L 2 L 2 C λ uniformly for λ λ 0, ε 0 = χe it g L 2 L 2 [0,T ]H 1/2 X C. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

10 Resolvent estimates and propagators: local smoothing A loss in the resolvent estimate gives local smoothing with loss. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

11 Resolvent estimates and propagators: local smoothing A loss in the resolvent estimate gives local smoothing with loss. χ( g λ iε) 1 χ L 2 L 2 C logλ λ uniformly for λ λ 0, ε 0 Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

12 Resolvent estimates and propagators: local smoothing A loss in the resolvent estimate gives local smoothing with loss. χ( g λ iε) 1 χ L 2 L 2 C logλ λ uniformly for λ λ 0, ε 0 = χe it g L 2 L 2 [0,T ]H 1/2 δ X C. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

13 Resolvent estimates and propagators: local smoothing A loss in the resolvent estimate gives local smoothing with loss. χ( g λ iε) 1 χ L 2 L 2 C logλ λ uniformly for λ λ 0, ε 0 = χe it g L 2 L 2 [0,T ]H 1/2 δ X C. Local smoothing in R n goes back to work of Kato, Sjölin, Vega, Constantin-Saut. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

14 Resolvent estimates and propagators: local smoothing A loss in the resolvent estimate gives local smoothing with loss. χ( g λ iε) 1 χ L 2 L 2 C logλ λ uniformly for λ λ 0, ε 0 = χe it g L 2 L 2 [0,T ]H 1/2 δ X C. Local smoothing in R n goes back to work of Kato, Sjölin, Vega, Constantin-Saut. Doi proved that a loss occurs when there are trapped geodesics on X. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

15 Resolvent estimates and propagators: local smoothing A loss in the resolvent estimate gives local smoothing with loss. χ( g λ iε) 1 χ L 2 L 2 C logλ λ uniformly for λ λ 0, ε 0 = χe it g L 2 L 2 [0,T ]H 1/2 δ X C. Local smoothing in R n goes back to work of Kato, Sjölin, Vega, Constantin-Saut. Doi proved that a loss occurs when there are trapped geodesics on X. If the loss in the resolvent estimate is only logarithmic (see Burq, Christianson, Nonnenmacher-Zworski, Datchev, Wunsch-Zworski, etc.) Burq-Guillarmou-Hassell show that in some settings one has lossless Strichartz estimates. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

16 Resolvent estimates and propagators: wave decay Meromorphic continuation of the resolvent and uniform estimates in a strip imply exponential wave decay. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

17 Resolvent estimates and propagators: wave decay Meromorphic continuation of the resolvent and uniform estimates in a strip imply exponential wave decay. λ 0 spectrum uniformly estimate up to here -Γ χ( g λ 2 ) 1 χ L 2 L 2 C λ k uniformly for Reλ λ 0, Imλ Γ Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

18 Resolvent estimates and propagators: wave decay Meromorphic continuation of the resolvent and uniform estimates in a strip imply exponential wave decay. λ 0 spectrum uniformly estimate up to here -Γ χ( g λ 2 ) 1 χ L 2 L 2 C λ k uniformly for Reλ λ 0, Imλ Γ = u(t,x) = w j (x)e itλ j t k j + R(t,x), j where R(t,x) e Γt uniformly on compact subsets of X. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

19 Resonant wave expansions u(t,x) = w j (x)e itλ j t k j + R(t,x), R(t,x) e Γt j Such resonant wave expansions have a long history, going back to Lax-Phillips and Vainberg. When X is nontrapping Γ may be taken arbitrarily large (Melrose-Sjöstrand,... Martinez). Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

20 Resonant wave expansions u(t,x) = w j (x)e itλ j t k j + R(t,x), R(t,x) e Γt j Such resonant wave expansions have a long history, going back to Lax-Phillips and Vainberg. When X is nontrapping Γ may be taken arbitrarily large (Melrose-Sjöstrand,... Martinez). [Figures by Laurent Guillopé] When trapping is present it governs the largest possible Γ (Gérard-Sjöstrand, Ikawa... Stefanov... Guillarmou-Naud, Nonnenmacher-Zworski, Petkov-Stoyanov, Melrose-Sá Barreto-Vasy, Wunsch-Zworski, Datchev, Dyatlov,...) Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

