RADIATION FIELDS ON ASYMPTOTICALLY EUCLIDEAN MANIFOLDS

Transcription

1 RADIATION FIELDS ON ASYMPTOTICALLY EUCLIDEAN MANIFOLDS ANTÔNIO SÁ BARRETO Abstract. F.G. Friedlander introduced the notion of radiation fields for asymptotically Euclidean manifolds. Here we answer some of the questions he proposed and apply the results to give a unitary translation representation of the wave group, and to obtain the scattering matrix for such manifolds. We also obtain a support theorem for the radiation fields. 1. Introduction and Statement of the Results Friedlander [1, ] introduced the forward radiation fields for asymptotically Euclidean manifolds as the limit, as times goes to infinity, of the forward fundamental solution of the wave operator along certain light rays. By reversing time one defines the backward radiation field. We will show that this leads to a unitary translation representation of the wave group and a dynamical definition of the scattering matrix. Similar results for asymptotically hyperbolic manifolds have been established in [13]. A smooth compact manifold X with boundary, X, is called asymptotically Euclidean, or scattering manifold see [9], when it is equipped with a Riemannian metric g, which is smooth in the interior of X, denoted by X, and can be written as 1.1) g = dx x 4 + H x near X. Here x C X) is a smooth defining function of X and H is a symmetric tensor which restricts to a smooth Riemannian metric h 0 on X. As observed in [8, 9], once 1.1) is known to exist, it determines x up to terms vanishing to second order at X, and hence it determines h 0 = H X. We will denote the dimension of X by n, and therefore X has dimension n 1. The Euclidean space R n can be viewed as such a manifold by compactification. As in [9], let S n + = {y R n+1 : y =1, y 1 > 0} denote the upper hemisphere of the unit sphere in R n+1. Its boundary is S n + = S n 1. Let SP : R n S n + ) 1 z z,. 1 + z ) z ) 1 The Euclidean metric when pushed forward to S n + has the form 1.1). Moreover, as observed in [10], any perturbation of the Euclidean metric which behaves like g ij = δ ij + 1 ) 1 z h ij z, z, as z z satisfies 1.1) when pushed forward to S n +. In odd dimensional Euclidean spaces, the forward radiation field is the modified Radon transform of Lax and Phillips, see Proposition 4.0 of [1] or Theorem.4 of chapter 4 of [7]. Therefore the radiation fields can be viewed as a generalization of such transform to asymptotically Euclidean manifolds. The methods developed here also provide new proofs, although rather difficult ones, of well known results 1

2 SÁ BARRETO about the Radon transform. In particular, Theorem 4. can be viewed as a generalization of the well known support theorem for Radon transform due to Helgason [4]. The methods we use to establish the relationship between the radiation fields and the scattering matrix are adaptations of the techniques of [1, ] and the construction of the resolvent of the Laplacian due to Hassell and Vasy [3]. It is shown in [6] that if g satisfies 1.1) there exists ɛ>0 and a product structure X X [0,ɛ)in which 1.) g = dx x 4 hx, y, dy) + x. These are the equivalent of boundary normal coordinates on a compact manifold with boundary. We will fix such a decomposition and x C X) will be as in 1.). Let be the positive) Laplace operator with respect to the metric g. It was shown in [9] that given f C X) andλ R, λ 0, there exists a u C X) satisfying 1.3) λ )u =0in X, u = x n 1 e i λ x F + x n 1 e i λ x G, F, G C X), F X = f. This leads to the stationary definition of the scattering matrix at energy λ. Melrose defined it in [9] as the operator 1.4) Aλ) :C X) C X) f G X. Melrose and Zworski showed in [10] that Aλ) is a classical Fourier integral operator of order zero associated to the geodesic flow at time π given by the metric h 0 on X. Here, as in [1,, 13], we will use the wave equation to define the radiation fields and arrive at an equivalent definition of the scattering matrix. Friedlander proved Theorem 1.1. [] For f 1,f C 0 1.5) X) compactly supported in the interior of X, let ut, z) C R + X ) satisfy D t ) ut, z) =0, on R X, u0,z)=f 1 z), D t u0,z)=f z). Let z =x, y) 0,ɛ) X be local coordinates near X in which 1.) hold. Let Ht) =1for t>0 and Ht) =0otherwise, denote the Heaviside function. Then there exist w k C R X), such that 1.6) In particular, x n 1 Hu)s + 1 x,x,y) x k w k s, y), as x 0. k=0 x n 1 Hu)s + 1 x,x,y) x=0 = w 0 s, y), is well defined. As any two boundary defining functions satisfying 1.) agree to second order at X, w 0 s, y) is independent of the choice of x.

