Propagation of polyhomogeneity, Diffraction and Scattering on Product Cones

We consider diffraction of waves on a product cone. We first show that diffractive waves enjoy a one-step polyhomogeneous asymptotic expansion, which is an improvement of Cheeger-Taylor's classical result of half-step polyhomogeneity of diffractive waves in [CT82a], [CT82b]. We also conclude that on product cones, the scattering matrix is the diffraction coefficient, which is the principal symbol of the diffractive half wave kernel, for strictly diffractively related points on the cross section. This generalize the result of Ford, Hassell and Hillairet in 2-dimensional flat cone settings [FHH18]. In the last section, we also give a radiation field interpretation of the relationship between the scattering matrix and the diffraction coefficient.


Introduction
In this paper, we study the diffraction coefficient of the wave equation u = 0 and the scattering matrix corresponding to the Helmholtz equation ∆ − λ 2 u = 0 on a cone C(N ).The diffraction refers to the effect that when a propagating wave encounters a corner of an obstacle or a slit, its wave front bends around the corner of the obstacle and propagates into the geometrical shadow region.When studying the wave equation on cones, we see that its singularities likewise split into two types after they encounter the cone point.One propagates along the natural geometric extensions of the incoming ray, while other singularities emerge at the cone point and start propagating along all outgoing directions as a spherical wave.The outgoing singularities are described to leading order by a diffraction coefficient, which is one of the central objects we study in this paper.
The (stationary) scattering theory of the wave equation gives an approach to studying the continuous spectrum of the Laplacian on non-compact manifolds.The scattering matrix, which, intuitively speaking, maps the incoming solution at the infinity of the (stationary) wave equation to the outgoing solution, is a central object of study.
In this paper, we focus on the diffraction and the scattering of the wave equation on cones.For notational purposes, we denote our n-dimensional cone by C(N ), which is R + ×N n−1 with metric dr 2 +r 2 h(θ, dθ) where h(θ, dθ) is the metric on the smooth manifold N n−1 .We consider the fundamental solution to the wave equation on C(N ) corresponding to the Friedrichs extension of the Laplacian.For t large enough, the singularities of the fundamental solutions consist of two parts by [CT82a] [CT82b].One lies on a sphere (up to reflection by the boundary of N ) of radius t to the initial point (r , θ ), while the other part lies on a sphere of radius t − r around the cone point and is conormal to {r = t − r }.We refer the latter to the diffractive wave front, and it is the main object of our interest in this paper.The former notion will be called the geometric wave front.See Figure 2 in Section 2 for an example of the geometric and diffractive waves.The diffraction coefficient is therefore defined by comparing the principal symbol of the incoming wave to the principal symbol of the diffractive wave, or equivalently, reading off the principal symbol of the diffractive half wave kernel.On the other hand, the scattering matrix is defined by considering the leading order behavior to the stationary wave equation ∆ − λ 2 u = 0 under certain boundary/asymptotic conditions; the solution u then has the leading order behavior u ∼ a + (θ)r − n−1 2 e iλr + a − (θ)r − n−1 2 e −iλr + O(r − n+1 2 ) as r → ∞ where a + (θ) is uniquely determined by a − (θ); the scattering matrix S(λ) is then defined by the unitary map from a + (θ) to a − (θ) for λ ∈ R\{0}.
The following theorem thus relates two central concepts of the theories of diffraction and scattering: Theorem 1.1.Away from the intersection of the geometric wave front and diffractive wave front, the kernel of the diffractive half wave propagator is a conormal distribution of the form: The principal symbol K 0 (r, r , θ, θ ) of the diffractive half wave kernel U D (t, r, r , θ, θ ) is related to the kernel of the scattering matrix S(λ, θ, θ ) by (1) K 0 (r, θ, r , θ ) = (2π) −1 (rr ) − n−1 2 S(λ, θ, θ ).
It is worth while to point out here that we are actually showing that the smooth part of the scattering matrix corresponds to the diffraction coefficient, while the singular part of the scattering matrix corresponds to the geometric wave.This was proved in a special case of 2-dimensional flat cones by Ford, Hassell and Hillairet [FHH18].
We also give a finer description of the structure of the diffractive wave by showing that it is one-step polyhomogeneous thus improving the result of half-step polyhomogeneity of diffractive waves that appears in Cheeger-Taylor [CT82a] [CT82b, Theorem 5.1, 5.3].Consequently, one half of the coefficients that appear in Cheeger-Taylor's expansion must vanish.
