Coherent States for infinite homogeneous waveguide arrays: Cauchy coherent states for $E(2)$

Perelomov coherent states for equally spaced, infinite homogeneous waveguide arrays with Euclidean E(2) symmetry are defined, and a new resolution of the identity is obtained. The key point to construct this novel resolution of the identity is the fact that coherent states satisfy the Helmholtz equation (in coherent states labels), and thus every coherent state belongs to a one-parameter family uniquely determined by the Cauchy initial data of the coherent state in a one-dimensional Cauchy set. For this reason we call \textit{Cauchy coherent} states to these initial data. The novel, non-local resolution of the identity in terms of Cauchy coherent states is provided using frame theory. It is also shown that Perelomov coherent states for the Eucliean E(2) group have a simple and natural physical realization in these waveguide arrays.