Essential spectra and eigenvalue asymptotics of Fourier blocks of the elastic Neumann-Poincar\'e operator on tori

Abstract

We study the spectral behavior of the Fourier blocks of the elastic Neumann-Poincar\'e operator on a torus. For the full elastic Neumann-Poincar\'e operator, it is known that eigenvalues accumulate at three points, namely zero and a symmetric pair of nonzero points determined by the Lam\'e 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.