Appendix A. Scattering Theory for Time-Dependent Potentials

We recall the basics of the scattering theory for time-dependent perturbations of the wave equation by restricting it to time-dependent potentials. We follow Cooper and Strauss [3, 4], where it was introduced for moving obstacles, and its adaptation to time-dependent potentials in [20]. Some of the statements below are new, however, like Theorem A.4 and Theorem A.5. This theory is a natural extension of (a part of) the Lax-Phillips scattering theory. We consider the wave equation (3) with a smooth time-dependent potential q(t, x) supported in the cylinder \(\mathbb \times \overline\). We assume \(n\ge 3\), odd to avoid working with the non-local translation representation when n is even.

1.1 Lax-Phillips Formalism About the Wave Equation

The natural Cauchy problem for the wave equation is the following

$$\begin (\partial _t^2-\Delta )u=0, \quad (u,u_t)|_ = (f_1,f_2). \end$$

(31)

We convert the wave equation into a system by setting \(\varvec(t)=(u,u_t)\); then

$$\begin \partial _t \varvec = A\varvec, \quad A:= \begin 0 & \quad \text \\ \Delta & \quad 0 \end. \end$$

(32)

We use boldface to denote vector-valued functions not necessarily of the type \((u,u_t)\) if there is no background scalar function u(t, x) present. In particular, \(\varvec(t)\) in Definition A.1 below is not necessarily of that form.

The natural energy space of states of finite energy is defined as the completion of \(C_0^\infty (\mathbb ^n) \times C_0^\infty (\mathbb ^n)\) under the energy norm

$$\begin \Vert \varvec\Vert ^2_}= \frac\int \left( |\nabla f_1|^2+|f_2|^2 \right) \textrm x, \quad \varvec:=(f_1,f_2). \end$$

In particular, the first term defines the Dirichlet space \(H_D(\mathbb ^n)\) with norm \(\Vert \nabla f\Vert _\). When \(n\ge 3\), they are locally in \(L^2\), as it follows from the Poincaré inequality. The operator A naturally extends to a skew-self-adjoint one (i.e., \(\textrmA\) is self-adjoint) on \(\mathcal \). Then, by Stone’s theorem, \(U_0(t) = e^\) is a well-defined strongly continuous unitary group, and the solution of (32) is given by \(\varvec(t) = U_0(t)\varvec\). The unitarity means energy conservation, in particular.

We define the local energy space \(\mathcal _}\) in the usual way:

$$\begin \mathcal _} = \: \; \phi \varvec \in \mathcal \text \phi \in C_0^\infty (\mathbb ^n)\}. \end$$

By the finite speed of propagation, the Cauchy problem (31) has a well-defined solution in \(\mathcal _}\) if the Cauchy data \(\varvec\) is in \(\mathcal _}\) only. We view those solutions as ones with (possibly) infinite energy but locally finite one. Then, \(\varvec\in C(\mathbb ;\; \mathcal _})\) and the wave equation is solved in distribution sense. One can easily extend this to distributions.

1.2 Existence of Dynamics

By [9], see also [17, X.12], the solution to

$$\begin (\partial _t^2-\Delta +q(t,x))u=0,\quad (u,u_t)|_=(f_1,f_2) \end$$

is given by \(\varvec(t)= U(t,s)\varvec\), where \(\varvec=(f_1,f_2)\) and U(t, s) is a two-parameter strongly continuous group of bounded operators with the properties

(i)

\(U(t,s)U(s,r) = U(t,r)\) for all t, s, r; and \(U(t,t)=\text \),

(ii)

\(\Vert U(t,s)\Vert \le \exp \left\^n}|q(\tau ,x)| \right\} \),

(iii)

for any \(\varvec\in D(A)\), we have \(U(t,s)\varvec\in D(A)\) and

$$\begin \frac} t} U(t,s)\varvec = (A-Q(t))U(t,s)\varvec, \quad \frac} s} U(t,s)\varvec = -U(t,s)(A-Q(s))\varvec, \end$$

(33)

where \(Q(t)\varvec= (0,q(t,\cdot )f_1)\) (and Q(t) is clearly bounded).

