PT-symmetric model with an interplay between kinematical and dynamical non-localities

A new family of non-Hermitian PT-symmetric quantum models is proposed in which the Hamiltonians $H=T+V$ are finite-dimensional and in which the dynamical-input potential $V$ is multi-parametric and non-local. The choice is supported by the exact solvability of Schr\"{o}dinger equation and by the well known fact that in PT-symmetric models a non-locality is already present due to the generic kinematical non-diagonality of the Hermitizing metrics $\Theta$. For a subfamily of our $H$s, also {\em all\,} of the eligible metrics $\Theta$ appear obtainable in closed form.