Stationary outgoing integral and flux

Actual source and local regularity: [F1] makes $u\in L^2(B_R)$. Therefore $g=Vu$, zero-extended outside the ball, belongs to $L^2\cap L^1$ with $\mathrm{\parallel }g{\mathrm{\parallel }}_2\le \mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }u{\mathrm{\parallel }}_2$ and $\mathrm{\parallel }g{\mathrm{\parallel }}_1\le \mathrm{\mid }{B}_R{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }g{\mathrm{\parallel }}_2$. On a fixed bounded $x$-region only bounded-radius values of $G_k$ and $\mathrm{\nabla }{G}_k$ enter their convolutions with $g$; those restricted kernels are in $L^1$, since $G_k=O(\mathrm{\mid }z{\mathrm{\mid }}^{-1})$ and $\mathrm{\nabla }{G}_k=O(\mathrm{\mid }z{\mathrm{\mid }}^{-2})+O(\mathrm{\mid }z{\mathrm{\mid }}^{-1})$ near zero. Apply the explicit estimate [F2] to the function and each distributional first derivative to obtain $G_k\ast g\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$. Radial differentiation away from zero gives $(-\mathrm{\Delta }-k^2)G_k=0$, and integration over a shrinking sphere gives the delta mass $1$ from $-{\mathrm{\partial }}_r(1\mathrm{/}(4\pi r))$; thus $(-\mathrm{\Delta }-k^2)G_k={\delta }_0$ distributionally. Consequently $(-\mathrm{\Delta }-k^2)\mathrm{\Psi }=-g$. Both $g$ and $\mathrm{\Psi }$ are locally square integrable, so $\mathrm{\Delta }\mathrm{\Psi }=-k^2\mathrm{\Psi }+g\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$ and [F2] gives $\mathrm{\Psi }\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$. The defining equation $(I+K_k)u=e^{ikd\cdot x}{\mathrm{\mid }}_{B_R}$ identifies $\mathrm{\Psi }=u$ on $B_R$, hence $g=V\mathrm{\Psi }$ everywhere and $(-\mathrm{\Delta }+V-k^2)\mathrm{\Psi }=0$. This proves (a), without an arbitrary-$L^1$ regularity assertion.

Uniform smooth far-field kernel: set $s=1\mathrm{/}r$ and $H(s,\mathrm{\Omega },y)=(1-2s\mathrm{\Omega }\cdot y+s^2\mathrm{\mid }y{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. For $0\le s\le (2R{)}^{-1}$ and $\mathrm{\mid }y\mathrm{\mid }\le R$, $H\ge 1\mathrm{/}2$; the quotient $(H-1)\mathrm{/}s$ extends smoothly to $s=0$ by the identity $(H-1)\mathrm{/}s=(-2\mathrm{\Omega }\cdot y+s\mathrm{\mid }y{\mathrm{\mid }}^2)\mathrm{/}(H+1)$, with value $-\mathrm{\Omega }\cdot y$ there. Define $A(s,\mathrm{\Omega },y)=H^{-1}\exp (ik(H-1)\mathrm{/}s)$ using that extension. Then $G_k(r\mathrm{\Omega }-y)=e^{ikr}A(s,\mathrm{\Omega },y)\mathrm{/}(4\pi r)$ and $A(0,\mathrm{\Omega },y)=e^{-ik\mathrm{\Omega }\cdot y}$. All fixed-order $s$ and angular derivatives of $A$ are uniformly bounded on this compact parameter set. The fundamental theorem in $s$ gives $A(s)-A(0)=s{\int }_0^1{\mathrm{\partial }}_{s}A(ts)dt$; the same formula after any $D_{\mathrm{\Omega }}$ bounds its angular derivatives by $C_{D}s$. Integrating against $g\in L^1$ proves smoothness of $f_k$ by dominated differentiation and gives $\mathcal{E}=-(4\pi {)}^{-1}{e}^{ikr}{r}^{-1}\int [A(1\mathrm{/}r)-A(0)]g(y)dy$ with all stated angular error bounds. Differentiating this exact formula in $r$ produces the three factors $ik\mathrm{/}r$, $-1\mathrm{/}{r}^2$ and $-1\mathrm{/}{r}^3$ multiplying respectively $A(1\mathrm{/}r)-A(0)$, that same difference, and ${\mathrm{\partial }}_{s}A(1\mathrm{/}r)$; their bounds give $\mathrm{\mid }{D}_{\mathrm{\Omega }}{\mathrm{\partial }}_r\mathcal{E}\mathrm{\mid }\le C_D{r}^{-2}$. This proves (b), including derivative control, directly from the actual compact source.

Direct scattered current: step 2.1 gives ${\mathrm{\Psi }}_{\mathrm{s}\mathrm{c}}=e^{ikr}{f}_k\mathrm{/}r+O(r^{-2})$ and ${\mathrm{\partial }}_r{\mathrm{\Psi }}_{\mathrm{s}\mathrm{c}}=ik{e}^{ikr}{f}_k\mathrm{/}r+O(r^{-2})$ uniformly on the sphere. Their product is $\overline{{\mathrm{\Psi }}_{\mathrm{s}\mathrm{c}}}{\mathrm{\partial }}_r{\mathrm{\Psi }}_{\mathrm{s}\mathrm{c}}=ik\mathrm{\mid }{f}_k{\mathrm{\mid }}^2\mathrm{/}{r}^2+O(r^{-3})$, since $f_k$ is bounded and each error is $O(r^{-2})$. Taking its imaginary part and multiplying by $\mathrm{\hslash }\mathrm{/}m$ yields the claimed $j_r^{\mathrm{s}\mathrm{c}}$. Independently, for ${\mathrm{\Psi }}_{\mathrm{i}\mathrm{n}}=e^{ikd\cdot x}$ the vector current is $(\mathrm{\hslash }\mathrm{/}m)\mathrm{Im}(\overline{{\mathrm{\Psi }}_{\mathrm{i}\mathrm{n}}}\mathrm{\nabla }{\mathrm{\Psi }}_{\mathrm{i}\mathrm{n}})=(\mathrm{\hslash }k\mathrm{/}m)d$, so incident flux through a plane normal to $d$ is $J_{\mathrm{i}\mathrm{n}}=\mathrm{\hslash }k\mathrm{/}m$. Dividing the scattered flux per asymptotic solid angle by that incident density gives $d\sigma \mathrm{/}d\mathrm{\Omega }=\mathrm{\mid }{f}_k{\mathrm{\mid }}^2$. Uniform convergence allows integration over the finite-area sphere, giving $\sigma ={\int }_{S^2}\mathrm{\mid }{f}_k{\mathrm{\mid }}^{2}d\sigma <\mathrm{\infty }$. No incident/scattered interference term enters this scattered-component computation, and no assertion that such terms average away is needed. Parts (a)-(c) follow.