Theorem 1.2.The symbol K(r, r , θ, θ ; λ) of the diffractive half wave kernel U D (t) is one-step polyhomogeneous in λ for λ > 0, i.e. time delay that arrive at infinity.We define the forward radiation field R + as the limit, as time goes to infinity, of the derivative of the forward fundamental solution of the wave equation along certain light rays.By reversing time one can define the backward radiation field R − .The forward/backward radiation field R ± is related to the scattering matrix S(λ) in the following formula: (3) This was first introduced by Friedlander [Fri80] in R n , and was proved later by Sá Barreto [SB03] for smooth asymptotically Euclidean manifolds.The intuition and motivation of Theorem 1.1 also come from the following facts: In principle, the scattering operator: is given by the Fourier conjugation of the leading symbol to the forward fundamental solutions [Fri01]; the scattering matrix is the Fourier conjugation of the scattering operator [SB03].This suggests that the scattering matrix and the diffraction coefficient should be the same up to some constant or scaling in radial variables.
We combine Cheeger-Taylor's functional calculus on cones and Melrose-Wunsch's propagation of conormality to give a simpler calculation of the diffractive coefficient and the one-step polyhomogeneity.As for determining the scattering matrix, we consider it modeby-mode to reduce the original equation to a Bessel equation.
The outline of this paper is as follows.In section 2, we prove a characterization of the one-step polyhomogeneous solutions to wave equations on cones and the propagation of one-step polyhomogeneity for the diffractive wave.These will be used in section 3 to determine the diffraction coefficient.Then in section 3, we compute the diffraction coefficient using the functional calculus on cones and the propagation of conormality.In section 4, we focus on computing the scattering matrix and give the diffraction-scattering relation, Theorem 1.1, in the end.Finally in section 5, we give an interpretation of this result using the radiation field.
Acknowledgment.The author wants to thank Jared Wunsch for proposing this interesting topic and genuinely thank him for his guidance and many helpful discussions as the author's advisor.The author also wants to thank Dean Baskin, Jeremy Marzuola and Antônio Sá Barreto for helpful discussions about the radiation field during the author's staying in MSRI.

Conic Diffraction Geometry
In this section we recall some basic notions on the geometry of product cones together with the geometric and diffractive wave on it.
Again, we denote the product cone with total dimension n by C(N ) ∼ = R + × N n−1 with metric dr 2 + r 2 h(θ, dθ), where N is a smooth manifold and h is the metric restricted to N .This is a particular case of general cones (C(Y ), g) with metric g = dr 2 + r 2 h(r, dr, θ, dθ), where in our case the metric h does not have r and dr dependence.Sometimes we also use X to denote the product cone for simplicity when the cross section N is not involved explicitly in the discussion.Without loss of generality we assume N has one connected component since otherwise we can restrict to a single component.
The Laplacian the Fourier normalization of r-derivative and ∆ θ is the Laplacian on N .Here we set ∆ to be the Friedrichs extension of the Laplacian acting on the domains C ∞ c ( X).The Friedrichs domain is defined as < ∞ , and D s denotes the corresponding domain of ∆ s/2 .Later in this paper, we will use L 2 (R t ; D s ) to denote the regularity on spacetime R × X.The following proposition from [MW04, Proposition 3.1] gives a characterization of domains of Friedrichs extension: Proposition 2.1.(Domains of the Friedrichs extension) For n > 4, . For n = 3, the same is true for w ∈ (−1, 0).
. In all cases, ∆ is an unbounded operator: and Dom(∆) coincides with the domain of the Friedrichs extension.
For a more detailed discussion we refer to [MW04] and [MVW08], though it is worth pointing out the following corollary to the proposition: Corollary 2.1.1.If u ∈ E (X o ), i.e., a compactly supported distribution in the interior of the cone X, then u ∈ D s is equivalent to u ∈ H s (X).
The d'Alembertian acting on the spacetime R × X is = D 2 t − ∆ and we also define the half-wave propagator U (t) as We now consider the diffraction of waves with respect to the cone point, which has been studied in detail by Cheeger and Taylor in [CT82a] and [CT82b] for product cones.There are two different notions of geodesics on cones, one more restrictive than the other.We can see that these notions on product cones are special cases of [MW04] on general cones with metric dr 2 + r 2 h(r, θ, dθ) within a small neighborhood of the cone point.
Definition 2.1.Suppose γ : (− , + ) → C(N ) is a piecewise geodesic on the cone C(N ) hitting the cone point only at time t = 0, then: • The curve γ is a diffractive geodesic if the intermediate terminal point γ(0 − ) and the initial point γ(0 + ) lie on the boundary {0} × N .
• The curve γ is a geometric geodesic if it is a diffractive geodesic such that the intermediate terminal point γ(0 − ) and the initial point γ(0 + ) are connected by a geodesic of length π on the boundary {0} × N (with respect to the boundary metric h).• The curve γ is a strictly diffractive geodesic if it is a diffractive geodesic but not geometric geodesic.