The two-parameter semi-group admits the expansion

$$\begin U(t,s) = U_0(t-s)+\sum _^\infty V_k(t,s), \end$$

(34)

where

$$\begin V_k(t,s)\varvec&= (-1)^k \int _s^t \textrm s_1 \int _s^ \textrm s_k \dots \int _s^}\textrm s_k\\&\quad \times U_0(t-s_1)Q(s_1) \dots U_0(s_-s_k) Q(s_k)U_0(s_k-s)\varvec, \quad k\gg 1. \end$$

This expansion is an iterated version of the Duhamel’s formula

$$\begin U(t,s)&= U_0(t-s)+ \int _s^t U(t,\sigma )Q(\sigma )U_0(\sigma -s) \,\textrm \sigma \nonumber \\&= U_0(t-s)+ \int _s^t U_0(t-\sigma )Q(\sigma )U(\sigma ,s)\,\textrm \sigma . \end$$

(35)

The convergence of (34) follows from the estimate

$$\begin \Vert V_k(t,s)\Vert \le \frac \left( \sup _\Vert Q(\tau )\Vert \right) ^k. \end$$

In particular, we get that we still have the finite speed of propagation property:

$$\begin \,}}U(t,s)\varvec\subset \,}}\varvec+B(0,|t-s|). \end$$

As before, the finite speed of propagation allows us to extend U(t, s) to the space \(\mathcal _}\) by a partition of unity.

Finally, notice that when q is time independent, then U(t, s) depends on the difference \(t-s\) only, i.e., \(U(t,s)=U(t-s)\) where U is a group. It is not unitary, however (unless \(q=0\)), in the space \(\mathcal \). If we redefine the energy norm by

$$\begin \Vert \varvec\Vert ^2_^q}= \int \left( |\nabla f_1|^2+q|f_1|^2+ |f_2|^2 \right) \textrm x, \end$$

(we need to know that it is a norm, however, and \(q\ge 0\) suffices for that), then U(t) is unitary in \(\mathcal ^q\).

1.3 Plane Waves, Translation Representation and Asymptotic Wave Profiles of Free Solutions

The plane waves

$$\begin \delta (t-\omega \cdot x) \end$$

solve the wave equation, obviously. They can be thought of as plane waves propagating in the direction \(\omega \) with speed one. If we replace t by \(t+s\) there, we can think of s as the delay time. The plane wave above is the Schwartz kernel of the Radon transform

$$\begin Rf(s,\omega ) = \int \delta (s-\omega \cdot x)f(x)\,\textrm x= \int _f(x)\,\textrm S_x. \end$$

For any density \(g(\omega ,s)\) (which can be a distribution as well), the superposition

$$\begin u(t,x) := \int _\times S^}\delta (t+s-\omega \cdot x)g(s,\omega )\,\textrm s\,\textrm \omega = \int _} g(\omega \cdot x-t,\omega )\,\textrm\omega \end$$

is still a solution of the wave equation. The expression above can be recognized as the transpose \(R'\) of the Radon transform applied to \(g_t(s,\omega ):= g(s-t,\omega )\). It turns out that all solutions of the free wave equation in the energy space have that form.

Indeed, in [13], Lax and Phillips defined the free translation representation \(\mathcal : \mathcal \rightarrow L^2(\mathbb \times S^)\) as follows

$$\begin k(s,\omega ) = \mathcal \varvec(s,\omega ) = c_n(-\partial _s^ Rf_1+\partial _s^ Rf_2), \end$$

where R is the Radon transform and \(c_n=2^(2\pi )^\), \(c_n^-= 2^(-2\pi )^\). The inverse is given by

$$\begin \mathcal ^k(x) = 2c_n^-\int _}\left( -\partial _s^ k(x\cdot \omega ,\omega ), \; \partial _s^k(x\cdot \omega ,\omega )\right) \textrm\omega . \end$$

(36)

The map \(\mathcal \) is unitary, and \((\mathcal U_0(t)\mathcal ^ k)(s,\omega ) = k(s-t,\omega )\), which explains the name. We also set

$$\begin u^\sharp (s,\omega ) = (-1)^ k(s,\omega ) \end$$

(37)

and call \(u^\sharp \) the asymptotic wave profile of the solution \(\varvec(t)= U_0(t)\varvec\). This name is justified by the theorem below, and it is the analog of the far free pattern for solutions of the free wave equation.

