Gluing semiclassical resolvent estimates via propagation of singularities

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 2010 1 / 18

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 2010 2 / 18

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 2010 2 / 18

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 2010 3 / 18

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 2010 3 / 18

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 2010 3 / 18

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 2010 4 / 18

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 2010 4 / 18

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 2010 4 / 18

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 2010 5 / 18

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 2010 5 / 18

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 2010 5 / 18

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 2010 6 / 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 Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December 2010 6 / 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 2010 6 / 18

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 2010 7 / 18

Resolvent estimates and propagators To summarize: Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December 2010 8 / 18

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 2010 8 / 18

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 2010 8 / 18

Theorem (joint with András Vasy). Let (X,g 0 ) be Euclidean outside of a compact set. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December 2010 9 / 18

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 2010 9 / 18

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 2010 9 / 18

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 2010 9 / 18

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 2010 9 / 18

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 2010 10 / 18

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 2010 10 / 18

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 2010 11 / 18

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 2010 11 / 18

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 2010 11 / 18

{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 2010 12 / 18

{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 2010 12 / 18

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 2010 12 / 18

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 2010 12 / 18

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 2010 12 / 18

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 2010 13 / 18

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 2010 13 / 18

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 2010 14 / 18

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 2010 14 / 18

Propagation of singularities Our third assumption is that singularities are propagated forward along geodesics. Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December 2010 15 / 18

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 2010 15 / 18

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 2010 15 / 18

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 2010 16 / 18

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 2010 16 / 18

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 2010 16 / 18

(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 2010 17 / 18

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 2010 17 / 18

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 2010 17 / 18

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 2010 17 / 18

(P λ)(f FA 0 FA 1 ) = Id A 1 A 0 + O(h ) Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December 2010 18 / 18

(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 2010 18 / 18