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This work is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level quantum models we demonstrate the phenomenon of fast transitions developed in the PT-symmetric quantum brachistochrone problem may in fact be attributed to the non- Hermiticity of evolution operator used, rather than to its invariance under PT operation. When it comes to Hilbert spaces of infinite dimension, we show how Lie algebras and Groenewold Moyal can be explored to construct isospectral Hermitian partners. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. Classically, our efforts were concentrated on integrable models presenting PT-symmetry. We use the Painleve test to check whether deformations of integrable systems preserve integrability and we study the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT-symmetric way.

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Paulo Assis works as a Post-Doctoral Research Associate at the University of Kent. He obtained his PhD degree in Mathematical Physics at City University London and both his MSc and BSc at the University of Sao Paulo.

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