21 Resolvent estimates and propagators To summarize: Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

22 Resolvent estimates and propagators To summarize: Resolvent estimates up to the spectrum imply local smoothing for the Schrödinger propagator. uniformly estimate up to here λ 0 spectrum Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

23 Resolvent estimates and propagators To summarize: Resolvent estimates up to the spectrum imply local smoothing for the Schrödinger propagator. uniformly estimate up to here λ 0 spectrum Meromorphic continuation and resolvent estimates in a strip beyond the spectrum imply exponential wave decay. λ 0 spectrum uniformly estimate up to here -Γ Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

24 Resolvent estimates and propagators To summarize: Resolvent estimates up to the spectrum imply local smoothing for the Schrödinger propagator. uniformly estimate up to here λ 0 spectrum Meromorphic continuation and resolvent estimates in a strip beyond the spectrum imply exponential wave decay. λ 0 spectrum uniformly estimate up to here Losses in these estimates come from trapped geodesics. -Γ Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

25 Theorem (joint with András Vasy). Let (X,g 0 ) be Euclidean outside of a compact set. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

26 Theorem (joint with András Vasy). Let (X,g 0 ) be Euclidean outside of a compact set. Suppose we know either that or that χ( g0 λ iε) 1 χ L 2 L 2 C logλ λ uniformly for λ λ 0, ε 0, χ( g0 λ 2 ) 1 χ L 2 L 2 C λ k uniformly for Reλ λ 0, Imλ Γ. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

27 Theorem (joint with András Vasy). Let (X,g 0 ) be Euclidean outside of a compact set. Suppose we know either that or that χ( g0 λ iε) 1 χ L 2 L 2 C logλ λ uniformly for λ λ 0, ε 0, χ( g0 λ 2 ) 1 χ L 2 L 2 C λ k uniformly for Reλ λ 0, Imλ Γ. Let g be a metric which agrees with g 0 on a large compact set, but is asymptotically hyperbolic in the sense of Vasy s talk yesterday. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

28 Theorem (joint with András Vasy). Let (X,g 0 ) be Euclidean outside of a compact set. Suppose we know either that χ( g0 λ iε) 1 χ L 2 L 2 C logλ λ uniformly for λ λ 0, ε 0, or that χ( g0 λ 2 ) 1 χ L 2 L 2 C λ k uniformly for Reλ λ 0, Imλ Γ. Let g be a metric which agrees with g 0 on a large compact set, but is asymptotically hyperbolic in the sense of Vasy s talk yesterday. Then the same resolvent estimates hold for g, possibly with worse constants C,λ 0. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

29 Theorem (joint with András Vasy). Let (X,g 0 ) be Euclidean outside of a compact set. Suppose we know either that χ( g0 λ iε) 1 χ L 2 L 2 C logλ λ uniformly for λ λ 0, ε 0, or that χ( g0 λ 2 ) 1 χ L 2 L 2 C λ k uniformly for Reλ λ 0, Imλ Γ. Let g be a metric which agrees with g 0 on a large compact set, but is asymptotically hyperbolic in the sense of Vasy s talk yesterday. Then the same resolvent estimates hold for g, possibly with worse constants C,λ 0. The theorem also works with the roles of g and g 0 reversed. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

30 This result is a special case of the following heuristic principle: Given a compact trapped set, high energy resolvent estimates are the same for any nontrapping infinity. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

31 This result is a special case of the following heuristic principle: Given a compact trapped set, high energy resolvent estimates are the same for any nontrapping infinity. We prove in fact a more general abstract theorem. Let X be a compact manifold with boundary such that X is the interior of X, and let x be a boundary defining function, i.e. x C (X : [0, )), x = 0 precisely on X, and dx 0 on X. {x = 0} = X = "infinity" {x = 1}... Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

