Metrics and isospectral partners for the most generic cubic PT-symmetric non-Hermitian Hamiltonian

Abstract

We investigate properties of the most general PT-symmetric non-Hermitian Hamiltonian of cubic order in the annihilation and creation operators as a ten parameter family. For various choices of the parameters we systematically construct an exact expression for a metric operator and an isospectral Hermitian counterpart in the same similarity class by exploiting the isomorphism between operator and Moyal products. We elaborate on the subtleties of this approach. For special choices of the ten parameters the Hamiltonian reduces to various models previously studied, such as to the complex cubic potential, the so-called Swanson Hamiltonian or the transformed version of the from below unbounded quartic -x^4-potential. In addition, it also reduces to various models not considered in the present context, namely the single site lattice Reggeon model and a transformed version of the massive sextic x^6-potential, which plays an important role as a toy modelto identify theories with vanishing cosmological constant.

arXiv comment21 pages

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• Metrics and isospectral partners for the most generic cubic PT-symmetric non-Hermitian Hamiltonian
• Complete Set of Inner Products for a Discrete PT-symmetric Square-well Hamiltonian
• $J$-self-adjoint operators with $\mathcal{C}$-symmetries: extension theory approach
• From Pseudo-bosons to Pseudo-Hermiticity via multiple generalized Bogoliubov transformations
• Pseudo supersymmetric partners for the generalized Swanson model
• Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding
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• The Quantum Effective Mass Hamilton-Jacobi Problem
• Non-Hermitian multi-particle systems from complex root spaces
• Time-dependent C-operators as Lewis-Riesenfeld invariants in non-Hermitian theories
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• Metric operators for non-Hermitian quadratic su(2) Hamiltonians
• Perturbative approach for strong and weakly coupled time-dependent non-Hermitian quantum systems
• Non-standard quantum algebras and infinite-dimensional PT-symmetric systems
• Isospectral Hermitian counterpart of complex non Hermitian Hamiltonian $p^2-g{x}^4+a/x^2$
• Infinite series of time-dependent Dyson maps
• Swanson Hamiltonian: non-PT-symmetry phase