3 RADIATION FIELDS 3 Consider the forward fundamental solution of the wave operator Ut, z, z ) which satisfies 1.7) D t ) Ut, z, z )=δt)δz,z ), on R X X Ut) =0, if t<0. where δz,z ) is the delta function with respect to the natural Riemannian measure given by the metric g. Theorem 1.1 shows that the limit of the Schwartz kernel n 1 1 lim x Us + x 0 x,x,y,z )=R + s, y, z ) exists, but it does not give any information about what type of distribution it is. We observe that, since the Lorentzian metric associated to g is the surfaces σ = dt dx x 4 hx, y, dy) x = dt + 1 x )dt 1 hx, y, dy) ) x x, {t 1 x = C}, {t + 1 x = C} are characteristic for the wave operator, and thus a point t,z ),z =x,y ), has a past domain of dependence, t,z ) satisfying 1.8) t,z ) {t, x, y) : t 1 x t 1 x, t+ 1 x t + 1 x }.. Radiation Fields And The Scattering Matrix For w 0,w 1 C0 X) the energy norm of w =w 0,w 1 ) is defined by w E = 1.1) dw0 g + w 1 ) dvol g, X where dw 0 g denotes the length of the co-vector with respect to the metric induced by g on T X, and define H E X) as the closure of C0 X) C0 X) with the norm.1). Let W t) be the map defined by.) W t) :C0 X) C0 X) C0 X) C0 X), W t)f 1,f )=ut, z),d t ut, z)), t R. The conservation of energy gives, see for example the proof of Proposition.4 of [1], that is a strongly continuous group of unitary operators. Theorem 1.1 defines a map W t) :H E X) H E X), t R,.3) R + : C 0 X) C 0 X) C R X) R + f 1,f )s, y) =x n 1 Dt Hu)s + 1 x,x,y) x=0 = D s w 0 s, y), which will be called the forward radiation field. Similarly one can prove that if u satisfies 1.5) and H t) =H t) n 1 lim x H u )s 1 x 0 x,x,y)=w 0 s, y)

4 4 SÁ BARRETO exists, and thus define the backward radiation field as.4) R : C 0 X) C 0 X) C R X) R f 1,f )s, y) =x n 1 Dt H u )s 1 x,x,y) x=0 = D s w0 s, y). By changing t t τ, the variable s then changes to s s + τ therefore R ± satisfy.5) R ± W τ)f)y, s) =R ± fy, s + τ), τ R. So Theorem 1.1 shows that R ± are twisted translation representations of the group W t). That is, if one sets R ± f)y, s) =R ± fy, s), then.6) R ± W τ)) = T τ R±, where T τ denotes the right translation by τ in the s variable. So R ± are translation representers in the sense of Lax and Phillips. We will use the results of [1,, 3] to prove Theorem.1. The maps R ± extend to isometries R ± : H E X) L X R). The scattering operator is defined to be the map S = R + R 1. It is clearly a unitary in L X R), and in view of.5), it commutes with translations in s. Therefore the Schwartz kernel Ss, y, s,y )ofs is completely determined by its values at s = s. In fact it satisfies s + s Ss, y, s,y )=S,y, s + ) s,y. The scattering matrix is defined by conjugating S with the partial Fourier transform in the s variable A = FSF 1. A is a unitary operator in L X R). Since S acts as a convolution in the variable s, if λ denotes the dual variable to s, AF is a multiplication in λ, i.e AFF )λ, y) = Aλ, y, y )FF λ, y )dvol h. X It is a consequence of the results of section 9 of [3] that the stationary and dynamical definitions of the scattering matrix coincide. More precisely: Theorem.. The Schwartz kernel of the map Aλ) defined by 1.4) is equal to Aλ, y, y ). 3. Proofs of Theorems.1 and. Friedlander proved that R + is a partial isometry. We put together the results from Lemma 3., Proposition 3.4 and Theorem 3.5 of [1] in Theorem 3.1. [1]) The space HE 1 = {f H E : R + f L R X) = f HEX)} is a closed subspace of H E, and for f H E and g H 1 E, Moreover f,g HE = R + f,r + g L R X) R + : H 1 E L R X)