Theorem A.1

(Lax-Phillips, [13]). Let \(\varvec(t)= U_0(t)\varvec\), \(\varvec\in \mathcal \). Then,

$$\begin \int \Big | u_t-|x|^u^\sharp \Big (|x|-t,\frac\Big ) \Big |^2\, \textrm x \rightarrow 0, \quad \text |t|\rightarrow \infty . \end$$

Remark A.1

In [13], the factor \((-1)^\) is missing from (37), i.e., \(u^\sharp = k\). Cooper and Strauss in [3] found out that this factor must be present in (37).

1.4 Outgoing Solutions and Their Asymptotic Wave Profiles

We follow here [2, 3]. Given u(t, x) (and only then), recall the notation \(\varvec(t) :=(u(t,\cdot ),u_t(t,\cdot ))\), see (32).

Definition A.1

The function \(\varvec(t)\in C(\mathbb ;\;\mathcal _})\) is called outgoing if \(\lim _ (\varvec(t),U_0(t) \varvec)=0\) for each \(\varvec\in C_0^\infty (\mathbb ^n) \times C_0^\infty (\mathbb ^n)\).

In this definition, \(\varvec(t)\) does not need to be a solution of the wave equation (anywhere). On the other hand, if u(t, x) solves the wave equation in \(|x|>\rho \) for some \(\rho >0\), then, see [2, 3], u is outgoing if and only if for any \(T\in \mathbb \), \(U_0(t-T)\varvec(T)=0\) in the forward cone \(|x|<t-T-\rho \).

One simple example of non-trivial outgoing solutions (for \(|x|>\rho )\) is the following. Let \(p\in L^1_}(\mathbb ;\;L^2(\mathbb ^n))\) satisfy \(p=0\) for \(t<t_0\), where \(t_0\) is fixed. Solve

$$\begin (\partial _t^2-\Delta )u=p(t,x) \quad \text \mathbb \times \mathbb ^n. \end$$

(38)

with Cauchy data

$$\begin (u,u_t)|_=(0,0). \end$$

By Duhamel’s formula,

$$\begin \varvec(t)= \int _^t U_0(t-s)\varvec(s)\,\textrm s, \quad \varvec(s):= (0,p(s,\cdot )). \end$$

The latter is well defined in \(\mathcal _}\) by finite speed of propagation. The solution for \(t<t_0\) is just zero. Then, \(\varvec\) is outgoing in a trivial way. Moreover, this is the unique outgoing solution of (38). Indeed, take the difference v of any two. Then, \(\varvec(t)=U_0(t)\varvec\), where \(\varvec\) is the initial condition. Then, \(0=\lim _ (\varvec(t),U_0(t) \varvec)= (\varvec,\varvec)\), for any test function \(\varvec\); therefore, \(\varvec=0\) and then \(\varvec=0\).

This can be generalized as follows.

Theorem A.2

([2, 3]). Let \(p\in L^1_}(\mathbb ;\;L^2(\mathbb ^n))\) and assume that for each t,

$$\begin \lim _\int _^t U_0(-s)\varvec(s)\, \textrm s \quad \text \mathcal _}, \quad \varvec(s) := (0,p(s,\cdot )). \end$$

(39)

Then, there exists a unique outgoing solution \(\varvec\in C(\mathbb ;\;\mathcal _})\) of (38) given by

$$\begin \varvec(t)= \int _^t U_0(t-s)\varvec(s)\,\textrm s. \end$$

Remark A.2

Clearly, \(p\in L^1((-\infty ,a);\;L^2(\mathbb ^n))\) for any a would guarantee the regularity assumption on p and (39). Also, the assumptions on p in the next theorem are enough.

