2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/127557The 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 $N=2K$).18 pp., 2 figsMathematical Physics81Q10; 13P10; 14J81; 14M12; 46C20; 47B50text