Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem

Abstract

In a previous paper (arXiv:math-ph/0604055) we introduced a very simple PT-symmetric non-Hermitian Hamiltonian with real spectrum and derived a closed formula for the metric operator relating the problem to a Hermitian one. In this note we propose an alternative formula for the metric operator, which we believe is more elegant and whose construction -- based on a backward use of the spectral theorem for self-adjoint operators -- provides new insights into the nature of the model.

LaTeX, 6 pages

References in corpus (6)

• Making Sense of Non-Hermitian Hamiltonians
• Closed formula for the metric in the Hilbert space of a PT-symmetric model
• Delta-Function Potential with a Complex Coupling
• Isospectral Hamiltonians from Moyal products
• Moyal products -- a new perspective on quasi-hermitian quantum mechanics
• Differential Realization of Pseudo-Hermiticity: A quantum mechanical analog of Einstein's field equation

Cited by in corpus (10)

• Gegenbauer-solvable quantum chain model
• Fundamental length in quantum theories with PT-symmetric Hamiltonians
• Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs
• Quantum star-graph analogues of PT-symmetric square wells
• Confluences of exceptional points and a systematic classification of quantum catastrophes
• Resonant alteration of propagation in guiding structures with complex Robin parameter and its magnetic-field-induced restoration
• Solvable quantum lattices with nonlocal non-Hermitian endpoint interactions
• Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions
• Quantum star-graph analogues of PT-symmetric square wells. II: Spectra
• Hidden symmetries in non-self-adjoint graphs