5 RADIATION FIELDS 5 is a surjective isometry and H E = H 1 E KerR + ), KerR + )={f H E : R + f =0},. These results were proved in [1] for a metric perturbation of the Euclidean metric, but in fact no intrinsic properties of R n were used and, as Friedlander pointed out in the paragraph after equation 44) of [], they can be easily extended to the asymptotically Euclidean case. Actually his approach in [] can be used to simplify the proofs in [1]. We will show that Theorem.1 follows from the results of [3]. Proof. To see this, first observe that if ut, z) satisfies 1.5), then D t )Hu)t, z) =if z)δt) if 1 z)d t δt). Taking partial Fourier-Laplace transform in t gives λ ) Hu)λ, z) =iλf1 z) if z), Iλ <0 So, in terms of the resolvent Rλ) = λ ) 1, Hu)λ, z) =Rλ)iλf 1 if )z), Iλ <0. On the other hand, we observe that the Fourier transform in s of v + x, s, y) =x n 1 Hu)s + 1 x,x,y)is 3.1) v + x, λ, y) =ix n 1 e i λ x Rλ)λf1 f )x, y), Iλ <0. 3.) Similarly, if v x, s, y) =x n 1 H u )s 1 x,x,y) From Theorem 3.1 we know that v x, λ, y) =ix n 1 e i λ x Rλ) iλf1 + f )x, y), Iλ >0. λŵ 0 λ, y) =λ v + 0,λ,y), L R X) and λŵ 0 λ, y) =λ v 0,λ,y) L R X). AsshownintheproofofLemma5.of[3], n 1 lim x e i i λ x Rλ) = x 0,Iλ 0 λ P λ), where P λ) is the adjoint of the Poisson operator P λ), which is the map that takes f C X) into the solution to 1.3). Therefore the limits of 3.1) and 3.) are and hence λŵ 0 λ, y) = 1 P λ) λf 1 f ), λŵ 0 λ, y) =1 P λ) λf 1 f ). 3.3) FR + f 1,f ))λ, y) = 1 P λ) λf 1 f ), FR f 1,f ))λ, y) = 1 P λ) λf 1 f ). Next we appeal to the results of section 9 of [3]. As we have no discrete spectrum, it follows from Proposition 9.1 of [3], and the fact that P ±λ) =λ P ±λ), that FR + f 1,f )) L R X) = 1 0 P λ) f dλ + 1 π f L X) + π f 1, f 1 L X) =π f 1,f ) E. 0 P λ) λ f 1 dλ =

6 6 SÁ BARRETO One should notice that the integral on the left hand side is taken over R, and the cross terms involving f 1 and f are odd, so their integral is zero. This proves Theorem.1. It is also proved in section 9 of [3] that the scattering matrix Aλ) is the unitary operator on L X) that intertwines these operators, i.e Hence from 3.3) Aλ)P λ) = P λ) Aλ)FR = FR + This gives Theorem.. 4. A support Theorem for R + The proof relies on Hörmander s uniqueness theorem for the Cauchy problem across a strongly pseudoconvex surface and the following uniqueness theorem, which is a particular case of a result due to Tataru [14], see also [11, 1, 15]. Theorem 4.1. [14]) Letg be a smooth Riemannian metric on a smooth manifold and let g its Laplacian. Let L be a first order smooth operator independent of t and let ut, z) C R X) satisfy D t g L ) ut, z) =0 ut, z) =0 t, z) [ T,T] {z; dz,z 0 ) <δ}, δ > 0. Then ut, z) =0for t + dz,z 0 ) <T,where dz,z 0 ) denotes the distance between z and z 0 with respect to g. Lemma 4.1. Let f,g) C0 X) be supported in {z =x, y) :x> }. Let ut, x, y) be the solution of 1.5). If 0 >s 0 > 1 and R + f,g)s, y) =0for s<s 0, then ut, x, y) =0for 0 x and 0 t s Proof. We will show that if 4.1) vx, s, y) =x n 1 1 us + x,x,y) satisfies s v0,s,y)=0fors<s 0, then vx, s, y) =0intheset{x, s, y) : 0 x, 1 <s<s 0 }. Translating this back to the variables x, y, t), and using finite speed of propagation, this gives the statement of the Lemma. n 1 Let P = x Dt n 1 )x. Taking s = t 1 x, it follows that Pv =0and P = + x x s x + h + A s + x + x A ) ) ) 4.) n 1 3 n x + + xa, Here Ax, y) = x log h and h is the negative) Laplacian with respect to the metric h, i.e h = 1 h y i h ij h y j. By finite speed of propagation, 4.3) vx, s, y) =0in{ x + 1 <s< 1 },

