Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
Abstract
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
- Publication:
-
Journal of Mathematical Physics
- Pub Date:
- January 2002
- DOI:
- 10.1063/1.1418246
- arXiv:
- arXiv:math-ph/0107001
- Bibcode:
- 2002JMP....43..205M
- Keywords:
-
- 11.30.Er;
- 98.80.Hw;
- 95.30.Sf;
- 04.20.-q;
- 11.30.Pb;
- 11.10.Ef;
- Charge conjugation parity time reversal and other discrete symmetries;
- Relativity and gravitation;
- Classical general relativity;
- Supersymmetry;
- Lagrangian and Hamiltonian approach;
- Mathematical Physics;
- High Energy Physics - Theory;
- Quantum Physics
- E-Print:
- Revised version, to appear in J. Math. Phys