Proof

The absolute convergence of the integral in \(^0_}\) follows from the assumptions. To show that u is outgoing, for \(\varvec \in C_0^\infty (\mathbb ^n)\times C_0^\infty (\mathbb ^n)\), consider

$$\begin (\varvec(t), U_0(t)\varvec)= \int _^t\left( U_0(t-s)\varvec(s), U_0(t)\varvec \right) \textrm s = \int _^t\left( U_0(-s)\varvec(s), \varvec \right) \textrm s. \end$$

The latter converges to 0, as \(t\rightarrow -\infty \) by assumption. \(\square \)

Theorem A.3

([2, 3]). Let \(n\ge 3\) be odd. Let \(p\in L^1_}(\mathbb ;\;L^2(\mathbb ^n))\) with \(p(t,x)=0\) for \(|x|>\rho \). Let u be the unique outgoing solution of (38).

(a)

Then, there is a unique function \(u^\sharp \in L_}^2(\mathbb \times S^)\) such that for all \(R_1<R_2\) we have

$$\begin \int _ \left| u_t(t,x) - |x|^ u^\sharp \left( |x|-t,\frac\right) \right| ^2\textrm x\rightarrow 0, \quad \text t\rightarrow \infty . \end$$

(b)

If \(p\in C_0^\infty (\mathbb ^)\),

$$\begin u^\sharp (s,\omega ) = c_n^- \partial _s^\int p(\omega \cdot x-s,x)\,\textrm x. \end$$

(40)

(c)

The map \(p\rightarrow u^\sharp \) is continuous.

Remark A.3

For general p as in the theorem, \(u^\sharp \) is still given by (40) but the derivative is in distribution sense; by (b), the result is in \(L_}^2(\mathbb \times S^)\). Another way to write (40) is

$$\begin \int u^\sharp (s,\omega )\phi (s)\,\textrm s = c_n \iint p(t,x) (\partial _s^ \phi ) (\omega \cdot x-s,x)\,\textrm t\, \textrm s, \quad \forall \psi (s)\in C_0^\infty (\mathbb ). \end$$

Proof

Motivated by Theorem A.2, for fixed \(R_1<R_2\), set

$$\begin \varvec = \int _^U_0(-\tau )\varvec(\tau )\,\textrm \tau , \quad \varvec(t) = U_0(t)\varvec. \end$$

By Huygens’ principle, \(\varvec(t)=\varvec(t)\) for \(R_1+t< |x|<R_2+t\). Therefore, \(\varvec\) does have an asymptotic wave profile, and \(v^\sharp (s,\omega ) = u^\sharp (s,\omega )\) for \(R_1<s<R_2\). On the other hand, we have a formula for \(v^\sharp \), (36) and (37) which says

$$\begin v^\sharp (s,\omega )&= (-1)^(\mathcal \varvec)(s,\omega ) \\&= c_n^-\partial _s^ \int _^ \int _ p(x\cdot \omega -s,x)\, \textrm S_x\, \textrm\tau . \end$$

Then,

$$\begin u^\sharp (s,\omega )|_ = v^\sharp (s,\omega )= c_n^- \partial _s^\int p(x\cdot \omega -s,x)\, \textrm x. \end$$

Since \(R_1<R_2\) are arbitrary, this, combined with Theorem A.2, proves (a); and (b) for \(p\in C_0^\infty \).

The proof of (c) is straightforward: use (40) and take Fourier transform w.r.t. s. In particular, we get that the map \(p\rightarrow u^\sharp \) can be extended continuously in those spaces. \(\square \)

Remark A.4

We call \(u^\sharp \) in the theorem the asymptotic wave profile of the unique outgoing solution of (38). Note that there are two cases where we defined such profiles: for free solutions in the energy space, see Theorem A.2, and in Theorem A.3 above, where u is in the energy space locally only, and solves (38) instead.

