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

中文翻译：

---

一类强非Hermitian实矩阵哈密顿量的奇异点和唯一性域

假设一个封闭的（即unit）量子系统的现象学哈密顿量具有一个由无扰动对角矩阵部分$ H ^ {（N）} _ 0 $和a组成的$ N $ x $ N $实矩阵形式。三对角矩阵摄动$ \ lambda \，W ^ {（N）}（\ lambda）$。系统发展的统一性（即对角化性和频谱的现实性）的要求自然地将矩阵元素的可变性限制在“物理”域$ {\ cal D} ^ {[ N]} \ subset \ mathbb {R} ^ d $。我们固定了无扰动的矩阵（模拟了一个非等距的方阱型无扰动谱），并且只允许最大非赫米特式的非对称矩阵扰动。这样就产生了一个隐含的Hermitian模型，它具有摄动量$ \ lambda $以及$ {= cal $} ^ {[N]} $内的$ d = N $矩阵元素，自由可变。我们的目的是描述失去系统统一性的量子相变边界$ \ partial {\ cal D} ^ {[N]} $（别名为例外点边界）。我们主要关注稳定性的强耦合极端，例如，加藤的特殊订货点$ N $（EPN）和边界$ \ partial {\ cal D} ^ { [N]} $在他们附近。我们构造的可行性是基于将高精度算术与计算机辅助符号处理（特别是包括Grbner基础消除技术）结合使用的。我们主要关注稳定性的强耦合极端，例如，加藤的特殊订货点$ N $（EPN）和边界$ \ partial {\ cal D} ^ { [N]} $在他们附近。我们构造的可行性是基于将高精度算术与计算机辅助符号处理（特别是包括Grbner基础消除技术）结合使用的。我们主要关注稳定性的强耦合极端，例如，加藤的特殊订货点$ N $（EPN）和边界$ \ partial {\ cal D} ^ { [N]} $在他们附近。我们构造的可行性是基于将高精度算术与计算机辅助符号处理（特别是包括Grbner基础消除技术）结合使用的。