PT对称性与能量本征值存在的充要条件
PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues
摘要：
尽管它在量子理论中被广泛使用，但哈密顿量的狄拉克厄米性的数学要求足以保证能量本征值的真实性，但不是必要的。本文通过建立三个定理，给出了充要的物理条件。首先，它表明，如果长期方程是真实的，哈密尔顿算子必然是PT对称的。其次，如果引入一个线性算子C，它服从两个方程[C，H]=0和C^2=1，那么一个可对角化的PT对称哈密顿量的能量本征值只有当这个C算子与PT交换时才是实数。第三，具有不可对角化的Jordan块形式的PT对称哈密顿量的能量本征值是实数。这些定理适用于任何维数的矩阵哈密尔顿算子。
英文摘要：
Despite its common use in quantum theory, the mathematical requirement of Dirac Hermiticity of a Hamiltonian is sufficient to guarantee the reality of energy eigenvalues but not necessary. By establishing three theorems, this paper gives physical conditions that are both necessary and sufficient. First, it is shown that if the secular equation is real, the Hamiltonian is necessarily PT symmetric. Second, if a linear operator C that obeys the two equations [C,H]=0 and C^2=1 is introduced, then the energy eigenvalues of a PT-symmetric Hamiltonian that is diagonalizable are real only if this C operator commutes with PT. Third, the energy eigenvalues of PT-symmetric Hamiltonians having a nondiagonalizable, Jordan-block form are real. These theorems hold for matrix Hamiltonians of any dimensionality.
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