Asymptotics of generalized eigenfunctions on manifold with Euclidean and/or hyperbolic ends

Transcription

1 Asymptotics of generalized eigenfunctions on manifold with Euclidean and/or hyperbolic ends Kenichi ITO (University of Tokyo) joint work with Erik SKIBSTED (Aarhus University) 3 July 2018

2 Example: Free Laplacian on R 2 We discuss a generalized eigenfunction expansion associated with H 0 = 1 2 on R2. Consider the generalized (i.e. non-l 2 ) eigenequation (H 0 λ)φ = 0 with λ > 0, and we apply the method of separation of variables as φ(x, y) = R(r)Θ(θ); (x, y) = (r cos θ, r sin θ), (r, θ) (0, ) S 1. Then the eigenequation is separated as, for some µ R, 1 2 R 1 2r R (λ µ ) r 2 R = 0, 1 2 Θ µθ = 0. 1

3 The second equation has the following family of solutions: Θ n (θ) = e inθ, µ n = n2 for n Z. 2 Then the first equation has a general solution R λ,n (r) = C 1 H (1) n ( 2λ r) + C 2 H (2) n ( 2λ r), where H (j) n are the Hankel functions. To have φ = RΘ smooth at (x, y) = 0 we need to impose C 1 = C 2. Now we have a family of normalized generalized eigenfunctions φ λ,n (x, y) = 1 [ 2 H (1) n ( 2λ r) + H (2) n ( ] 2λ r) e inθ 2π for λ > 0, n Z. 2

4 Note that, as r, [ 1 φ λ,n (x, y) 2π 4 2λ e i( 2λ r nπ/2 π/4) + e i( 2λ r nπ/2 π/4) ] e inθ. r Define the Agmon Hörmander spaces B, B0 as { } B = ψ L 2 loc (R2 ); sup 2 n n N 2 ψ 2 dxdy <, n 1 r<2 n } B0 {ψ = L 2 loc (R2 ); lim 2 n ψ 2 dxdy = 0, n 2 n 1 r<2 n respectively. Note that φ λ,n B \ B0, and that L 2, 1/2 (R 2 ) B 0 B L 2, s (R 2 ) for s > 1/2. 3

5 Theorem. Let λ > 0, and set E λ = { φ B ; (H 0 λ)φ = 0 }, G = L 2 (S 1 ). 1. There is no generalized eigenfunction in B0 λ, i.e., E λ B0 = {0}. with eigenvalue 2. For any φ E λ there exist unique ξ ± G such that φ(x, y) 1 r [e i 2λ r ξ + (θ) + e i 2λ r ξ (θ) ] B 0. ( ) 3. Conversely, for any ξ ± G there exist unique φ E λ and ξ G, respectively, such that ( ) holds. The map ξ ξ + is called the stationary scattering matrix. 4

6 Theorem. Define (F 0 u)(λ, n) = R 2 u(x, y)φ λ,n(x, y) dxdy. 1. F 0 is unitary as L 2 (R 2 ) L 2( (0, ) Z, d(λ #) ). In particular, u(x, y) = n Z 0 (F 0 u)(λ, n)φ λ,n (x, y) dλ for u L 2 (R 2 ). 2. F 0 unitarily diagonalizes H 0 as F 0 H 0 F 0 = M λ, where M λ is a multiplication operator by λ. 5

7 Riemannian manifold with ends Let (M, g) be a connected and complete Riemannian manifold. An open subset E M is an end, if there is a diffeomorphism Ē = [2, ) S for some connected and closed manifold S. The metric g is of warped-product type on E, if g(r, θ) = dr dr + f(r)h αβ (θ) dθ α dθ β ; (r, θ) (2, ) S for some Riemannian metric h on S. 6

8 Assumption A. (M, g) has ends E n, n = 1,..., N, of warpedproduct types such that 1. For each n {1,..., N} one of the following holds: f n (r) = r δ with δ > 1. (Asympt. Euclidean) f n (r) = exp(κr δ ) with κ > 0, 0 < δ < 1. (Asympt. Euclidean) f n (r) = exp(κr) with κ > 0. (Asympt. hyperbolic) 2. M \ (E 1 E N ) is compact. Typical examples are R d and H d. We extend r C (E) onto M such that r C (M) and r 1 on M. Remark. We can treat much more general manifolds with ends, but we do not discuss it for simplicity. 7

9 The Schrödinger operator We study the Schrödinger operator H = H 0 + V on H = L 2( M, det g dx ), where H 0 is the Laplace Beltrami operator given as H 0 = 1 2 = p i gij p j = 2 det g p ig ij The effective potential q is given by det gp j ; p j = i j. q = V ( r)2 = V + (d 1)2 32 ( f n f n ) 2 on E n. Assumption B. There exists a splitting q = q 1 + q 2 such that q q 1 Cr 1 ɛ, q 2 Cr 1 ɛ for some ɛ > 1/2. 8

10 Critical energy Define the critical energy by λ H = lim sup r q Kumura ( 97) proved that [λ H, ) σ ess (H). If f(r) = exp(κr) and V 1 0, we have Recall that, if M = H d, then λ H = (d 1)2 κ f(r) = (sinh r) 2 exp(2r), σ(h 0 ) = [ (d 1) 2 8 ),. 9

11 Function spaces Define the weighted spaces as H s = r s H for s R. Set the dyadic annuli Ω ν = { x M; 2 ν r(x) < 2 ν+1} for ν 0, and define the associated Agmon Hörmander spaces as { } B = ψ L 2 loc (M); ψ B = 2 ν/2 χ Ων ψ H <, ν=0 { } B = ψ L 2 loc (M); ψ B = sup 2 ν/2 χ Ων ψ H <, ν 0 } B0 {ψ = B ; lim 2 ν/2 χ ν Ων ψ H = 0. Note that for any s > 1/2 H s B H 1/2 H H 1/2 B 0 B H s. 10

