CPT-Frames for Non-Hermitian Hamiltonians
CPT-Frames for Non-Hermitian Hamiltonians
复制标题
非厄米哈密顿量的 CPT 框架
DOI：
10.1088/0253-6102/60/3/12
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发表时间：
2013-09-15
影响因子：
3.1
通讯作者：
Chen Zheng-Li
中科院分区：
文献类型：
--
作者：
Cao Huai-Xin;Guo Zhi-Hua;Chen Zheng-Li
基于PT-对称量子理论，提出了Hilbert空间K上的PT-框架、PT-对称算子和CPT-框架以及K上算子的概念。证明了PT-对称线性算子的谱和点谱都是关于实轴对称的，且连续PT-对称算子的特征值是实的。对于CD上的线性算子H，证明了H具有不破的PT对称性当且仅当它有d个不同的本征值，且相应的本征态是PT的本征态。给定K上的CPT-框架，诱导出K上的一个新的正内积，称为CPT-内积。导出了稠定线性算子的CPT-伴随与Dirac伴随之间的关系，证明了具有有界CPT-框架的算子是CPT-Hermite的当且仅当它是T-对称的，在这种情况下，它类似于Hermite算子。讨论了由CPT框架组成的算子C的存在性。这些概念和结果将为PT对称量子力学的数学讨论服务。
Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on Cd, it is proved that H has unbroken PT-symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed. These concepts and results will serve a mathematical discussion about PT-symmetric quantum mechanics.