1.5 Scattering Solutions

The scattering solutions \(u^-\) and \(u^+\) were introduced in Sect. 3 as the solutions of (3), and (5), respectively. Since they involve distributions, not necessarily in the energy spaces (even locally), we proceed as follows. We can think of \((u(t,x;s,\omega ), u_t(t,x;s,\omega ))\) as distribution in the \((s,\omega )\) variables with values in \(\mathcal _}\). It is more convenient, however, to do the following. Let \(h_j(t)=h(t)t^j/j!\), \(j=1,2\dots \), where h is the Heaviside function; and we also set \(h_=\delta \) . Then, \(h_j'=h_\), \(j=0,1,2,\dots \). To define \(u^-\) eventually, we solve

$$\begin (\partial _t^2-\Delta +q(t,x))\Gamma =0, \quad \Gamma |_= h_1(t+s-x\cdot \omega ) \end$$

(41)

first (notice that \( h_1(t+s-x\cdot \omega ) \) is locally in the energy space now), set

$$\begin \Gamma _} = \Gamma - h_1(t+s-x\cdot \omega ), \end$$

compute the asymptotic wave profile \(\Gamma ^\sharp (s',\omega ';s,\omega )\) of \(\Gamma _}\), and differentiate the result twice w.r.t. s to get the analog of the scattering amplitude. In particular, then

$$\begin u(t,x;s,\omega ) = \partial _s^2\Gamma (t,x;s,\omega ), \quad u_}^-(t,x;s,\omega ) = \partial _s^2\Gamma _}(t,x;s,\omega ). \end$$

(42)

will be well defined as distributions.

In a similar way, one can construct the scattering solutions \(u^+\) which look like plane waves as \(t\rightarrow +\infty \), instead of \(t\rightarrow -\infty \). They would solve (5). Performing the change of variables \(\tilde=-t\) (time reversal), \(\tilde=-s\), \(\tilde=-\omega \), we see that \(u^+(t,x;s,\omega ) =\tilde^+(-t,x;-s,-\omega ) \), where \(\tilde\) is related to \(\tilde(t,x):=q(-t,x)\). The regularized version, \(\Gamma ^+\), can be constructed as in (41) with the condition \(\Gamma ^+ = h_1(-t-s+x\cdot \omega )\) for \(t>-s+\rho \). The right-hand side of this condition then would be supported outside \(\mathbb \times B(0,R)\) for \(t>-s+\rho \). Then, we define \(u^+\) and \(u^+_}\) as in (42).

By the finite speed of propagation property

$$\begin \,}}u^-(\cdot ,\cdot ;s,\omega ) \subset \, \quad \,}}u^+(\cdot ,\cdot ;s,\omega ) \subset \. \end$$

(43)

Next theorem generalizes Theorem A.3.

Theorem A.4

Let p be as in Theorem A.3. Set

$$\begin \varvec(t): = \int _^t U(t,s) \varvec(s)\,\textrm s. \end$$

Then, \(\varvec\in C(\mathbb ;\;\mathcal _})\) is outgoing and has an asymptotic wave profile \(u^\sharp (s,\omega )\) given by

$$\begin u^\sharp (s,\omega ) = c_n^- \partial _s^\int p(t,x) u^+ (t,x;s,\omega )\,\textrm t\,\textrm x. \end$$

Proof

By (35),

$$\begin \varvec(t)&= \int _^t U_0(t-s) \varvec(s)\,\textrm s + \int _^t\int _s^t U_0(t-\sigma )Q(\sigma ) U(\sigma ,s) \varvec(s)\,\textrm \sigma \,\textrm s\nonumber \\&= \int _^t U_0(t-s) \varvec(s)\,\textrm s + \int _^t\int _^\sigma U_0(t-\sigma )Q(\sigma ) U(\sigma ,s) \varvec(s)\,\textrm s\,\textrm \sigma . \end$$

(44)

Then, we are in the situation of Theorem A.3 with \(\varvec(t)\) there replaced by

$$\begin \tilde}(t):= \varvec(t)+ \varvec_1(t), \quad \varvec_1(t):= Q(t)\int _^t U(t,s) \varvec(s)\,\textrm s= Q(t)\varvec(t). \end$$

(45)

Then, \(\varvec\) has an asymptotic wave profile \(u^\sharp (s',\omega ')\) satisfying

$$\begin u^\sharp (s',\omega ') = c_n^- \partial _s^\int \tilde(t,x)\delta (t+s'-\omega '\cdot x)\,\textrm x\,\textrm t. \end$$

(46)

