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Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians. I linear amplification

by L. F. Alves da Silva, R. A. Dourado, and M. H. Y. Moussa

Submission summary

Authors (as registered SciPost users):

Rodrigo Dourado · Miled Moussa · Luís da Silva

Abstract

In this work we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for a TD non-Hermitian Hamiltonian H(t). Assuming H(t) to be simultaneously ρ(t)-pseudo-Hermitian and Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t)=Ξ⁻¹(t)ρ(t), which retrieves an important result obtained by Mostafazadeh [J. Math, Phys. 43, 3944 (2002)] for the time-independent (TI) scenario. We apply our method for the derivaton of the symmetry associated with a TD non-Hermitian linear Hamiltonian: a cavity field under linear amplification. The computed TD symmetry operator is then particularized for the equivalent TI linear Hamiltonian and its PT-symmetric restriction. In this TI scenario we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator: I^{2k}=1 with k odd [J. Phys. A 35, L467 (2002)]. The results here derived together with those in the sequel, where we extend our analysis for a TD non-Hermitian quadratic Hamiltonian, enables us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians.