We say that $\Gamma$, the boundary of a bounded Lipschitz domain, is locally dilation invariant if, at each $x\in \Gamma$, $\Gamma$ is either locally $C^1$ or locally coincides (in some coordinate system centred at $x$) with a Lipschitz graph $\Gamma_x$ such that $\Gamma_x=\alpha_x\Gamma_x$, for some $\alpha_x\in (0,1)$. In this paper we study, for such $\Gamma$, the essential spectrum of $D_\Gamma$, the double-layer (or Neumann-Poincar\'e) operator of potential theory, on $L^2(\Gamma)$. We show, via localisation and Floquet-Bloch-type arguments, that this essential spectrum %of $D_\Gamma$ %on such $\Gamma$ is the union of the spectra of related continuous families of operators $K_t$, for $t\in [-\pi,\pi]$; moreover, each $K_t$ is compact if $\Gamma$ is $C^1$ except at finitely many points. For the 2D case where, additionally, $\Gamma$ is piecewise analytic, we construct convergent sequences of approximations to the essential spectrum of $D_\Gamma$; each approximation is the union of the eigenvalues of finitely many finite matrices arising from Nystr\"om-method approximations to the operators $K_t$. Through error estimates with explicit constants, we also construct functionals that determine whether any particular locally-dilation-invariant piecewise-analytic $\Gamma$ satisfies the well-known spectral radius conjecture, that the essential spectral radius of $D_\Gamma$ on $L^2(\Gamma)$ is $<1/2$ for all Lipschitz $\Gamma$. We illustrate this theory with examples; for each we show that the essential spectral radius is $<1/2$, providing additional support for the conjecture. We also, via new results on the invariance of the essential spectral radius under locally-conformal $C^{1,\beta}$ diffeomorphisms, show that the spectral radius conjecture holds for all Lipschitz curvilinear polyhedra.

翻译：称有界Lipschitz区域的边界$\Gamma$为局部膨胀不变的，若在每个$x\in \Gamma$处，$\Gamma$要么局部$C^1$光滑，要么（在某个以$x$为中心的坐标系中）局部与某个Lipschitz图$\Gamma_x$重合，且存在$\alpha_x\in (0,1)$使得$\Gamma_x=\alpha_x\Gamma_x$。本文研究此类$\Gamma$在$L^2(\Gamma)$上势理论的双层（或Neumann-Poincaré）算子$D_\Gamma$的本质谱。通过局部化与Floquet-Bloch型论证，我们证明该本质谱是算子族$K_t$（$t\in [-\pi,\pi]$）相关连续族谱的并集；此外，若$\Gamma$除有限点外为$C^1$光滑，则每个$K_t$为紧算子。对于二维情形，当$\Gamma$额外满足分段解析性时，我们构造收敛的$D_\Gamma$本质谱近似序列；每个近似由有限个矩阵的特征值并集给出，这些矩阵源于$K_t$算子Nyström方法近似。通过显式常数误差估计，我们构造泛函以判定特定局部膨胀不变的分段解析$\Gamma$是否满足著名的谱半径猜想（即对所有Lipschitz区域$\Gamma$，$D_\Gamma$在$L^2(\Gamma)$上的本质谱半径$<1/2$）。通过实例验证该理论，每个例子均显示本质谱半径$<1/2$，为猜想提供额外支持。此外，利用本质谱半径在局部共形$C^{1,\beta}$微分同胚下的不变性新结果，我们证明谱半径猜想对所有Lipschitz曲线多面体成立。