Construction of maximally non-Hermitian potentials under unbroken PT-symmetry constraint

A family of discrete Schr\"{o}dinger equations with imaginary potentials $V(x)$ is studied. Inside the domain ${\cal D}$ of unitarity-compatible values of $V(x)$, the reality of all of the bound-state energies survives up to the ``exceptional-point'' (EP) maximally non-Hermitian spectral-degeneracy boundaries $\partial {\cal D}$. The computer-assisted localization of the EP limits is performed showing that the complexity of the task grows quickly with the number $N$ of grid points $x$.