As pointed out in [MW04], the geometric geodesics are those that are locally realizable as limits of families of ordinary geodesics in the interior X o .Figure 1 gives geometric and diffractive geodesics at the cone point.In this paper, we focus on the diffraction coefficient and the scattering matrix away from the points that are related by the geometric geodesics, i.e. we consider the pair (r, θ) and (r , θ ) with d h (θ, θ ) = π for the study of the diffraction coefficient and the smooth part of scattering matrix.At the intersection of the geometric and diffractive fronts, the structure of the singularities is more complicated.This can be seen intuitively from the following picture of diffraction by a slit in Figure 2, which is equivalent to a product cone of angle 4π.In the case of 2-dimensional flat cone, the wave kernel close to the intersection is then a singular Fourier integral operator in a calculus associated to two intersecting Lagrangian submanifolds.We refer to [FHH18] for a detailed discussion.

Figure 2. Geometric and Diffractive Front
Diffraction by a slit in R 2 which is a cone with cross section N = (0, 2π).The wave source is marked on the picture.The diffractive front D is the boundary of region III (the red circle), while the geometric front G is part of the boundary of region II (the blue arcs).In this picture, the intersection of these two wave fronts G ∩ D consists of two points.

Propagation of Polyhomogeneity
In this section, we briefly discuss the propagation of one-step polyhomogeneity on product cones in order to prove our diffractive symbol estimates.This also gives a onestep polyhomogeneity of the diffractive wave.Prior to these, we give a characterization of one-step polyhomogeneous solution to the wave equation on cones.Without loss of generality, we assume that u is the spherical wave hitting the cone point at time t = 0.
We first introduce the polyhomogeneous symbols: ) is called (one-step) polyhomogeneous of order m if it admits an asymptotic symbol expansion: where a k are homogeneous symbols of order m − k, i.e., We denote the polyhomogeneous symbol class of order m by S m phg (X; R l ).We introduce the radial and tangential operators for later reference.Let (4) R = tD t + rD r denote the radial vector field on R × X.And we also define tangential operators on N : (5) Note that for the above operators and the d'Alembertian , we have a group of commutator relations: which will be useful later to prove the propagation of polyhomogeneity.These commutator relations are motivated by Melrose and Wunsch's argument on propagation of conormality: from [MW04, Theorem 4.8], these commutator relations imply that if u = 0 and u is conormal to {t + r = 0} with respect to D s for t < 0 then u is conormal to {t − r = 0} with respect to D s for t > 0. The conormality here is characterized by the operators Y s , R and through the following definition: with respect to D s if the iterative regularity: for all i, j, k ∈ N and t ≶ 0. We use ID s to denote this conormality, where I stands for iterative.
For a more detailed discussion on the conormal distribution, we refer to [Hör09].By the Hörmander-Melrose theory, on the product cone X of total dimension n, the iterative regularity ID −m−1/2− for any > 0, with order that we will discuss later, is equivalent to the oscillatory integral definition of conormality of order m − (n − 1)/4 which is defined in the following sense: of order k if it locally admits an oscillatory integral representation for t ≶ 0: denote the space of all distributions on R × X that conormal to {t ± r = 0} of order k, where N * {t ± r = 0} is the conormal bundle of {t ± r = 0}.
Remark 3.1.We use S m (X; R λ ) to denote the Kohn-Nirenberg symbol class on the cone X of the order m hereafter.
Following [Hör09], we have the following equivalent relation between the previous two definitions of conormality.
Theorem 3.1.The iterative regularity definition of conormality is equivalent to the oscillatory integral definition, more precisely, we have the following inclusions of conormal distributions on R × X: We also need the following interpolation lemma to raise the iterative conormal regularity from the Sobolev regularity and the lower iterative regularity.We refer the readers to [MVW08, Lemma 12.2] for a detailed proof of the lemma.It is presented in the form of coisotropic regularity there but the essence is the same.
Suppose now that u = 0, and for t ≶ 0, We can thus write, for t ≶ 0, u = e i(t±r)λ a(r, θ, λ) dλ mod C ∞ for some a ∈ S m (X; R λ ).From this, recall also that u ∈ L 2 (R t ; D s ) for all s < −m − 1/2.We employ the notation u ∈ L 2 (R t ; D −m−1/2−0 ) to denote this type of space hereafter.For the propagation of conormality, note that the symbols of Y 1 , and R consist of the defining functions of N * {t ± r = 0}; by showing the iterative regularity: for any i, j, k ∈ N, t ≶ 0 and for some order s, we can show that the conormality is therefore preserved.For the propagation of polyhomogeneity, we need stronger conditions.In fact, in addition to the above preservation of iterative tangential regularity, we need that applying the radial vector field with particular shifts improves the regularity by one-step at each time.We show later that this actually leads to a complete characterization of polyhomogeneous distributions.Before proceeding to the proof of the complete characterization, we start by showing a characterization of the leading order polyhomogeneity.This characterization is due to Baskin and Wunsch [BW19].From now on, we use the notation ID s−0 to denote that u lies in the iterative regularity class ID s− for any > 0.