The first term on the right-hand side of (44) is handled by Theorem A.3. We analyze the second term below, which we call \(\varvec_1(t)\). By Theorem A.3 again, its asymptotic wave profile is given by

$$\begin u_1^\sharp (s',\omega ')&= c_n^- \partial _s^\int \Big [ Q(t,x)\int _^t U(t,x;s,y) \varvec(s,y)\,\textrm s\,\textrm y\Big ]_2\, \nonumber \\ &\quad (t+s'-\omega '\cdot x)\,\textrm t\,\textrm x\nonumber \\&= c_n^- \partial _s^ \int K(s',\omega ';s,y) p(s,y)\,\textrm s\,\textrm y, \end$$

(47)

where the last identity defines K, i.e.,

$$\begin K(s',\omega ';s,y)= \int ^_s \int q(t,x) U_(t,x;s,y) \delta (t+s'-\omega '\cdot x)\,\textrm x\,\textrm t. \end$$

(48)

By (33),

$$\begin (-\partial _s + A_y'- Q'(s)) U'(t,x;s,y)=0, \end$$

where A is the operator defined by (32) and the primes denote transpose operators in distribution (not in energy space) sense. This equality can be written also as

$$\begin \begin -\partial _s & \quad \Delta _y-q(s)\\ \text & \quad -\partial _s \endU'(t,x;s,y) =0. \end$$

In particular,

$$\begin (\partial _s^2-\Delta _y+q(s)) U_(t,x;s,y)=0. \end$$

(49)

Differentiate K in (48) to obtain

$$\begin \partial _s K(s',\omega ';s,y)= \int ^_s \int q(t,x) \partial _s U_(t,x;s,y) \delta (t+s'-\omega '\cdot x)\,\textrm x\,\textrm t \end$$

because \(U_(s,x;s,y)=0\). Differentiate again:

$$\begin \partial _s^2 K(s',\omega ';s,y)&= \int ^_s \int q(t,x) \partial _s^2 U_(t,x;s,y) \delta (t+s'-\omega '\cdot x)\,\textrm x\,\textrm t \\&\quad -q(s,x)\delta (s+s'-\omega '\cdot y). \end$$

Then, by (49) and (48),

$$\begin (\partial _s^2-\Delta _y+q(s)) K(s',\omega ';s,y)= -q(s,x)\delta (s+s'-\omega '\cdot y). \end$$

(50)

On the support of the integrand in (48), we have \(t+s'<\rho \), \(s<t\). Therefore,

$$\begin K(s',\omega ';s,y)|_=0. \end$$

(51)

Therefore, K solves (50), (51), which is the same problem solved by \(u^+_}(s',\omega ';s,y)\), see (5). Therefore, \(K=u^+_}\).

Going back to (46) and (45), we see that

$$\begin u^\sharp (s',\omega ')&= c_n^- \partial _s^\int \left( p(t,x)+p_1(t,x) \right) \delta (t+s'-\omega '\cdot x)\,\textrm x\,\textrm t\\&= c_n^- \partial _s^\int p(t,x) ) \left( \delta (t+s'-\omega '\cdot x)+ u^+_}(s',\omega ';t,x)\right) \,\textrm x\,\textrm t\\&= c_n^- \partial _s^\int p(t,x) ) u^+(s',\omega ';t,x) \,\textrm x\,\textrm t, \end$$

where we used (47) and the identity \(K=u^+_}\) we just derived. \(\square \)

1.6 The Scattering Amplitude and the Scattering Kernel

Let \(\Gamma \) solve (41). Since the Cauchy data \((h_1(t+s-x\cdot \omega ), h_0(t+s-x\cdot \omega ))\), for say, \(t=-s-\rho -1\), is in \(\mathcal _}\), a solution \((\Gamma ,\Gamma _t)\) with locally finite energy exists. Then, \(\Gamma _}\) is clearly outgoing. It solves the Cauchy problem

$$\begin (\partial _t^2-\Delta )\Gamma _}= -q\Gamma ,\quad \Gamma _}|_= 0. \end$$

