Perturbative approach for strong and weakly coupled time-dependent non-Hermitian quantum systems

Abstract

We propose a perturbative approach to determine the time-dependent Dyson map and the metric operator associated with time-dependent non-Hermitian Hamiltonians. We apply the method to a pair of explicitly time-dependent two dimensional harmonic oscillators that are weakly coupled to each other in a PT-symmetric fashion and to the strongly coupled explicitly time-dependent negative quartic anharmonic oscillator potential. We demonstrate that once the perturbative Ansatz is set up the coupled differential equations resulting order by order may be solved recursively in a constructive manner, thus bypassing the need for making any guess for the Dyson map or the metric operator. Exploring the ambiguities in the solutions of the order by order differential equations naturally leads to a whole set of inequivalent solutions for the Dyson maps and metric operators implying different physical behaviour as demonstrated for the expectation values of the time-dependent energy operator.

arXiv comment24 pages, 2 figures

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Cited by in corpus (6)

• An introduction to PT-symmetric quantum mechanics -- time-dependent systems
• IR/UV mixing from local similarity maps of scalar non-Hermitian field theories
• Time-dependent C-operators as Lewis-Riesenfeld invariants in non-Hermitian theories
• Real energies and Berry phases in all PT-regimes in time-dependent non-Hermitian theories
• Lewis-Riesenfeld invariants for PT-symmetrically coupled oscillators from two dimensional point transformations and Lie algebraic expansions
• Infinite series of time-dependent Dyson maps