Spectral and scattering properties at thresholds for the Laplacian in a half-space with a periodic boundary condition

TL;DR

This paper investigates the spectral and scattering characteristics of the Laplacian in a half-space with periodic boundary conditions, focusing on resolvent expansions, scattering matrix continuity, and wave operator formulas.

Contribution

It introduces new resolvent expansion formulas at embedded thresholds and establishes the continuity of the scattering matrix for this specific scattering system.

Findings

Derived resolvent expansions at embedded thresholds

Proved continuity of the scattering matrix

Established new formulas for wave operators

Abstract

For the scattering system given by the Laplacian in a half-space with a periodic boundary condition, we derive resolvent expansions at embedded thresholds, we prove the continuity of the scattering matrix, and we establish new formulas for the wave operators.