Photon position eigenstates in configuration space

The expressions of the eigenfunctions of the Hawton photon position operator in the configuration space are derived for several classes of wave function, including the Riemann-Silberstein and Landau-Peierls cases. Although these eigenfunctions have a simple form in momentum space, the explicit characterization of their representations in the configuration space is rather more involved. We provide closed expressions of these eigenfunctions in terms of linear combinations of the complete elliptic integrals $K(\kappa)$ and $E(\kappa)$ with modulus $\kappa$ depending on trigonometric functions of the polar angle. We show that they diverge not only at the value $\mathbf q$ of the position eigenvalue, but also on a plane containing $\mathbf q$ and that they decay as inverse powers of the distance from $\mathbf q$.