论文标题
3D中的Neumann-Poincaré操作员无限的许多嵌入式特征值
Infinitely many embedded eigenvalues for the Neumann-Poincaré operator in 3D
论文作者
论文摘要
本文构建了一个表面，其Neumann-Poincaré（NP）积分运算符具有无限的许多特征值嵌入其基本频谱中。表面是通过平滑附着圆锥形奇异性的球体的球体，从而赋予了必需的光谱。旋转对称性可以将操作员分解为傅立叶组件。构建了无限的许多傅立叶组件的特征值，以便它们位于其他傅立叶组件的基本范围内，因此位于完整NP操作员的基本范围内。证明要求扰动足够小，具有受控的曲率，并且圆锥形的奇异性足够平坦。
This article constructs a surface whose Neumann-Poincaré (NP) integral operator has infinitely many eigenvalues embedded in its essential spectrum. The surface is a sphere perturbed by smoothly attaching a conical singularity, which imparts essential spectrum. Rotational symmetry allows a decomposition of the operator into Fourier components. Eigenvalues of infinitely many Fourier components are constructed so that they lie within the essential spectrum of other Fourier components and thus within the essential spectrum of the full NP operator. The proof requires the perturbation to be sufficiently small, with controlled curvature, and the conical singularity to be sufficiently flat.
