Li Jun-Qing, Miao Yan-Gang, Xue Zhao
School of Physics, Nankai University, Tianjin, China.
School of Physics, Nankai University, Tianjin, China; Kavli Institute for Theoretical Physics China, Chinese Academy of Sciences, Beijing, China; Bethe Center for Theoretical Physics and Institute of Physics, University of Bonn, Bonn, Germany.
PLoS One. 2014 Jun 4;9(6):e97107. doi: 10.1371/journal.pone.0097107. eCollection 2014.
A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator η+ and defining the annihilation and creation operators to be η+ -pseudo-Hermitian adjoint to each other. The operator η+ represents the η+ -pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-PT-symmetric Hamiltonian with imaginary linear coordinate and linear momentum terms is constructed and analyzed in detail. The operator η+ is found, based on which, a real spectrum and a positive-definite inner product, together with the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution, are obtained for the non-Hermitian and non-PT-symmetric Hamiltonian. Moreover, this Hamiltonian turns out to be coupled when it is extended to the canonical noncommutative space with noncommutative spatial coordinate operators and noncommutative momentum operators as well. Our method is applicable to the coupled Hamiltonian. Then the first and second order noncommutative corrections of energy levels are calculated, and in particular the reality of energy spectra, the positive-definiteness of inner products, and the related properties (the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution) are found not to be altered by the noncommutativity.
通过找到一个厄米算符η⁺并定义湮灭算符和产生算符为彼此的η⁺ - 伪厄米共轭，提出了一种研究非厄米哈密顿量的可能方法。算符η⁺表示哈密顿量的η⁺ - 伪厄米性。作为一个例子，构造并详细分析了一个具有虚线性坐标和线性动量项的非厄米且非PT对称的哈密顿量。找到了算符η⁺，基于此，对于该非厄米且非PT对称的哈密顿量，得到了实谱、正定内积以及波函数的概率解释、本征态的正交性和时间演化的幺正性。此外，当该哈密顿量扩展到具有非对易空间坐标算符和非对易动量算符 的正则非对易空间时，它被证明是耦合的。我们的方法适用于耦合哈密顿量。然后计算了能级的一阶和二阶非对易修正，特别是发现能谱的实性、内积的正定性以及相关性质（波函数的概率解释、本征态的正交性和时间演化的幺正性）不会因非对易性而改变。
