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

A generic PT-symmetric Hamiltonian is assumed tridiagonalized and truncated to N < ∞ dimensions, H → H(chainmodel), and all its up-down symmetrized special cases with J = [N/2] real couplings are considered, H(chainmodel) → H(N). Using symbolic manipulation and extrapolation techniques we find out that in the strongly non-Hermitian regime the secular equation gets partially factorized at all N. This enables us to reveal a fine-tuned alignment of the dominant couplings implying an asymptotically sharply spiked shape of the boundary of the J-dimensional quasi-Hermiticity domain D(N) in which all the spectrum of energies E(N)n remains real and observable