32 Put P = h 2 g 1. Studying (P λ) 1 as h 0 for λ near 0 is equivalent to studying ( g λ) 1 for Reλ large. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

33 Put P = h 2 g 1. Studying (P λ) 1 as h 0 for λ near 0 is equivalent to studying ( g λ) 1 for Reλ large. {x = 0} = X = "infinity" {x = 1}... Let X 0 = {x < 4}, X 1 = {x > 1}. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

34 Put P = h 2 g 1. Studying (P λ) 1 as h 0 for λ near 0 is equivalent to studying ( g λ) 1 for Reλ large. {x = 0} = X = "infinity" {x = 1}... Let X 0 = {x < 4}, X 1 = {x > 1}. Define model operators P 0 and P 1 with P 0 X0 = P X0, P 1 X1 = P X1. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

35 {x = 0} = X = "infinity" {x = 1}... X 0 = {x < 4}, X 1 = {x > 1}. P 0 X0 = P X0, P 1 X1 = P X1. Suppose: Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

36 {x = 0} = X = "infinity" {x = 1}... X 0 = {x < 4}, X 1 = {x > 1}. P 0 X0 = P X0, P 1 X1 = P X1. Suppose: 1 Infinity is convex in the sense that x < 4, ẋ = 0 ẍ < 0 along geodesics. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

37 Suppose: {x = 0} = X = "infinity" {x = 1}... X 0 = {x < 4}, X 1 = {x > 1}. P 0 X0 = P X0, P 1 X1 = P X1. 1 Infinity is convex in the sense that x < 4, ẋ = 0 ẍ < 0 along geodesics. 2 The resolvents for the model operators are polynomially bounded: χ(p j λ) 1 χ a j (λ,h) h N for λ D [ E,E] ± i[0,γh]. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

38 Suppose: {x = 0} = X = "infinity" {x = 1}... X 0 = {x < 4}, X 1 = {x > 1}. P 0 X0 = P X0, P 1 X1 = P X1. 1 Infinity is convex in the sense that x < 4, ẋ = 0 ẍ < 0 along geodesics. 2 The resolvents for the model operators are polynomially bounded: χ(p j λ) 1 χ a j (λ,h) h N for λ D [ E,E] ± i[0,γh]. 3 The resolvents for the model operators propagate semiclassical singularities (lack of decay as h 0) forward along geodesics. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

39 Suppose: {x = 0} = X = "infinity" {x = 1}... X 0 = {x < 4}, X 1 = {x > 1}. P 0 X0 = P X0, P 1 X1 = P X1. 1 Infinity is convex in the sense that x < 4, ẋ = 0 ẍ < 0 along geodesics. 2 The resolvents for the model operators are polynomially bounded: χ(p j λ) 1 χ a j (λ,h) h N for λ D [ E,E] ± i[0,γh]. 3 The resolvents for the model operators propagate semiclassical singularities (lack of decay as h 0) forward along geodesics. Then χ(p λ) 1 χ Ch 2 a 2 0 a 1. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

40 Suppose: {x = 0} = X = "infinity" {x = 1}... X 0 = {x < 4}, X 1 = {x > 1}. P 0 X0 = P X0, P 1 X1 = P X1. 1 Infinity is convex in the sense that x < 4, ẋ = 0 ẍ < 0 along geodesics. 2 The resolvents for the model operators are polynomially bounded: χ(p j λ) 1 χ a j (λ,h) h N for λ D [ E,E] ± i[0,γh]. 3 The resolvents for the model operators propagate semiclassical singularities (lack of decay as h 0) forward along geodesics. Then χ(p λ) 1 χ Ch 2 a 2 0 a 1. If a 0 = C/h (typical for nontrapping infinities and real λ) we find χ(p λ) 1 χ Ca 1. The resolvent estimate is the same as for the model with infinity suppressed. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

41 Convexity of infinity The assumption x < 4, ẋ = 0 ẍ < 0 is satisfied for a wide range of asymptotically Euclidean and hyperbolic infinities (and can in fact be weakened). Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