7 RADIATION FIELDS 7 We also know that v is smooth up to x =0. So let s vx, s, y) v k s, y)x k, as x 0 denote its Taylor series at x =0. From 4.3) k=0 4.4) v k s, y) =0, s < 1. By assumption s v0,s,y)=0fors<s 0. So v 0 s, y) =0 fors<s 0. Let ) ) n 1 3 n Q = h +, Q = Q 0 + xq 1 + x Q +... A = A 0 + xa 1 + x A +... Then ) A 0 s + Q 0 v s v + +Q 0 + A 0 s 0 s v k +1) s v k+1 + kk +1)v k + k= j+m=k ) Q j v m + A j s v m v k =0fors< 1, k =0, 1,... ) v 1 + Q 1 + A 1 + j+m=k 1 ) ) s + A 0 v 0 x + A j v m + ma j v m ) x k, Since v 0 s, y) =0fors<s 0, then it follows that s v 1s, y) =0fors<s 0. From 4.4), v 1 s, y) =0for s<s 0. Proceeding the same way we find that v k s, y) =0fors<s 0 and k N. So v vanishes to infinite order at {x =0,s<s 0 } and {s = 1,x<ɛ 0 }, and therefore we can extend it as a solution to 4.) in a neighborhood of {s = 1, x =0}, which is equal to zero in x<0. More precisely the extension of v vanishes in {x <0,s<s 0 }, see figure 1. Let φx, s) = x δs+ 1 )+s+ 1 ). For δ>0small enough, this function is strongly pseudoconvex with respect to P at every point of the set Σ = {x =0,s= 1 }. Indeed, as P is of second order, and the level surface {φ =0} is non-characteristic at Σ, then, according to the equation after the first paragraph in section 8.4 of [5], strong pseudoconvexity reduces to the condition 4.5) H p φ ) 0, 1,y,ξ,σ,η) > 0, if p0, 1,y,ξ,σ,η)=H p φ)0, 1,y,ξ,σ,η)=0, ξ,σ,η) 0, where p is the principal symbol of P and H p its Hamiltonian. In this case p = ξσ x ξ hx, y, η), H p = σ + x ξ) x ξ s + and xξ + h x ) ξ H h,

