AI 中文总结
该研究探究环面弹性Neumann-Poincaré算子傅里叶块的谱结构，证明三类点被无穷多本征值逼近，给出单侧本征值计数渐近式，明确了傅里叶块的本质谱构成。
AI 中文摘要
我们研究环面上弹性Neumann-Poincaré算子的傅里叶块的谱行为。对于完整弹性Neumann-Poincaré算子，已知其本征值会在三个点处累积，即零点以及由Lamé参数确定的一对非零对称点。目前尚不清楚该谱结构是否在每个单独的傅里叶块中保持。我们证明，对于每个固定的傅里叶模，这三个点中的每一个都会被无穷多个离散本征值从两侧逼近。此外，我们建立了精确的单侧本征值计数渐近式，其主导系数明确且严格为正。因此，每个傅里叶块的本质谱恰好由这三个点给出。该证明结合了旋转坐标系下的傅里叶分解与Plemelj对称化原理，以得到每个块的自伴实现。三次多项式变换消除了零阶主部，将问题简化为-1阶的紧伪微分算子，随后通过这些约化算子的主符号确定本征值计数渐近式。
英文摘要
We study the spectral behavior of the Fourier blocks of the elastic Neumann-Poincaré operator on a torus. For the full elastic Neumann-Poincaré operator, it is known that eigenvalues accumulate at three points, namely zero and a symmetric pair of nonzero points determined by the Lamé parameters. It remains unclear whether this spectral structure persists within each individual Fourier block. We prove that, for every fixed Fourier mode, each of these three points is approached by infinitely many discrete eigenvalues from both sides. Moreover, we establish precise one-sided eigenvalue counting asymptotics with explicit and strictly positive leading coefficients. Consequently, the essential spectrum of every Fourier block is exactly given by these three points. The proof combines a rotating-frame Fourier decomposition and the Plemelj symmetrization principle to obtain a self-adjoint realization of each block. A cubic polynomial transformation removes the order-zero principal part and reduces the problem to compact pseudodifferential operators of order $-1$. The eigenvalue counting asymptotics are then determined by the principal symbols of these reduced operators.
