Maximal couplings in PT-symmetric chain-models with the real spectrum of energies

Abstract

The domain ${\cal D}$ of all the coupling strengths compatible with the reality of the energies is studied for a family of non-Hermitian $N$ by $N$ matrix Hamiltonians $H^{(N)}$ with tridiagonal and ${\cal PT}-$symmetric structure. At all dimensions $N$, the coordinates are found of the extremal points at which the boundary hypersurface $\partial {\cal D}$ touches the circumscribed sphere (for odd $N=2M+1$) or ellipsoid (for even $N=2K$).