Behavior of eigenvalues in a region of broken-PT symmetry
Behavior of eigenvalues in a region of broken-PT symmetry
复制标题
PT 对称破缺区域中特征值的行为
DOI：
10.1103/physreva.95.052113
复制
发表时间：
2017
影响因子：
2.9
通讯作者：
Zichao Wen
中科院分区：
文献类型：
--
作者：
C. Bender;Nima Hassanpour;Daniel W. Hook;Daniel W. Hook;S. Klevansky;Christoph Sünderhauf;Christoph Sünderhauf;Zichao Wen;Zichao Wen
对称量子力学始于对哈密顿$H={p}^{2}+{x}^{2}{(ix)}^{\ensuremath{\varepsilon}}$.的研究当确保数学为0时，该非厄米特哈密顿量的本征值是离散的、实数的和正的。参数空间的这一部分称为不间断的$\mathcal{pt}$对称性区域。在破坏数学对称性的区域，只有有限个特征值是实的，其余的特征值表现为复数共轭对。未破缺的数学{PT}对称性区域已被研究，但破缺的数学{PT}$对称性区域迄今仍未被探索。本文详细地用数值和解析的方法研究了4l-l0$的本征值行为。具体地说，它报告了在$\ensuremath{\varepsilon}=\ensuremath1$处发现了无穷阶例外点，在$\ensuremath{\varepsilon}=\ensuremath2$处从离散谱到部分连续谱的转变，在库仑值$\ensuremath{\varepsilon}=\ensuremath3$处的转变，以及特征值的行为接近保形极限$\ensuremath{\varepsilon}=\ensuremath{-}4$。
$\mathcal{PT}$-symmetric quantum mechanics began with a study of the Hamiltonian $H={p}^{2}+{x}^{2}{(ix)}^{\ensuremath{\varepsilon}}$. When $\ensuremath{\varepsilon}\ensuremath{\ge}0$, the eigenvalues of this non-Hermitian Hamiltonian are discrete, real, and positive. This portion of parameter space is known as the region of unbroken $\mathcal{PT}$ symmetry. In the region of broken $\mathcal{PT}$ symmetry, $\ensuremath{\varepsilon}l0$, only a finite number of eigenvalues are real and the remaining eigenvalues appear as complex-conjugate pairs. The region of unbroken $\mathcal{PT}$ symmetry has been studied but the region of broken $\mathcal{PT}$ symmetry has thus far been unexplored. This paper presents a detailed numerical and analytical examination of the behavior of the eigenvalues for $\ensuremath{-}4l\ensuremath{\varepsilon}l0$. In particular, it reports the discovery of an infinite-order exceptional point at $\ensuremath{\varepsilon}=\ensuremath{-}1$, a transition from a discrete spectrum to a partially continuous spectrum at $\ensuremath{\varepsilon}=\ensuremath{-}2$, a transition at the Coulomb value $\ensuremath{\varepsilon}=\ensuremath{-}3$, and the behavior of the eigenvalues as $\ensuremath{\varepsilon}$ approaches the conformal limit $\ensuremath{\varepsilon}=\ensuremath{-}4$.