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}=1I2k=1 with kk 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 \mathcal{PT}𝒫𝒯-symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising.
Highlights
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 )
The method we propose for the derivation of TD symmetries for TD non-Hermitian Hamiltonians applies indistinctly to linear or antilinear, unitary or nonunitary symmetries
In this work we have proposed a method for the derivation of general TD continuous symmetry operators for TD non-Hermitian Hamiltonians
Summary
In the last two decades, since the seminal contributions of Bender and Boettcher [1] and Mostafazadeh [2–4], PT -symmetric Hamiltonians —invariant under parity (P) and timereversal (T ) symmetry— have been investigated in practically all domains of physics, from low to high energies, revealing to be an increasingly autonomous and thought-provoking field. In this work we propose a method to derive a general time-dependent (TD) continuous symmetry operator for a TD non-Hermitian Hamiltonian This will be done in the broader scenario of non-autonomous Hamiltonians, and for this reason we revisit the TD non-Hermitian Hamiltonians of a cavity field under linear [11] and parametric [12] amplifications. From this relation we retrieve the Mostafazadeh’s theorems for the TI scenario [4]. In this TI scenario we retrieve the Bender-Berry-Mandilara [29] result, and when considering a PT -symmetric TI non-Hermitian Hamiltonian, we verify that the TI symmetry again reduces to the PT operator.
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