Sliced Spectral Analysis and Geometric Mechanisms of Radiation for Periodic Elliptic Operators

Abstract

Radiation in higher-dimensional periodic media is fundamentally more difficult than in one dimension because the multidimensional spectral parameterization no longer admits the one-dimensional analytic structure underlying classical complex-analytic methods. We introduce a sliced spectral analysis (SSA), which incorporates the observation direction as an explicit parameter and decomposes the Brillouin-zone integral into an outer integration over spectral slices and an inner one-dimensional spectral problem on each directional slice. This restores, on every slice, the analytic structure of the one-dimensional theory and thereby provides a unified analytic framework for radiation in arbitrary dimensions. Making the observation direction explicit has a second, independent consequence. It makes direction-dependent spectral geometry accessible and, in particular, enables the definition of a grazing set, a genuinely higher-dimensional geometric object separating the real and complex Fermi surfaces and characterizing the transition between propagating and evanescent modes. This geometric structure naturally leads to a decomposition of the limiting absorption solution into evanescent, non-grazing, and grazing components, each associated with a distinct radiation mechanism. Finally, we establish the necessity of the three-component decomposition through explicit examples. The grazing contribution may become the leading asymptotic term. The Helmholtz Green's function provides a degenerate example in which the limitation of the classical propagating--regular decomposition becomes explicit: the classical decomposition assigns separate leading-order contributions to the propagating and regular parts, whereas these contributions are naturally associated with the grazing mechanism and cancel only after recombination.