2002/11/26 by Carl M Bender, Carl M. Bender, Peter N. Meisinger +2 · 8 citations
Engineering · Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Control and Stability of Dynamical Systems #Quantum Mechanics and Non-Hermitian Physics #quant-ph
paper · pdf · doi:10.1088/0305-4470/36/4/312
arxiv created 2002/11/26 · openalex publication_date 2003/01/15 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
It is shown that if a Hamiltonian H is Hermitian, then there always exists an operator having the following properties: (i) is linear and Hermitian; (ii) commutes with H ; (iii) 2 = 1; (iv) the n th eigenstate of H is also an eigenstate of with eigenvalue (−1) n . Given these properties, it is appropriate to refer to as the parity operator and to say that H has parity symmetry, even though may not refer to spatial reflection. Thus, if the Hamiltonian has the form H = p 2 + V ( x ), where V ( x ) is real (so that H possesses time-reversal symmetry), then it immediately follows that H has symmetry. This shows that symmetry is a generalization of Hermiticity: all Hermitian Hamiltonians of the form H = p 2 + V ( x ) have symmetry, but not all -symmetric Hamiltonians of this form are Hermitian.