Lemma 3.3 (Characterization of the Leading Polyhomogeneity).Assume u ∈ C ∞ , u is conormal to {t ± r = 0} and takes the oscillatory integral form (7) for t ≶ 0 (away from the cone point) with a ∈ S m (X; R λ ).Then Then we have (1) If a is polyhomogeneous of order m, then (2) Conversely, suppose that where a m is homogeneous of degree m and r m−1 ∈ S m−1+0 (X; R λ ), where Proof.Applying the operator R − iα to (7) and integrating by parts, we obtain (8) (R − iα)u ≡ e i(t±r)λ ((t ± r)λ + rD r − iα)a(r, θ, λ) dλ here we use ≡ to denote equivalence modulo C ∞ -errors.Now we also need to use the thus a must satisfy the transport equation where ∆ is the Laplacian on cones; in particular, since ir 2λ ∆a ∈ S m−1 (X; R λ ), this forces Plug this into the right side of (8) yields where e ∈ S m−1 (X; R λ ) is the remainder term.Thus if a ∈ S m (X; R λ ) is a polyhomogeneous symbol, then so is (λD λ + im)a ∈ S m−1 (X; R λ ) and we find that by the equivalence of two definitions of conormality.This proves the first part of the lemma.
Conversely, if (R − iα)u ∈ L 2 (R t ; D −m+1/2−0 ), by the commutator relations (6) and the fact that the symbol of R is one of the defining functions of {t ± r = 0}, we have (R − iα)u ∈ ID −m−1/2−0 by conormality of u.Thus by Lemma 3.2 (also see [MVW08,Lemma 12.2]), we know (R − iα)u is also conormal with iterative regularity This forces the symbol of (R − iα)u to be in the class S m−1+0 (X; R λ ).By the proof of the first part, consider the order of the symbol of (R − iα)u gives Equivalently, Integrating it to infinity yields This implies that we must have the leading asymptotic decomposition as in the statement of lemma.
For later reference we record a sharpening of the symbol computation above.In particular, note that if u ∈ C ∞ then we can compute the symbol of (R − iα)u explicitly by substituting the full transport equation ( 9) into (8) to obtain Therefore (R − iα) acting on u can be characterized by (−λD λ − im + r 2λ ∆) acting on its symbol a.We now generalize Lemma 3.3 to get a characterization of full polyhomogeneity by induction.The result given in the following lemma is similar to the characterization given by Joshi [Jos97] for polyhomogeneous Lagrangian distributions on smooth manifolds, though the Hamilton vector field of our operator R is not a multiple of the radial vector field of fiber variables as in [Jos97].
Lemma 3.4 (Characterization of the Complete Polyhomogeneity).Assume u ∈ C ∞ , u is conormal to {t ± r = 0} and takes the oscillatory integral form (7) for t ≶ 0 with a ∈ S m (X; R λ ).Set

Then we have
(1) If a is polyhomogeneous of order m, then (2) Conversely, suppose that for the above α 1 , ..., α k , where a m , a m−1 , ..., a m−k+1 are homogeneous symbols with the degrees same as their indices and r m−k ∈ S m−k+0 (X; R λ ).
Proof.First assume a ∈ S m (X; R λ ) is polyhomogeneous.By an integration by parts argument and the fact that u ∈ C ∞ , we see that in the Kohn-Nirenberg class.This is aided by the commutator relation (6) which makes k j=1 (R − iα j )u for all k ∈ N again a solution to the wave equation modulo smooth terms, so we can apply the transport equation substitution iteratively as we did in Lemma 3.3.At each step, −λD λ + i n+1 2 − iα j + r 2λ ∆ acting on polyhomogenous symbols still gives polyhomogeneous ones.Also note that r 2λ ∆ always lowers the symbol order by one, and the coefficient α j , when combining with R to annihilate the principal part of each step, only depends on the order of the symbol it acts on.
By Lemma 3.3, we know for Taking this as the new symbol b of the conormal distribution, then applying (R − iα 2 ) to the new distribution gives Applying this argument repeatedly with we proved that the first statement is true.