42 Convexity of infinity The assumption x < 4, ẋ = 0 ẍ < 0 is satisfied for a wide range of asymptotically Euclidean and hyperbolic infinities (and can in fact be weakened). We use it as follows: γ N γ Y The geodesic trajectory γ Y is allowed, but γ N is forbidden. In other words, a geodesic cannot come arbitrarily close to infinity and then turn back. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

43 Polynomially bounded model resolvents Our second assumption is χ(p j λ) 1 χ a j (λ,h) h N. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

44 Polynomially bounded model resolvents Our second assumption is χ(p j λ) 1 χ a j (λ,h) h N. For P 0 (the model near infinity) this is proven with a 0 = C/h for λ real for a wide range of asymptotically Euclidean and hyperbolic infinities by Cardoso-Vodev. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

45 Polynomially bounded model resolvents Our second assumption is χ(p j λ) 1 χ a j (λ,h) h N. For P 0 (the model near infinity) this is proven with a 0 = C/h for λ real for a wide range of asymptotically Euclidean and hyperbolic infinities by Cardoso-Vodev. For an analytic continuation with polynomial estimates stronger assumptions are needed. See for example Wunsch-Zworski, Melrose-Sá Barreto-Vasy, Vasy. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

46 Polynomially bounded model resolvents Our second assumption is χ(p j λ) 1 χ a j (λ,h) h N. For P 0 (the model near infinity) this is proven with a 0 = C/h for λ real for a wide range of asymptotically Euclidean and hyperbolic infinities by Cardoso-Vodev. For an analytic continuation with polynomial estimates stronger assumptions are needed. See for example Wunsch-Zworski, Melrose-Sá Barreto-Vasy, Vasy. For P 1 (the model near the trapped set) delicate assumptions on the trapped set are needed. It must have hyperbolic dynamics (negative curvature is sufficient) and be thin in a suitable sense. See for example Nonnenmacher-Zworski, Petkov-Stoyanov, Wunsch-Zworski. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

47 Propagation of singularities Our third assumption is that singularities are propagated forward along geodesics. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

48 Propagation of singularities Our third assumption is that singularities are propagated forward along geodesics. One way to say this is that if Op(a) f = O(h ) for a C 0 (T X) with suppa U with U closed under backward geodesic flow, then Op(a)(P j λ) 1 f = O(h ). Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

49 Propagation of singularities Our third assumption is that singularities are propagated forward along geodesics. One way to say this is that if Op(a) f = O(h ) for a C 0 (T X) with suppa U with U closed under backward geodesic flow, then Op(a)(P j λ) 1 f = O(h ). In other words, if u is nontrivial at some point ρ T X, then (P j λ)u must be nontrivial at some point in the backwards geodesic flowout of ρ. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

50 Propagation of singularities Our third assumption is that singularities are propagated forward along geodesics. One way to say this is that if Op(a) f = O(h ) for a C 0 (T X) with suppa U with U closed under backward geodesic flow, then Op(a)(P j λ) 1 f = O(h ). In other words, if u is nontrivial at some point ρ T X, then (P j λ)u must be nontrivial at some point in the backwards geodesic flowout of ρ. This type of result goes back to work of Hörmander. It was shown by Vasy-Zworski for scattering (asymptotically Euclidean) manifolds, and by Melrose-Sá Barreto-Vasy, Vasy for certain asymptotically hyperbolic manifolds. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

51 Proof. Let χ 1 = χ 1 (x) C (X) have χ 1 = 1 near {x 3} and χ 1 = 0 near {x 2}, and let χ 0 = 1 χ 1. As a first parametrix take F = χ 0 (x 1)(P 0 λ) 1 χ 0 (x) + χ 1 (x + 1)(P 1 λ) 1 χ 1 (x). Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

