The {C}alderon operator for the {M}axwell system in the exterior of an infinite cylinder in $\mathbb{R}^3$

We study the Calderon operator for the time-harmonic Maxwell system in the “exterior” $\Omega^+$ of an infinite cylinder in $x_3$-direction. The Calderon is the analogue of the Dirichlet-to-Neumann operator for the scalar Helmholtz equation. In the first part we study the case where the Calderon operator corresponds to solutions $u$ on $\Omega^+$ which, together with their curls, decay along $x_3$. In the second part we consider the case where the solution $u$ is assumed to be quasi-periodic with respect to $x_3$. In both cases we derive properties of the Calderon operator with respect to coercivity and compactness. These properties are useful for the investigation of problems in all of $\mathbb{R}^3$ if one uses the Calderon operator to reduce the problem to the “interior” of the cylinder. The proofs rely heavily on properties of the Hankel functions which are studied in detail in the appendix.