8 8 SÁ BARRETO where h = i,j hij η i η j is the principal symbol of h, and H h denotes the Hamiltonian of h, which only includes derivatives in y and η. So p0, 1,y,ξ,σ,η)= ξσ h0,y,η), H p φ)0, 1,y,ξ,σ,η)=σ +δξ, H p φ ) 0, 1,y,ξ,σ,η)=8ξ +δ h,y,η). If p0, 1,y,ξ,σ,η)=H p φ)0, 1,y,ξ,σ,η)=0, then σ = δξ, h0,y,η)=δξ, and Hpφ0, 1,ξ,σ,η)= h0,y,η)+δ h ) δ,y,η). As h0,y,η) is positive definite, δ can be chosen so that H pφ0, 1,ξ,σ,η) C η, η 0. Thus, for this choice of δ, 4.5) is satisfied. Since v = 0in {x, s, y) : φx, s) > 0}, for x, s small enough, Hörmander s uniqueness theorem, Theorem of [5], implies that v = 0 in a neighborhood of {x =0,s= 1 }. Therefore, there exists ɛ>0 such that vx, s, y) =0 if ɛ<x<ɛ, 1 ɛ<s< 1 + ɛ. Now we apply the same argument used above to show that vx, s, y) = 0 in a neighborhood of {x = 0, s = 1 + ɛ}. Using the compactness of [ 1,s 0 ], the argument above can be repeated a finite number of times to show that there exists ɛ 0 > 0 such that vx, s, y) =0 if ɛ 0 <x<ɛ 0, x + 1 <s<s 0. Given x 1 0, ), we can use a compactness argument and the same method as above to show that The smoothness of v then guarantees that vx, s, y) =0 if 0<x<x 1, x + 1 <s<s 0. vx, s, y) =0 if 0<x, 1 <s<s 0. See figure 1. The following is the main result of this section: Theorem 4.. Let f C0 X) and let s 0 < 0. If R0,f)s, y) =0or Rf,0) = 0) for all s<s 0 then f =0in the set {x, y) X : x< 1 s 0 }. Proof. Since f is compactly supported in the interior of X, there exists such that fx, y) =0for x<. If > 1 s 0, there is nothing to be proved. So we assume that < 1 s 0. It follows from Lemma 4.1, and the observation that ut, z) is odd in t, that ut, z) =0forx< and s 0 1 <t< s Thus it follows from Theorem 4.1 that for X x0 = {,y)}, u0,z)=d t u0,z)=0intheset {z : dz,x x0 ) <s }. In this metric this distance is xz) 1 dz,x x0 )= τ dτ = 1 1 xz). Thus if dz,x x0 ) <s it follows that xz) < 1 s 0.

9 RADIATION FIELDS 9 s s 0 x 0 1/s 0 s= /x+1/ s= 1/x x v=0 1/x 0 φ=0 Figure 1. Region of uniqueness. Theorem.4 in chapter 4 of [7], see also Proposition 4.0 of [1], states that, in the odd dimensional Euclidean space R n, ) n 1 R + 0,f)s, ω) = Rfs, ω), s where Rfs, ω) = fx) dh x x,ω =s is the Radon transform. Thus, in this particular case, Theorem 4. is the equivalent of the support theorem for Radon transform, see [4], which says that if f C0 R n )andrfs, ω) =0for s >ρ,then fz) =0for z >ρ.however, the proof given here is by no means a simplification of Helgason s proof. 5. Acknowledgments I am grateful for the financial support I received from the NSF under grant DMS I thank the two referees for carefully reading the first version of this paper and for making numerous suggestions and corrections. References [1] F.G. Friedlander. Radiation fields and hyperbolic scattering theory. Math. Proc. Camb. Phil. Soc., 88, , 1980). [] F.G. Friedlander. Notes on the wave equation on asymptotically Euclidean manifolds. J. of Func. Anal. 184, no. 1, 1-18, 001). [3] A. Hassell and A. Vasy. The spectral projections and the resolvent for scattering metrics. J. Anal. Math ). [4] S. Helgason. The Radon Transform. Birkhauser, nd edition, 1999). [5] L. Hörmander. The analysis of linear partial differential operators IV. Springer-Verlag, [6] M.S. Joshi and A. Sá Barreto. Recovering asymptotics of metrics from fixed energy scattering data. Invent. Math. 137, , 1999) [7] P. Lax and R. Phillips. Scattering theory. Academic Press, 1989, Revised edition. [8] R.B. Melrose. Geometric scattering theory. Cambridge Univ. Press, [9] R.B. Melrose. Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces. Spectral and Scattering Theory. M. Ikawa, ed.) Marcel Dekker, [10] R.B. Melrose and M. Zworski. Scattering metrics and geodesic flow at infinity. Invent. Math. 14, ). [11] L. Robbiano. Théorème d unicité adapté au contrôle des solutions des problémes hyperboliques. Comm. in P.D.E 16, , 1991).

10 10 SÁ BARRETO [1] L. Robbiano and C. Zuilly. Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients. Invent. Math. 131, , 1998). [13] A. Sá Barreto. Radiation fields and inverse scattering on asymptotically hyperbolic manifolds. Preprint, 00. [14] D. Tataru. Unique continuation for solutions to PDE s: between Hörmander s theorem and Holmgren s theorem. Comm. in P.D.E. 0, , 1995). [15] D. Tataru. Unique continuation for operators with partially analytic coefficients. J. Math. Pures Appl. 78, , Department of Mathematics Purdue University, West Lafayette IN 47907, U.S.A. address: sabarre@math.purdue.edu