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–Poincaré) 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 Nyström-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.

如果在每个\(x\in \Gamma \)处，\ (\Gamma \)是局部\(C^1 \)或局部重合（在以x为中心的某个坐标系中）与 Lipschitz 图\(\Gamma _x\)使得\(\Gamma _x=\alpha _x\Gamma _x\)，对于某些\(\alpha _x\在 (0,1)\) 中。在本文中，我们研究，对于这样的\(\Gamma \) ， \(D_\Gamma \)的本质谱，势论的双层（或 Neumann–Poincaré）算子，在\(L^2(\伽玛）\）. 我们通过定位和 Floquet–Bloch 类型的论证表明，这个基本谱是相关连续运算符族\(K_t\)的谱的并集，对于\(t\in [-\pi ,\pi ]\ ) ; 此外，如果\(\Gamma \)是\(C^1\)，则每个\(K_t \)都是紧致的，但在有限多个点除外。此外，对于\(\Gamma \)是分段解析的二维情况，我们构建了\(D_\Gamma \)的基本谱的收敛近似序列；每个近似值都是由 Nyström 方法对运算符\(K_t\)的近似值产生的有限多个有限矩阵的特征值的并集. 通过使用显式常数的误差估计，我们还构造了泛函来确定任何特定的局部膨胀不变分段解析\(\Gamma \)是否满足众所周知的谱半径猜想，即\(D_\Gamma的基本谱半径\)在\(L^2(\Gamma )\)上是\(<1/2\)对于所有 Lipschitz \(\Gamma \)。我们用例子来说明这个理论；对于每一个，我们都表明基本光谱半径是 \(<1/2\)，为猜想提供了额外的支持。我们还通过关于局部共形\(C^{1,\beta }\)下基本谱半径不变性的新结果微分同胚，表明谱半径猜想适用于所有 Lipschitz 曲线多面体。