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Abstract

We demonstrate that the existence of a Hermitian time-dependent intertwining operator that maps the non-Hermitian time-dependent energy operator to its Hermitian conjugate and its right to its left eigenstates guarantees the reality of the instantaneous energies. This property holds throughout all three $\cal{PT}$-regimes, in the time-independent scenario referred to as the $\cal{PT}$-symmetric regime, the exceptional point and the spontaneously broken $\cal{PT}$-regime. We also propose a modified adiabatic approximation consisting of an expansion of the wavefunctions in terms the instantaneous eigenstates of the energy operator, instead of the usually used eigenfunctions of the Hamiltonian. We show that this proposal always leads to real Berry phases. We illustrate the working of our general proposals with two explicit examples for a time-dependent non-Hermitian spin model.

Publication Type:	Article
Additional Information:	This is an author-created, un-copyedited version of an article published in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1751-8121/acbe80.
Subjects:	Q Science > QA Mathematics
Departments:	School of Science & Technology > Mathematics
SWORD Depositor:	Symplectic Administrator

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