Appropriate Inner Product for PT-Symmetric Hamiltonians
Appropriate Inner Product for PT-Symmetric Hamiltonians
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PT 对称哈密顿量的适当内积
DOI：
10.1103/physrevd.97.045001
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发表时间：
2017
影响因子：
5
通讯作者：
P. Mannheim
中科院分区：
文献类型：
--
作者：
P. Mannheim
一个不是厄米的哈密顿量H仍然可以有一个真实的完整的能量本征谱，如果它是PT对称的。对于这样的哈密顿三种可能的内积已被认为是在文献中，$V$规范，$PT$规范，和$C$规范。这里$V$是实现$VHV^{-1}=H^{\dagger}$的算子，$PT$范数是状态与其$PT$共轭的重叠，$C$是一个离散线性算子，对于任何可以对角化的哈密顿量都存在。在这里，我们表明，它是$V$范数是最基本的，因为它总是由理论本身选择。此外，我们表明，$V$范数总是等于$PT$范数，如果一个定义的$PT$共轭的状态包含其内在的$PT$相位。我们讨论的条件下，$V$范数符合$C$算子范数，并表明，在一般情况下，不应该使用线性$C$算子，但它的目的是使用一个可以代替使用反线性$PT$算子本身。
A Hamiltonian $H$ that is not Hermitian can still have a real and complete energy eigenspectrum if it instead is $PT$ symmetric. For such Hamiltonians three possible inner products have been considered in the literature, the $V$ norm, the $PT$ norm, and the $C$ norm. Here $V$ is the operator that implements $VHV^{-1}=H^{\dagger}$, the $PT$ norm is the overlap of a state with its $PT$ conjugate, and $C$ is a discrete linear operator that always exists for any Hamiltonian that can be diagonalized. Here we show that it is the $V$ norm that is the most fundamental as it is always chosen by the theory itself. In addition we show that the $V$ norm is always equal to the $PT$ norm if one defines the $PT$ conjugate of a state to contain its intrinsic $PT$ phase. We discuss the conditions under which the $V$ norm coincides with the $C$ operator norm, and show that in general one should not use the linear $C$ operator but for the purposes that it is used one can instead use the antilinear $PT$ operator itself.