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A generic PT-symmetric Hamiltonian is assumed tridiagonalized and truncatedto N dimensions, and its up-down symmetrized special cases with J=[N/2] realcouplings are considered. In the strongly non-Hermitian regime the secularequation gets partially factorized at all N. This enables us to reveal afine-tuned alignment of the dominant couplings implying an asymptoticallysharply spiked shape of the boundary of the J-dimensional quasi-Hermiticitydomain in which all the spectrum of energies remains real and observable.

Tridiagonal PT-symmetric N by N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime