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.

arXiv commentLaTeX, 6 pages

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• Fundamental length in quantum theories with PT-symmetric Hamiltonians
• PT-symmetric models in curved manifolds
• Non-self-adjoint graphs
• Complete Set of Inner Products for a Discrete PT-symmetric Square-well Hamiltonian
• On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators
• The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics
• Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs
• Quantum star-graph analogues of PT-symmetric square wells
• Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding
• Resonant alteration of propagation in guiding structures with complex Robin parameter and its magnetic-field-induced restoration
• Confluences of exceptional points and a systematic classification of quantum catastrophes
• Theory of response to perturbations in non-Hermitian systems using five-Hilbert-space reformulation of unitary quantum mechanics
• Complex magnetic fields: An improved Hardy-Laptev-Weidl inequality and quasi-self-adjointness
• Equivalent Hermitian operator from supersymmetric quantum mechanics
• Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions
• Solvable quantum lattices with nonlocal non-Hermitian endpoint interactions
• Guiding structures with multiply connected cross-sections: evolution of propagation in external fields at complex Robin parameters
• Quantum star-graph analogues of PT-symmetric square wells. II: Spectra
• Hidden symmetries in non-self-adjoint graphs
• Contact interactions and Kronig-Penney Models in Hermitian and PT-symmetric Quantum Mechanics
• New Concept of Solvability in Quantum Mechanics