2012/04/30 by Junqing Li, Jun-Qing Li, Qian Li +1 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic number #Hamiltonian (control theory) #Hermitian matrix #Hilbert space #Mathematical analysis #Mathematical physics #Mathematics #Operator (biology) #Physics #Pure mathematics #Quantum Mechanics and Non-Hermitian Physics #Quantum mechanics #Self-adjoint operator #Symmetric space #hep-th #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1088/0253-6102/58/4/08
published as Commun. Theor. Phys. 58 (2012) 497-503 · 13 pages, no figures; v2: 14 pages, typos corrected, clarifications and references added, this version accepted for publication in Commun. Theor. Phys
arxiv created 2012/07/10 · openalex publication_date 2012/10/01 · openalex created_date 2016/06/24 · arxiv updated 2018/01/17 · openalex updated_date 2026/08/05
Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones. Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged. In order to give the positive definite inner product for the PT-symmetric systems, a new operator V, instead of C, can be introduced. The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics, however, it can be constructed, as an advantage, directly in terms of Hamiltonians. The spectra of the two non-Hermitian PT-symmetric systems are obtained, which coincide with that given in literature, and in particular, the Hilbert spaces associated with positive definite inner products are worked out.