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 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：我们称有界Lipschitz区域的边界Γ为局部膨胀不变的，如果对每个x∈Γ，Γ要么局部C¹光滑，要么局部（在某个以x为中心坐标系下）与一个Lipschitz图Γ_x重合，且存在α_x∈(0,1)使得Γ_x=α_xΓ_x。本文研究此类Γ上势论中的双层（或Neumann-Poincaré）算子D_Γ在L^2(Γ)上的本质谱。通过局部化和Floquet-Bloch型论证，我们证明该本质谱是相关连续算子族K_t（t∈[-π,π]）谱的并集；此外，若Γ除有限点外均为C¹光滑，则每个K_t为紧算子。对于二维情形，当Γ进一步为分段解析时，我们构造了D_Γ本质谱的收敛逼近序列；每个逼近由有限个矩阵的特征值并集给出，这些矩阵源自K_t算子的Nyström方法近似。通过显式常数误差估计，我们构造了泛函以判定任意局部膨胀不变分段解析Γ是否满足著名的谱半径猜想（即对一切Lipschitz区域Γ，D_Γ在L^2(Γ)上的本质谱半径<1/2）。通过实例验证该理论，每个例子均表明本质谱半径<1/2，为猜想提供了额外支持。此外，利用本质谱半径在局部共形C^{1,β}微分同胚下的不变性的新结果，我们证明该谱半径猜想对所有Lipschitz曲线多面体成立。