52 Proof. Let χ 1 = χ 1 (x) C (X) have χ 1 = 1 near {x 3} and χ 1 = 0 near {x 2}, and let χ 0 = 1 χ 1. As a first parametrix take Then F = χ 0 (x 1)(P 0 λ) 1 χ 0 (x) + χ 1 (x + 1)(P 1 λ) 1 χ 1 (x). (P λ)f = Id+[χ 0 (x 1),P 0 ](P 0 λ) 1 χ 0 (x) + [χ 1 (x + 1),P 1 ](P 1 λ) 1 χ 1 (x) = Id+A 0 + A 1 Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

53 Proof. Let χ 1 = χ 1 (x) C (X) have χ 1 = 1 near {x 3} and χ 1 = 0 near {x 2}, and let χ 0 = 1 χ 1. As a first parametrix take Then F = χ 0 (x 1)(P 0 λ) 1 χ 0 (x) + χ 1 (x + 1)(P 1 λ) 1 χ 1 (x). (P λ)f = Id+[χ 0 (x 1),P 0 ](P 0 λ) 1 χ 0 (x) + [χ 1 (x + 1),P 1 ](P 1 λ) 1 χ 1 (x) = Id+A 0 + A 1 The errors A 0 and A 1 may be large in h (as large as h N ), but they are microlocally concentrated on incoming, resp. outgoing geodesics. A 0 A 1 Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

54 (P λ)f = Id+A 0 + A 1, A j = [χ j (x ± 1),P j ](P j λ) 1 χ j (x). Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

55 We solve away these errors using F. (P λ)f = Id+A 0 + A 1, A j = [χ j (x ± 1),P j ](P j λ) 1 χ j (x). (P λ)(f FA 0 FA 1 ) = Id+A 0 + A 1 A 0 A 2 0 A 1 A 0 A 1 A 0 A 1 A 2 1 = Id A 1 A 0 + A 0 A 1 Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

56 We solve away these errors using F. (P λ)f = Id+A 0 + A 1, A j = [χ j (x ± 1),P j ](P j λ) 1 χ j (x). (P λ)(f FA 0 FA 1 ) = Id+A 0 + A 1 A 0 A 2 0 A 1 A 0 A 1 A 0 A 1 A 2 1 = Id A 1 A 0 + A 0 A 1 The A j terms cancel, and the A 2 j terms vanish because the support of dχ j (x ± 1) does not overlap that of χ j. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

57 We solve away these errors using F. (P λ)f = Id+A 0 + A 1, A j = [χ j (x ± 1),P j ](P j λ) 1 χ j (x). (P λ)(f FA 0 FA 1 ) = Id+A 0 + A 1 A 0 A 2 0 A 1 A 0 A 1 A 0 A 1 A 2 1 = Id A 1 A 0 + A 0 A 1 The A j terms cancel, and the A 2 j terms vanish because the support of dχ j (x ± 1) does not overlap that of χ j. The A 0 A 1 term is microlocally concentrated on geodesics like γ N, which are ruled out by the convexity assumption, and hence this term is O(h ). γ N γ Y Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

58 (P λ)(f FA 0 FA 1 ) = Id A 1 A 0 + O(h ) Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

59 (P λ)(f FA 0 FA 1 ) = Id A 1 A 0 + O(h ) Another iteration of the same error-term-removing procedure replaces the A 1 A 0 term by one which is microlocally concentrated on geodesics ruled out by the convexity assumption. This gives (P λ)(f FA 0 FA 1 + FA 1 A 0 ) = Id+O(h ) Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18

60 (P λ)(f FA 0 FA 1 ) = Id A 1 A 0 + O(h ) Another iteration of the same error-term-removing procedure replaces the A 1 A 0 term by one which is microlocally concentrated on geodesics ruled out by the convexity assumption. This gives (P λ)(f FA 0 FA 1 + FA 1 A 0 ) = Id+O(h ) Writing out F FA 0 FA 1 + FA 1 A 0 in terms of the model resolvents and applying the estimates χ(p j λ) 1 χ a j and [χ j (x ± 1),P j ] Ch gives χ(p λ) 1 χ Ch 2 a 2 0a 1. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December / 18