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 ...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.展开更多
这篇译文译自《Fifteen Papers on Functional Aflulysis》,American Mathema-tical Society Translations Series 2.Ⅴ124。(《泛函分析十五篇论文集》),它在抽象边界条件意义下对正定对称算子的扩张进行了描述性讨论,同时对算子的根...这篇译文译自《Fifteen Papers on Functional Aflulysis》,American Mathema-tical Society Translations Series 2.Ⅴ124。(《泛函分析十五篇论文集》),它在抽象边界条件意义下对正定对称算子的扩张进行了描述性讨论,同时对算子的根向量系的完备性也进行了讨论;它对研究算子理论的读者无疑是有一定帮助的。展开更多
基金Supported by the National Natural Science Foundation of China under Grant No.11171197
文摘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.
文摘这篇译文译自《Fifteen Papers on Functional Aflulysis》,American Mathema-tical Society Translations Series 2.Ⅴ124。(《泛函分析十五篇论文集》),它在抽象边界条件意义下对正定对称算子的扩张进行了描述性讨论,同时对算子的根向量系的完备性也进行了讨论;它对研究算子理论的读者无疑是有一定帮助的。