By Theorem A.3, \(\Gamma _}\) has an asymptotic wave profile \(\Gamma _}^\sharp \) given by

$$\begin \Gamma _}^\sharp (s',\omega ';s,\omega )&= -c_n^-\partial _^ \int q (x\cdot \omega '-s' ,x )\Gamma (x\cdot \omega '-s' ,x;s,\omega )\, \textrm x\\&= -c_n^-\partial _^ \int q (t ,x )\Gamma (t ,x;s,\omega )\delta (t+s'- x\cdot \omega ' )\, \textrm t\,\textrm x. \end$$

Differentiate twice w.r.t. s, see (42), to get

$$\begin u^_} (s',\omega ';s,\omega )= -c_n^-\partial _^ \int q (t,x )u^-(t,x;s,\omega )\delta (t+s'- x\cdot \omega ' )\, \textrm t\,\textrm x. \end$$

Definition A.2

The scattering amplitude \(A^\sharp \) is given by

$$\begin A^\sharp (s',\omega ';s,\omega ) = \int q (t ,x )u^-(t ,x;s,\omega )\delta (t+s'- x\cdot \omega ' )\, \textrm t\,\textrm x, \end$$

where \(u^-\) solves (3).

By the finite speed of propagation, \(u^-(t,x;s,\omega ) = 0\) for \(x\cdot \omega >t+s\). Therefore, the integrand vanishes outside of the region \(x\cdot (\omega -\omega ')\le s-s'\). The l.h.s. has a lower bound \(-2\rho \) on \(\,}}q\); therefore,

$$\begin \,}}A^\sharp \subset \\subset \. \end$$

Note that \(A^\sharp \) and \(u^\sharp _} = -c_n^-\partial _^ A^\sharp \) can be reconstructed from each other thanks to that support property.

Since the perturbed dynamics is a two-parameter group, we need to generalize the notion of the wave operators and the scattering operator.

Definition A.3

The wave operators \(\Omega _-\) and \(W_+\) in \(\mathcal \) are defined as the strong limits

$$\begin \Omega _- = \text \lim _U(0,t)U_0(t), \quad W_+ \varvec= \lim _U_0(-t)U(t,0)\varvec; \quad \varvec\in \text \;\Omega _-, \end$$

if they exist and define continuous operators. In the latter case, the scattering operator S is defined by

$$\begin S= W_+\Omega _-. \end$$

This definition also makes sense for \(\varvec \in \mathcal _}\), where \(\mathcal _}\) denotes the subspace of \(\mathcal \) consisting of compactly supported functions, with \(S\varvec\) taking values possibly in \(\mathcal _}\).

Theorem A.5 (a)

The wave operator \(\Omega _-: \mathcal _} \rightarrow \mathcal \) exists and

$$\begin U(t,0)\Omega _-\varvec = 2c_n^-\int _\times S^} \varvec^-(t,x;s,\omega ) \partial _s^ (\mathcal \varvec)(s,\omega )\,\textrm s\,\textrm\omega . \end$$

(52)

(b)

The wave operator \(W_+: \mathcal \rightarrow \mathcal _}\) exists.

(c)

The scattering operator \(S:\mathcal _}\rightarrow \mathcal _} \) exists.

Proof

Choose \(\varvec\in \mathcal _}\), so that \(\varvec(x)=0\) for \(|x|>R\) with some \(R>0\). Let \(k=\mathcal \varvec\). Then, \(k(s,\omega )=0\) for \(|s|>R\). For \(t<-R-\rho :=t_0\), \(U(0,t)U_0(t)\varvec= U(0,t_0)U_0(t_0)\varvec\). In particular, the limit defining \(\Omega _-\varvec\) exists trivially and \(U(t,0)\Omega _-\varvec = U(t,t_0)U_0(t_0)\varvec\). The r.h.s. of the latter solves the perturbed wave equation and equals \(U_0(t_0)\varvec= \mathcal ^ k(\cdot -t_0,\cdot )\) for \(t=t_0\). To prove (52), we need to show that the r.h.s. of (52), call it \(\varvec(t)\), has the same initial condition for \(t\le t_0\).

