One of the simplest non-Hermitian Hamiltonians first proposed by Schwartz (1960 {\it Commun. Pure Appl. Math.} \tb{13} 609) which may possess a spectral singularity is analyzed from the point of view of non-Hermitian generalization of quantum mechanics. It is shown that $η$ operator, being a second order differential operator, has supersymmetric structure. Asymptotic behavior of eigenfunctions of a Hermitian Hamiltonian equivalent to the given non-Hermitian one is found. As a result the corresponding scattering matrix and cross section are given explicitly. It is demonstrated that the possible presence of the spectral singularity in the spectrum of the non-Hermitian Hamiltonian may be detected as a resonance in the scattering cross section of its Hermitian counterpart. Nevertheless, just at the singular point the equivalent Hermitian Hamiltonian becomes undetermined.
Gap Probabilities for Double Intervals in Hermitian Random Matrix Ensembles as $τ$-Functions -- Spectrum Singularity case
从一个间隔$(-x,x)$对称的厄米特随机矩阵，其中的Fredholm内核是一种类型的贝塞尔内核参数$ a $（广义的正弦内核在批量缩放的情况下）的起源的特征值的排除概率被认为是。它表明，这个概率是一个$τ$函数的平方，在冈本的意义下，Painlevé系统\PIII。然后，这导致概率的因式分解为Painlevé系统\PIIIdash的两个$τ$ -函数的乘积。以前的研究已经给出了这种类型的公式，但涉及具有不同参数的\PIIIdash系统，从而意味着$τ$ -函数的乘积或等价的Hamilton和之间的同一性。
Asymptotic non-Hermitian degeneracy phenomenon and its exactly solvable simulation
The Czech Academy of Sciences, Nuclear Physics Institute（捷克科学院核物理研究所）；Department of Physics, Faculty of Science, University of Hradec Králové（赫拉德茨-克拉洛韦大学理学院物理系）；School for Data Science and Computational Thinking, Stellenbosch University（斯坦伦布什大学数据科学与计算思维学院）
Phase transitions in quasi-Hermitian quantum models at exceptional points of order four
The Czech Academy of Sciences, Nuclear Physics Institute（捷克科学院核物理研究所）；Department of Physics, Faculty of Science, University of Hradec Králové（赫拉德茨克拉洛韦大学理学院物理系）；School for Data Science and Computational Thinking, Stellenbosch University（斯坦伦布什大学数据科学与计算思维学院）；Institute of System Science, Durban University of Technology（德班理工大学系统科学研究所）
