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arXiv:2207.02477

Non-Hermitian Hamiltonian Beyond Pt -Symmetry for Time-Dependant Su(1,1) and Su(2) Systems --- Exact Solution and Geometric Phase in Pseudo-Invariant Theory

Nadjat Amaouche, Maroua Sekhri, Rahma Zerimeche, Mustapha Maamache, J.-Q. Liang

Published Jan 1, 2022• 2 citations
arXivHTMLPDF
DOI
10.2139/ssrn.4173978

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arXiv:2308.13834•Aug 26, 2023•0 citations
In this paper we investigate time-dependent non-Hermitian Hamiltonians, which consist of SU(1,1) and SU(2) generators. The former Hamiltonian is PT symmetric but the latter one is not. A time-dependent non-unitary operator is proposed to construct the non-Hermitian invariant, which is verified as pseudo-Hermitian with real eigenvalues. The exact solutions are obtained in terms of the eigenstates of the pseudo-Hermitian invariant operator for both the SU(1,1) and SU(2) systems in a unified manner. Then, we derive the Lewis–Riesenfeld (LR) phase, which can be separated into the dynamic and the geometrical phases. The analytical results are well consistent with those of the corresponding Hermitian Hamiltonians reported in the literature.
In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo PT symmetry , i.e. the non-Hermitian Hamiltonian H is related to its adjoint H^{†} via the relation, H^{†}=PTHPT . We propose a derivation of pseudo PT symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new metric η(t)=PTη(t) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1,1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new metric and discuss a concrete physical applications of our results.