For \(t\le t_0\), \(u(t,x;s,\omega )= \delta (t+s-x\cdot \omega )\). Then, by (36),

$$\begin v(t) = 2c_n^-\int _\times S^} \delta (t+s-x\cdot \omega )\partial _s^k(s,\omega )\,\textrm s\,\textrm\omega = (\mathcal ^k)_1(\cdot -t,\cdot ), \end$$

which proves (a).

To prove the existence of \(W_+\) in (b), fix first \(R>0\) and let \(\varvec_\) be the characteristic function of that ball. By (35),

$$\begin \varvec_U_0(-t) U(t,s) = \varvec_U_0(-s)+ \varvec_ \int _s^t U_0(-\sigma )Q(\sigma )U(\sigma ,s)\,\textrm \sigma . \end$$

By Huygens’ principle, \(\varvec_ U_0(-\sigma )Q(\sigma )=0\) for \(\sigma >R+\rho \). For \(t>R+\rho \), then the integral above is independent of t and therefore the strong limit \(\varvec_W_+\) exists in a trivial way, defining a unique element in \(\mathcal _}\).

Part (c) follows from (a) and (b). \(\square \)

The scattering operator S on \(\mathcal \) exists (as a bounded operator) under some conditions, see the references in [20, sec. 3]. Then, \(-c_n^-\partial _^ A^\sharp (s',\omega '; s,\omega ) \) is the Schwartz kernel of \(\mathcal (S-\text )\mathcal ^\). In the general case, we can consider the latter as the Schwartz kernel of the operator mapping asymptotic wave profiles instead of translation representations, see (37), as shown [20, Proposition 3.1].

Proposition 1

([2, 3, 20]). Let \(\varvec\in C_0^\infty (\mathbb ^n) \times C_0^\infty (\mathbb ^n)\). Let \(\varvec_0(t)= U_0(t)\varvec\), and let \(\varvec(t)\) be the solution of (1) which equals \(\varvec_0(t)\) for \(t\ll 0\). Then, we have

$$\begin v^\sharp (s',\omega ') = v_0^\sharp (s',\omega ')-c_n^-\partial _^\int _\times S^} A^\sharp (s',\omega '; s,\omega ) v_0^\sharp (s,\omega )\,\textrm s\,\textrm\omega . \end$$

The proof of the proposition is done by taking the asymptotic wave profile of (52) and applying Duhamel’s formula (35) first.

Finally, we will prove Proposition 3.1

Proof of Proposition 3.1

Start with

$$\begin U_1(t,s)-U_2(t,s) = \int _s^t U_1(t,\sigma )(Q_1(\sigma )-Q_2(\sigma ))U_2(\sigma ,s)\,\textrm \sigma , \end$$

which can be obtained by applying the Fundamental Theorem of Calculus to \(F(\sigma )= U_1(t,\sigma ) U_2(\sigma ,s) \) in the interval \(\sigma \in [s,t]\). Apply \(U_0(-s)\varvec\) on the right-hand, and take the (strong) limit \(s\rightarrow -\infty \) to get

$$\begin&U_1(t,0)\Omega _\varvec -U_2(t,0)\Omega _\varvec\nonumber \\ &\quad = \int _^t U_1(t,\sigma )(Q_1(\sigma )-Q_2(\sigma )) \left[ U_2(\sigma ,0) \Omega _\varvec\right] \,\textrm \sigma . \end$$

(53)

For the left-hand side, and for the expression in the square brackets we will apply Theorem A.5(a). We take the asymptotic wave profile of (53) next applying Theorem A.4. Then, (6) is just that expression, written as a composition of Schwartz kernels, eventually applied to \(\mathcal \varvec\). \(\square \)

Appendix B. A Weighted Radon Transform

We recall the definition of the weighted Euclidean Radon transform

$$\begin R_\mu f (p,\omega ) = \int _ \mu (x,\omega ) f(x) \textrmx = \int _ \mu (p\omega + y, \omega ) f(p\omega + y) \textrmy. \end$$

where \(\mu \in C^\infty (\mathbb \times \mathbb ^)\) is a weight. We will use the next result:

Theorem B.1

Let \(\Omega \) be an open, bounded set and