We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing $\Gamma$ for the boundary of the obstacle, the relevant integral operators map $L^2(\Gamma)$ to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth $\Gamma$ and are sharp, and the bounds on the norm of the inverse are valid for smooth $\Gamma$ and are observed to be sharp at least when $\Gamma$ is curved. Together, these results give bounds on the condition number of the operator on $L^2(\Gamma)$; this is the first time $L^2(\Gamma)$ condition-number bounds have been proved for this operator for obstacles other than balls.

中文翻译：

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Helmholtz 外 Neumann 问题边界积分算子的高频估计

我们研究了一个常用的二类边界积分方程来解决高频亥姆霍兹外诺依曼问题，其中，障碍物的边界用$\Gamma$表示，相关的积分算子映射$L^2(\Gamma )$ 给自己。我们证明了积分算子及其逆的范数的新频率显式边界。范数的边界对于分段平滑的 $\Gamma$ 有效并且是尖锐的，而逆范数的边界对于平滑的 $\Gamma$ 有效并且至少在 $\Gamma$ 为弯曲。总之，这些结果给出了 $L^2(\Gamma)$ 上运算符的条件数的界限；这是第一次证明 $L^2(\Gamma)$ 条件数边界对于除球以外的障碍物的该算子。