For the second part, we can work by induction.In the previous lemma, we have showed that (R − iα 1 ) raises regularity almost by one implies the leading one-step polyhomogeneity.Assume the conclusion in the second part of this lemma is true for up to k-terms one-step polyhomogeneity, i.e., assume with a m , a m−1 , ..., a m−k+1 homogeneous symbols and r m−k ∈ S m−k+0 (X; R λ ).Thus as in the proof of the first part.Now we consider This has the oscillatory integral form: Now we apply Lemma 3.2, since by the assumption conormal to {t ± r = 0} and thus Since r 2λ ∆ lowers the symbol order by one, we have: Plugging in α k+1 defined above, using the same argument as in the proof of Lemma 3.
Considering that the action of −λD λ + i n+1 2 − iα j + r 2λ ∆ for 1 ≤ j ≤ k on the symbol a gives b k , together with a being k-term one-step polyhomogeneous (12), a must take the form where a m , a m−1 , ..., a m−k are homogeneous with degrees given by their indices and r m−k−1 ∈ S m−k−1+0 (X; R λ ) (c.f.[Jos97, Proposition 2.1]).We thus finished our induction.
Remark 3.2.It is worth to point out that this lemma cannot be generalized directly to give a similar characterization of one-step polyhomogeneity on general non-product cones (R × Y, dr 2 + r 2 h(r, θ, dθ)).This is due to the fact that the commutator equation is now [ , R] = −2i + E with E an error term.The existence of this error term makes even (R − iα 1 )u no longer a solution to the wave equation, for which being a solution is an essential property for us to build our characterization.
Recall that our ultimate goal in this section is to develop the propagation of polyhomogeneity.Prior to this we use the foregoing results to obtain a propagation of leading order polyhomogeneity which will be used later to give a diffraction symbol estimate.Then we can show the propogation of full polyhomogeneity as a corollary.
First, we recall from [MW04, Theorem 4.8] the more basic results on propagation of conormality.These follow easily in the situation at hand by commutation of R k and Y s through the equation, together with the observation that the symbols of R, Y 1 , and form a set of defining functions to the conormal bundle of {t ± x = 0}.The continuity of the evolution map asserted in the following proposition follows from the proof of [MW04, Theorem 4.8] or, as usual, from the Inverse Mapping Theorem; the essence of the direct proof is that norms of powers of the test operators are conserved relative to domains of powers of the Laplacian which agree with Sobolev spaces away from r = 0; converting these estimates to estimates in symbol spaces requires a Sobolev embedding step, which loses at most a fixed number of derivatives (which can then be interpolated away up to an ).For brevity, we abbreviate the restriction to a time interval by a (c,d) ≡ a| t∈(c,d) .
(2) The map from negative time data to positive-time data is continuous in the following sense: for any a < b < 0 < c < d, any > 0, and any M, there exist M and C such that if we write u in the form (7) with symbol a then Here the symbol norm S m M is given by Remark 3.3.We will abbreviate the existence of symbol estimates of this type as "a satisfies effective estimates" in what follows.The effective estimates here is crucial to prove the propagation of polyhomogeneity and to compute the diffraction coefficient using the mode-by-mode solutions.
(R − iα)u = e i(t+r)λ b(r, θ, λ)dλ for t < 0, while u = e i(t+r)λ a(r, θ, λ)dλ for a ∈ S m phg (X; R λ ).Then for t > 0, since by commutator relations (6) in the beginning of this section (R − iα)u = 0, Proposition 3.5 implies that for t > 0 and for all > 0, with symbol seminorms depending on those of b.By definition of conormal distribution, this implies (R − iα)u ∈ L 2 (R t ; D −m+1/2−0 ) for t > 0, Thus by Lemma 3.3 we have the corresponding symbol decomposition of a and we therefore proved the first part of our proposition.
To prove the quantitative estimates of the second part, we note that by (10) and the quantitative propagation of conormality from Proposition 3.5 we have for t > 0 for all > 0, together with symbol estimates: whenever α − < β − < 0 < α + < β + , for all M, for M .Now without loss of generality we can assume m = 0, otherwise we use to make the order of a zero.This yields again enjoying the same type of effective estimates as b.Thus (14) D λ a = −λ −1 b + r 2λ 2 ∆a ∈ S −2− (X; R λ ) for t ∈ (α + , β + ), again the RHS enjoys effective estimates since r/(2λ 2 )∆ is a continuous map from symbols of order s to symbols of order s − 2, and multiplication by powers of λ is a continuous symbol map.In particular, then Integrating ( 14) from λ = ±∞ with C(r, θ, sgn λ) as constant of integration (cf.[Jos97, Propasition 2.1] for this strategy) yields Here the term C(r, θ, sgn(λ)) corresponds to the homogeneous term in a, while the integral term e(r, θ, λ) is a remainder that lies in S −1+ , which corresponds to r m−1 , and satisfies effective estimates directly following from the above two estimates (15) and ( 16).
