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A phenomenological Hamiltonian of a closed (i.e., unitary) quantum system isassumed to have an $N$ by $N$ real-matrix form composed of a unperturbeddiagonal-matrix part $H^{(N)}_0$ and of a tridiagonal-matrix perturbation$\lambda\,W^{(N)}(\lambda)$. The requirement of the unitarity of the evolutionof the system (i.e., of the diagonalizability and of the reality of thespectrum) restricts, naturally, the variability of the matrix elements to a"physical" domain ${\cal D}^{[N]} \subset \mathbb{R}^d$. We fix the unperturbedmatrix (simulating a non-equidistant, square-well-type unperturbed spectrum)and we only admit the maximally non-Hermitian antisymmetric-matrixperturbations. This yields the hiddenly Hermitian model with the measure ofperturbation $\lambda$ and with the $d=N$ matrix elements which are, inside${\cal D}^{[N]}$, freely variable. Our aim is to describe the quantumphase-transition boundary $\partial {\cal D}^{[N]}$ (alias exceptional-pointboundary) at which the unitarity of the system is lost. Our main attention ispaid to the strong-coupling extremes of stability, i.e., to the Kato'sexceptional points of order $N$ (EPN) and to the (sharply spiked) shape of theboundary $\partial {\cal D}^{[N]}$ in their vicinity. The feasibility of ourconstructions is based on the use of the high-precision arithmetics incombination with the computer-assisted symbolic manipulations (including, inparticular, the Gr\"{o}bner basis elimination technique).

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