The combined-field operator for smooth-boundary Helmholtz scattering

Let ${\mathrm{\Omega }}^{-}$ be a bounded connected $C^2$ domain in ${\mathbb{R}}^3$ with connected boundary $\mathrm{\Gamma }$, let ${\mathrm{\Omega }}^{+}={\mathbb{R}}^3\setminus \overline{{\mathrm{\Omega }}^{-}}$ be connected, and let $n$ point outward from ${\mathrm{\Omega }}^{-}$. Fix $k>0$ and $\eta \in \mathbb{R}\setminus \{0\}$. The outgoing Helmholtz kernel is $G_k(x,y)=\frac{e^{ik\mathrm{\mid }x-y\mathrm{\mid }}}{4\pi \mathrm{\mid }x-y\mathrm{\mid }},x\mathrm{\ne }y\mathrm{.}$ For $\varphi \in C^{\mathrm{\infty }}(\mathrm{\Gamma })$ and $x\mathrm{\notin }\mathrm{\Gamma }$, define $S_k\varphi (x)={\int }_{\mathrm{\Gamma }}{G}_k(x,y)\varphi (y)d{S}_y$ and $D_k\varphi (x)={\int }_{\mathrm{\Gamma }}{\mathrm{\partial }}_{n_y}{G}_k(x,y)\varphi (y)d{S}_y$. The combined-field potential is $u_{\varphi }=D_k\varphi -i\eta S_k\varphi$, and its exterior trace defines the smooth-density operator $A_{k,\eta }\varphi ={\gamma }^{+}{u}_{\varphi }$. This definition makes no assertion that $A_{k,\eta }$ extends boundedly or Fredholmly to a Sobolev space; these analytical properties are proved in separate theorems. For this smooth-boundary branch, ${\gamma }^{+}$ denotes the trace from ${\mathrm{\Omega }}^{+}$ and uses the same fixed normal $n$ as the interior trace. For this $C^2$ surface, fix a finite graph atlas and subordinate partition as in the general Lipschitz chart-space convention of Lipschitz polyhedral boundary, orientation, and trace spaces, using graph surface measure. Use its square-sum Hilbert norm, real spaces, complexification and conjugate-dual convention for $H^{\pm 1\mathrm{/}2}(\mathrm{\Gamma })$. This imports the space convention only; the $C^2$ trace/lift and layer theorems are proved separately and do not invoke the polyhedral trace theorem.

The Sobolev extension, compact perturbation and actual bounded invertibility are proved in Smooth-boundary trace lifts and Sobolev Helmholtz layer extensions, The smooth Helmholtz combined field is a compact perturbation of half the identity and Bounded invertibility of the smooth-boundary Helmholtz combined field. The latter retains its historical identifier.