We analyze the well posedness of certain field-only boundary integral equations (BIE) for frequency domain electromagnetic scattering from perfectly conducting spheres. Starting from the observations that (1) the three components of the scattered electric field $\mathbf{E}^s(\mathbf{x})$ and (2) scalar quantity $\mathbf{E}^s(\mathbf{x})\cdot\mathbf{x}$ are radiative solutions of the Helmholtz equation, novel boundary integral equation formulations of electromagnetic scattering from perfectly conducting obstacles can be derived using Green's identities applied to the aforementioned quantities and the boundary conditions on the surface of the scatterer. The unknowns of these formulations are the normal derivatives of the three components of the scattered electric field and the normal component of the scattered electric field on the surface of the scatterer, and thus these formulations are referred to as field-only BIE. In this paper we use the Combined Field methodology of Burton and Miller within the field-only BIE approach and we derive new boundary integral formulations that feature only Helmholtz boundary integral operators, which we subsequently show to be well posed for all positive frequencies in the case of spherical scatterers. Relying on the spectral properties of Helmholtz boundary integral operators in spherical geometries, we show that the combined field-only boundary integral operators are diagonalizable in the case of spherical geometries and their eigenvalues are non zero for all frequencies. Furthermore, we show that for spherical geometries one of the field-only integral formulations considered in this paper exhibits eigenvalues clustering at one -- a property similar to second kind integral equations.

翻译：本文分析了完美导电球体频域电磁散射中某些仅含场量的边界积分方程（BIE）的适定性。基于以下观察：（1）散射电场$\mathbf{E}^s(\mathbf{x})$的三个分量，（2）标量量$\mathbf{E}^s(\mathbf{x})\cdot\mathbf{x}$均为亥姆霍兹方程的辐射解，利用格林恒等式作用于上述量及散射体表面边界条件，可推导出完美导电障碍物电磁散射的新型边界积分方程公式。这些公式的未知量为散射电场三个分量的法向导数及散射体表面散射电场的法向分量，因此称为仅含场量的边界积分方程。本文在仅含场量的边界积分方程方法中引入Burton-Miller组合场方法，推导出仅含亥姆霍兹边界积分算子的新型边界积分公式，并证明该公式在球面散射体情况下对所有正频率均具有适定性。基于球面几何中亥姆霍兹边界积分算子的谱性质，我们证明了组合场仅含场量边界积分算子在该几何条件下可对角化，且其本征值对所有频率均非零。此外，本文还表明对于球面几何，本文考虑的某一仅含场量积分公式具有本征值聚集于1的特性——与第二类积分方程相似的特性。