Theorems.ScatteringMain / ScatteringMain.operatorValuedScattering

For every medium $M$—that is, every complex coefficient $a(x,t)$ with positive spatial radius $R$, positive time radius $T$, and positive decay $\gamma$, vanishing whenever $R<|x|$ or $T<|t|$, strongly measurable in $x$ at every $t$, infinitely smooth in $t$, and with every time derivative uniformly bounded in $x,t$—and for every real $R_1$, the following implication is asserted. If $R<R_1$, there exist complex-linear maps $\mathcal T,\mathcal R$ on upper-half-plane signals and complex-linear maps $\mathcal T_t,\mathcal R_t$ on arbitrary time signals satisfying all of the following.

For every $\alpha\in\mathbb R$, there is a finite extended nonnegative constant $C_\alpha$ such that whenever a signal $f:\mathbb C\to\mathbb C$, zero when $\operatorname{Im}z\le0$, has $z\mapsto(z+i)f(z)$ holomorphic on $\operatorname{Im}z>0$ and

$$\sup_{\sigma>0}e^{-2\sigma\alpha} \int_{\mathbb R}|(\xi+i\sigma+i)f(\xi+i\sigma)|^2\,d\xi<\infty,$$

the function $z\mapsto z(\mathcal T f)(z)$ is holomorphic and has finite analogous norm at weight $\alpha+2R_1$, with its squared norm at most $C_\alpha$ times the displayed input norm. The identical assertion, with its own constant for each $\alpha$, holds for $\mathcal R$. The maps are fixed before $\alpha$ and $f$, while the constants may depend on $\alpha$.

For every $\alpha\in\mathbb R$ and every $f$ satisfying that incoming Hardy condition, there exists a frequency field $u(x,z)$, zero when $\operatorname{Im}z\le0$, such that each $z\mapsto u(x,z)$ is holomorphic on the upper half-plane; every upper-half-plane spatial slice is almost-everywhere strongly measurable; and for some real weight $\beta$, independent of the positive radius $r$,

$$\sup_{\sigma>0}e^{-2\sigma\beta} \int_{\mathbb R}\int_{[-r,r]} |(\xi+i\sigma)u(x,\xi+i\sigma)|^2\,dx\,d\xi<\infty$$

for every $r>0$. It satisfies the integrable weak equation

$$\int_{\mathbb R}\bigl( -u(x,z)\phi''(x)-z^2u(x,z)\phi(x)+(A_Mu)(x,z)\phi(x) \bigr)\,dx=0$$

for every upper-half-plane $z$ and every smooth compactly supported $\phi$, where

$$(A_Mu)(x,z)=\frac1{2\pi}\int_{\mathbb R} \widehat a_M(x,\operatorname{Re}z-\xi) \frac{w}{w+i\gamma}u(x,w)\,d\xi,\qquad w=\xi+i\operatorname{Im}z,$$

and $\widehat a_M(x,\eta)=\int e^{i\eta t}a(x,t)\,dt$; this action is spatially almost-everywhere strongly measurable and its $\xi$-integrand is integrable for almost every $x$. The solution obeys

$$u(x,z)=f(z)e^{izx}+(\mathcal Rf)(z)e^{-izx}\quad(x<-R), \qquad u(x,z)=(\mathcal Tf)(z)e^{izx}\quad(x>R).$$

Any other weak frequency solution satisfying these same prescribed exterior identities equals $u$ almost everywhere in $x$, separately at each upper-half-plane $z$. No condition is imposed at $x=\pm R$ or inside $[-R,R]$, and the exceptional set in uniqueness may depend on $z$.

For every smooth compactly supported $g:\mathbb R\to\mathbb C$ whose topological support lies in $(R,\infty)$ and for which $e^{izt}g(t)$ is integrable for every upper-half-plane $z$, both $\mathcal T_tg$ and $\mathcal R_tg$ have the same transform integrability, and

$$\widehat{\mathcal T_tg}(z)=\mathcal T(\widehat g)(z), \qquad \widehat{\mathcal R_tg}(z)=\mathcal R(\widehat g)(z)$$

for every $\operatorname{Im}z>0$, where $\widehat g(z)=\int e^{izt}g(t)\,dt$ in the upper half-plane and is canonically zero elsewhere.

For every such admissible $g$, there exists an energy field consisting of $u,u_t,u_x$. The named derivatives satisfy

$$\int u_t\phi=-\int u\,\partial_t\phi,\qquad \int u_x\phi=-\int u\,\partial_x\phi$$

for all smooth compactly supported spacetime tests, with the relevant products integrable. At every $t$, all three spatial slices are almost-everywhere strongly measurable and square-integrable; $u$ and $u_x$ are jointly continuous in the sum of their squared $L^2_x$ distances, and $u_t$ is continuous in squared $L^2_x$ distance. For almost every $x$, the causal memory integral

$$K_Mu(t,x)=\int_{s\le t}e^{-\gamma(t-s)}(-i\,u_t(s,x))\,ds$$

is integrable, and its spatial slice is almost-everywhere strongly measurable. For every smooth compactly supported spacetime test $\phi$, the integrable weak expression satisfies

$$\int_{\mathbb R^2} \bigl(u_t\partial_t\phi-u_x\partial_x\phi-a(x,t)K_Mu\,\phi\bigr) \,dt\,dx=0.$$

Pointwise for every $t<0$ and every $x$, $u(t,x)=g(t-x)$. Its exterior values satisfy

$$u(t,x)=g(t-x)+(\mathcal R_tg)(t+x)\quad(x<-R), \qquad u(t,x)=(\mathcal T_tg)(t-x)\quad(x>R).$$

Every other causal energy solution for the same $g$, whether or not it is separately assumed to have these exterior values, agrees almost everywhere in $x$ at every $t$ in its value and both named weak derivatives.

The operator quadruple may depend on $R_1$, and no uniqueness of the operators is asserted. The input-existence clauses are conditional and require no nonzero datum; the zero frequency signal belongs to every incoming class and the zero time signal is admissible. The theorem is automatically satisfied at each $R_1\le R$ because the defining statement begins with an implication, but its universal quantifier over $R_1$ also includes all $R_1>R$. In every nonvacuous branch, $R_1$ appears in the Hardy shift $2R_1$, whereas all exterior regions and time-data support conditions use $R$.