Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians

Abstract

In this work we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) I ( t ) for a TD non-Hermitian Hamiltonian H(t) H ( t ) . Assuming H(t) H ( t ) to be simultaneously \rho(t) ρ ( t ) -pseudo-Hermitian and \Xi(t) Ξ ( t ) -anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t)=\Xi^{-1}(t)\rho(t) 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 derivation of the symmetries associated with TD non-Hermitian linear and quadratic Hamiltonians. The computed TD symmetry operators for both cases are then particularized for their equivalent TI Hamiltonians and PT -symmetric restrictions. In the TI scenario we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator: I^{2k}=1 I 2 k = 1 with k k odd [J. Phys. A 35, L467 (2002)]. The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians. From this relation the TD metric is automatically derived from the TD symmetry of the problem. Then, when placed in perspective with the antilinear symmetry I(t)=\Xi^{-1}(t)\rho(t) I ( t ) = Ξ − 1 ( t ) ρ ( t ) , the symmetry-metric relation finally allow us to derive the \Xi(t) Ξ ( t ) -anti-pseudo-Hermitian operator. Our results reinforce the prospects of going beyond