We finally state the propagation of one-step polyhomogeneity as a corollary of our previous results, which we summarized as Theorem 1.2 in the introduction: Corollary 3.6.1 (Propagation of One-Step Polyhomogeneity).Suppose that u = 0 and that for t < 0, u ∈ I m−(n−1)/4 phg (R × X; N * {t + r = 0}).Then for t > 0, u ∈ I m−(n−1)/4 phg (R × X; N * {t − r = 0}) and has an oscillatory integral representation of the form (7) where the symbol a is polyhomogeneous.
Remark 3.4.This corollary in particular shows that the diffractive wave enjoys one-step polyhomegeneity, which improves the result of half-step polyhomogeneity given implicitly in Cheeger-Taylor [CT82a]

Diffraction Coefficient on Product Cones
Assume t > r .The half wave propagator can be decomposed as where the first part is the geometric wave propagator and the second is the diffractive wave propagator with WF U (t)\WF U D (t) ⊂ {bicharacteristics of the geometric geodesics} .
On the diffracted front D and away from the geometric front G (see Figure 2), we can write the kernel of the diffractive half wave propagator as where K(r, θ; r , θ ; λ) is a polyhomogeneous symbol of order 0, and U D (t) is a conormal distribution to {r + r = t}.Here we confuse the propagator with its Schwartz kernel and the assumption t > r is to ensure the existence of the diffractive front.This expression is due to Cheeger-Taylor [CT82a] [CT82b] and Melrose-Wunsch [MW04], and it can be seen as a consequence of conormality of the diffractive wave.We show later in this section that the symbol of the diffractive half wave kernel has the form: where ϕ j is the j-th Fourier mode of ∆ θ on N and σ T j (t, r, r ; λ) is the total symbol of j-th mode of diffractive fundamental solution E j (t, r, r ).We define σ P j (t, r, r ; λ) to be the principal symbol of σ T j .And the diffraction coefficient is defined to be K 0 (r, θ; r , θ ) := j σ P j (t, r, r ; λ)ϕ j (θ)ϕ j (θ ).
The idea of this section is the following.We first consider diffractions of spherical waves.By [MW04, Theorem 4.8], the spherical diffractive wave is cornormal to {r + r = t} for t > r .Then we consider a mode-by-mode decomposition of the diffractive fundamental solution, and shows that the principal symbol of the diffractive half wave kernel is given by the sum of the principal symbol of the diffractive wave of each Fourier mode.This reduces the computation of the diffraction coefficient to each mode.
The construction is based on the functional calculus [Tay13] on product cones.We first consider the exact solution (on a single mode) to the half wave equation.Recall the Laplacian on a product cone is and we define ν j := µ 2 j + α 2 with α = − n−2 2 , where µ 2 j , ϕ j denote eigenvalues and eigenfunctions of ∆ θ .If we take This can be seen by reducing (∆ − λ 2 )(g j ϕ j ) = 0 to a Bessel equation by change of variables.For a detailed treatment on this solution on a single Fourier mode, we refer to [BY19].By the functional calculus on product cones, for g(r, θ) = j g j (r)ϕ j (θ) ∈ L 2 (C(N ); C).We define the operator ν on N by Using (19), we take g(r, x) to be one single mode of spherical wave δ(r − r )ϕ j (θ) and the operator as half wave operator for fixed r > 0, where H is the Heaviside function.
From the previous discussion we have modulo the singularities at {r = r + t} for the first equation.Now we compute the diffractive fundamental solution E D (t, r, r , θ, θ ).We first regularize it by averaging it angularly to instead study for an arbitrary ϕ ∈ C ∞ c (N ) supported close to a single point θ 0 ∈ N. Using the Plancherel theorem, Fourier expanding ϕ in N , i.e., taking the eigenfunction expansion gives Now what we can compute by the asymptotic expansion of Bessel functions is the principal symbol of the conormal solution u ν j at N * {r + r = t} (as we will do in the later part of this section); for now we write these principal symbols σ P j (t, r, r , λ)ϕ j (θ).Formally, the principal symbol of u ϕ is the sum of the principal symbols of u ν j , though we have to be careful to show that the subprincipal symbols of u ν j will not add up and contribute to the principal symbol of u ϕ .This leads to the following theorem: Theorem 4.1 (Principal Symbols of Diffraction).The principal symbol σ P of u ϕ is equal to the sum of principal symbols of mode-by-mode solutions: i.e., the subprincipal symbols of u ν j won't add up and contribute to principal symbol of u ϕ .
Proof.The convergence of this sum is due to the fact that c ν j are rapidly decaying with respect to ν j as j → ∞, which comes from the fact that ϕ is C ∞ so the Fourier coefficient rapidly decays; we can check that the series of principal symbols converges.However, we need information about the growth rate of the symbol remainders: the equality of conormal distributions tells us that really where we use σ T to denote the total symbol, i.e., the full amplitude of the conormal distribution with the canonical choice of phase function φ(t, r, λ) = (t ± r)λ.Thus, it remains to check that the sum of remainder terms converges in the topology of symbols of order m − 1 + .By Proposition 3.6 we find that any desired symbol semi-norm of σ T j − σ m,j is bounded by some symbol semi-norm of the symbol of solution u ν j for t 0. Examination of the initial data shows that each of these norms grows at most polynomially in ν j (with the growth arising from θ derivatives).Thus, since c ν j decays rapidly, the series (22) does indeed converge in every symbol seminorm with respect to the S m−1+ topology, and the subprincipal terms cannot affect the principal symbol of the sum.2Following Theorem 4.1, we now construct the principal symbol of the diffractive fundamental solution.We fix χ ∈ C ∞ (N ) equal to 1 near θ 0 and use the above results for all ϕ supported on {χ = 1}.We have established that Now let ϕ approach δ(θ ) in the sense of distribution (with θ not geometrically related to θ), so that its Fourier coefficients c ν j approach ϕ j (θ ).We then obtain in the limit, in a neighborhood of any pair θ and θ that are related by strictly diffractively geodesics, i.e., for θ, θ with d h (θ, θ ) = π, σ P (E D (t, r, r , θ, θ )) = σ P j (t, r, r , λ)ϕ j (θ)ϕ j (θ ), as desired.
In order to get the diffraction coefficient, now it remains to compute the principal symbol of the diffactive fundamental solution for each mode: σ P j (t, r, r , λ).We employ the functional calculus on product cones and the conormality of diffractive waves.
Note that for positive ν and z, the Bessel function J ν (z) is the real part of Hankel function H (1) ν (z).Thus using the asymptotic formulas of Hankel functions [DLMF, 10.17.5], we can extract the leading part of J ν j (λr) as the principal symbol with phase variable λ: We now combine this Bessel asymptotics together with (23) to get the diffractive principal symbol.Thus, e iλ(r+r −t) e −i(ν j π+ π 2 ) dλ ϕ j (θ) modulo singularities at the conormal bundle N * {r = r + t} and the lower order singularities at the conormal bundle N * {r + r = t}.The second equality is due to the fact that diffractive wave is conormal to {r + r = t} [MW04, Theorem 4.8], so the only part in J ν j (λr) and J ν j (λr ) that contributes to the diffractive principal symbol is each of their first terms in the asymptotic expansion (24), and the remaining terms are smooth near N * {r + r = t}, hence will not contribute to the diffractive wave.Now, comparing the above equation with the general formula for the diffractive half wave kernel (17), we have the diffraction coefficient: where ν = ∆ θ + n−2 2 2 .

Scattering Matrix on Product Cones
Consider the leading order behaviors of the solutions of ∆ − λ 2 u = 0 under the asymptotic condition: where a − /a + is called the incoming/outgoing coefficient.Then a + (θ) is uniquely determined by a − (θ) and the scattering matrix S(λ) is the unitary map from a − (θ) to a + (θ) for λ ∈ R\{0}.This property is known for smooth asymptotically Euclidean manifolds [Mel94], and we show below in Proposition 5.1 it is also true on product cones.Meanwhile, we show that the scattering matrix on a product cone is , which is related to the diffraction coefficient (25) as Here we should note that we only consider the smooth part of the scattering matrix.By [MZ96], on smooth asymptotically Euclidean manifolds (smooth manifolds with large conical ends), the scattering matrix is a Fourier integral operator with the canonical relation given by geodesic flow at time π.In the previous section, we found the diffraction coefficient of the points on product cones which are strictly diffractively related.This corresponds to the smooth part of the scattering matrix, i.e.S(λ, θ, θ ) for d h (θ, θ ) = π, where S(λ, θ, θ ) is the kernel of the scattering matrix.From now on we use the name scattering matrix without saying that it means the smooth part.