12 Purely outgoing/incoming spherical waves Let G = L 2 (S 1 ) L 2 (S N ), Ē n = [2, ) Sn for n = 1,..., N. For any λ > λ H and ξ G we introduce purely outgoing/incoming spherical waves φ ± [ξ] B by φ ± [ξ](r, θ) η(r) = 4 2[λ q 1 (r, θ)] f(r) (d 1)/4 exp ( ±i r r 0 ) 2[λ q 1 (s, θ)] ds ξ(θ). Here η is a smooth cut-off function such that η = 1 for large r. 11

13 Asymptotics of generalized eigenfunctions Theorem (I. Skibsted). Let λ > λ H, and set E λ = { φ B ; (H λ)φ = 0 }. 1. E λ B0 =. 2. For any φ E λ there exist unique ξ ± G such that φ φ + [ξ + ] + φ [ξ ] B 0. ( ) 3. Conversely, for any ξ ± G there exist unique φ E λ and ξ G, respectively, such that ( ) holds. The stationary scattering matrix S(λ): G G is defined as ξ + = S(λ)ξ, cf. Melrose Zworski ( 96). S(λ) is a unitary operator. 12

14 Application: Channel scattering According to G = L 2 (S 1 ) L 2 (S N ), let us decompose S(λ) = [ S(λ) ij ]i,j. Now we can show the following off-diagonal injectivity property, which was conjectured by Hempel Post Weder ( 14). Corollary. For any i j the component S(λ) ij is injective. Proof. For any ξ G there exists φ E λ such that φ φ + [S(λ)ξ ] + φ [ξ ] B0. Let, e.g., ξ = (ξ,1, 0,..., 0). Then (S(λ)ξ ) 2,..., (S(λ)ξ ) N are all non-zero due to E λ B0 =, so that S(λ) 1j are injective. We can discuss similarly for the other components. 13

15 Motivation for the limiting resolvents We set the resolvent as It would be natural to expect R(z) = (H z) 1 for z C \ R. δ(h λ) = 1 [ ] R(λ + i0) R(λ i0). 2πi In fact, for any ψ B we have by Cauchy s theorem 1 [ ] lim R(λ + iɛ) R(λ iɛ) ψ dλ = ψ, ɛ +0 2πi R and we would also have lim (H ɛ +0 λ)[ R(λ + iɛ) R(λ iɛ) ] ψ = 0. 14

16 Limiting resolvents Let I = (λ H, ) and I ± = { λ ± iγ C; λ > λ H, Γ > 0 }. Theorem (I. Skibsted). For s > 1/2 there exist R(λ ± i0) = lim R(z) in L(H s, H s ) I ± z λ locally uniformly in λ I, respectively. Moreover, for any ψ H t with t > 1 there exist F ± (λ)ψ G such that respectively. R(λ ± i0)ψ φ ± [F ± (λ)ψ] B 0, 15

17 Commutator theory A proof depends on a commutator theory with respect to A = i[h 0, r] = Re p r = 1 pr + p 2( r). Lemma. As quadratic forms on C 0 (M), [H, ia] = p i ( 2 r) ij p j ( r q 1 ) 2 Im(q 2 A); 2 r = 1 2 f h. The usual Mourre conjugate operator is given by à = i[h 0, r 2 ] = Re(rp r ) = 1 2( rpr + p rr ). [H, ia] does not have a strict positivity in the radial direction, but this missing positivity can be recovered from weight functions. 16

18 To prove E λ B0 = {0} we first choose a weight function ( r ( Θ = χ m,n exp 2αr + 2β 1 + s/2 ν ) ) 1 δ ds with α, β 0, δ > 0, n > m 0, ν 0. Here χ m,n is a smooth cut-off function leaving 2 m r 2 n. Lemma. Let λ > λ H, and fix any α 0 0 and let δ > 0 be small. Then there exist β, c, C > 0, n 0 0 and a function γ such that uniformly in α [0, α 0 ], n > m n 0 and ν n 0 Im ( AΘ(H λ) ) cr 1 θ δ 0 Θ C( χ 2 m 1,m+1 + χ2 n 1,n+1 + Re ( γ(h λ) ). 0 ) r 1 e θ To complete the proof we need repeat similar arguments for another weight Θ = χ m,n e αr. 17

19 To prove uniform boundedness of R(z): B B we choose Θ = r/2 ν 0 (1 + s) 1 δ ds = [ 1 ( 1 + r/2 ν) δ ]/ δ; ν 0, δ > 0. Lemma. Let I (λ H, ) be compact, and δ > 0 small. There exist c, C > 0, n 0 and a uniformly bounded function γ such that uniformly in z I ± and ν 0 Im ( AΘ(H z) ) cθ + caθ A Cχ 2 nθ Re ( γ(h z) ). Theorem. Let I I be compact. Then there exists C > 0 such that for any z I ± and ψ B R(z)ψ B C ψ B. 18

20 Lemma. Let I I be compact, and δ, β > 0 be small. Then there exist c, C > 0 such that uniformly in z I I + and ν 0 Im ( (A a) Θ 2β (H z) ) c(a a) Θ Θ 2β 1 (A a) Cr 1 ɛ+2δ Θ 2β Re ( γθ 2β (H z) ), where γ is a certain function. Theorem. Let I I be compact, and β > 0 be small. there exists C > 0 such that for any ψ r β B and z I ± Then r β (A a)r(z)ψ B C r β ψ B, respectively. 19