Exceptional points and domains of unitarity for a class of strongly non-Hermitian real-matrix Hamiltonians

A family of non-Hermitian real and tridiagonal-matrix candidates H(N)(λ)=H0(N)+λW(N)(λ) for a hiddenly Hermitian (a.k.a. quasi-Hermitian) quantum Hamiltonian is proposed studied. Fairly weak assumptions are imposed upon the unperturbed matrix [the square-well-simulating spectrum H0(N) not assumed equidistant)] its maximally N-parametric antisymmetric-matrix perturbations [matrix W(N)(λ) even required to be PT-symmetric]. Despite that, “physical” parametric domain D[N] (constructively) shown exist, guaranteeing that in interior, remains non-degenerate, rendering evolution unitary. Among degeneracies occurring at boundary ∂D[N] stability, our main attention paid their extreme version corresponding Kato’s exceptional point order N (EPN). The localization EPNs and, vicinity, quantum-phase-transition boundaries found feasible, too large N, using computer-assisted symbolic manipulations, including, particular, Gröbner-basis elimination high-precision arithmetics.