We define the scattering matrix through the following proposition: Proposition 5.1 (The Scattering Matrix on Product Cones).The scattering matrix S(λ) on the product cone C(N ) for λ ∈ R\{0} is a unitary operator: where a + (θ) is the outgoing coefficient in the asymptotic expansion (26) and it is uniquely determined by the incoming coefficient a − (θ).Moreover, the scattering matrix on a product cone takes the form: Proof.Consider the homogeneous equation on C By Section 2, this is where the first space is defined via the volume form induced by the conic metric, and the latter spaces can be identified with the space L 2 (R + ; r n−1 dr).E j denotes the j-th eigenspace.

The equation (27) thus becomes
where µ 2 j is the eigenvalue to ∆ θ with the eigenfunction ϕ j .We thus reduce the equation ( 27) to for all j.By changing variable ρ with ρ = λr, we have Writing v j (r) = ρ σ ω j (ρ), we can replace the previous equation by Rewrite it into the following form Setting 2σ + (n − 1) = 1, i.e., σ = 1 − n 2 , the equation above then becomes This is a homogeneous Bessel equation ( 29) Its general solutions in terms of Bessel/Hankel functions are the linear combinations of H (1) ν (z) and H (2) ν (z).We can then construct solutions to (27) with the prescribed boundary condition using Bessel functions asymptotics as following.Consider the general solutions to (29) Noticing that from the above change of variables, v j = ρ 1− n 2 ω j (ρ) and ρ = λr, we can obtain general solutions to (28): ν j (λr) and thus they have general solutions to (27): For solutions to the wave equation, we have the energy norm • E : and define the wave group W (t) by We know by conservation of energy that W (t) is a strongly continuous group of unitary operators.
We now define a map which is called the forward radiation field.Its existence follows from Theorem 6.1.Similarly we can define the backward radiation field as Sá Barreto also proved in [SB03] that the forward and backward radiation fields are in fact unitary on smooth asymptotically Euclidean manifolds under the energy norm of the Cauchy data.It leads to the definition of the scattering operator which is essentially the Fourier conjugation of the scattering matrix.Theorem 6.2.The maps R ± extend to isometries The scattering operator defined by is unitary on L 2 (R × N ); the scattering matrix S is given by conjugating the scattering operator with the partial Fourier transform in the s-variable: Here we note that the proof of this theorem in [SB03] can be extended to product cones from smooth asymptotically Euclidean manifolds.This is because the proof relies on the fact that the Laplacian on product cones has purely absolute continuous spectrum so that we could apply the proposition that the Poisson operators give isometries from the absolute continuous spectral subspace of ∆ to L 2 (R × N ) [HV99, Proposition 9.1] to give an isometry between the energy norm of the initial data (f 0 , f 1 ) and the Fourier transform of the forward/backward radiation field as in [SB03].Otherwise, although there are still isometries between the absolute continuous spectral subspace of ∆ and L 2 (R × N ), the energy norm would need to include the contribution from the discrete eigenmodes.
In the rest of this section, we construct the scattering matrix from the diffractive coefficient using the radiation field, and we shall see that the scattering matrix agrees with the diffraction coefficient up to scaling.In other words, using the radiation field, we give a second proof to Theorem 1.1.
Then from (34), we get the kernel of the scattering operator S in terms of the fundamental solution: Now we compute the scattering operator using (36) and the kernel of the diffractive wave propagator.Since we showed the diffractive wave enjoys one-step polyhomogeneity in Section 3, by the formula of the kernel of the diffractive wave propagator, we can make a WKB style ansatz for the diffractive wave of E(z, z , t) as the following: E(z, z , t) ≡ (rr ) − n−1 2 e i(r+r −t)λ a(r, r , θ, θ , λ) dλ modulo singularities other than N * {r + r = t} with symbol a = ∞ k=0 āk (r)λ −k , where ā0 is the diffraction coefficient.Here the symbol a and each term āk in its expansion depend on (r, r , θ, θ ), though we only emphasis the r dependence for our purpose.
By the WKB type approximation, this will be reduced to the symbol expansion a k solving a series of transport equations.This leads to the result that the diffractive wave takes the form of polyhomogeneity in λr: (rr ) − n−1 2 e i(r+r −t)λ ∞ k=0 a k • (λr) −k dλ where a k does not depend on r or λ.Now the kernel of the scattering operator of the diffractive wave can be computed as the following using equation (36):

Figure 1 .
Figure 1.Diffractive and geometric geodesics The blow up pictures of cone C(N ) at cone point {0} × N : [C(N ); N ].On the right are the diffractive geodesics, while on the left are geometric geodesics and they are connected by geodesics in the boundary with length π.
3, this forces b k to take the form b k = bm−k +r m−k−1 , where bm−k is homogeneous of order m− k and rm [CT82b, Theorem 5.1, 5.3].This half-step polyhomogeneity is further explicitly pointed out by Ford and Wunsch in [FW17, Proof of Proposition 2.1].