Bound states of PT-symmetric separable potentials

All of the PT-symmetric potentials that have been studied so far have been local. In this paper nonlocal PT-symmetric separable potentials of the form $V(x,y)=iϵ[U(x)U(y)-U(-x)U(-y)]$, where $U(x)$ is real, are examined. Two specific models are examined. In each case it is shown that there is a parametric region of the coupling strength $ϵ$ for which the PT symmetry of the Hamiltonian is unbroken and the bound-state energies are real. The critical values of $ϵ$ that bound this region are calculated.