Non-Hermitian Hamiltonians of Lie algebraic type

TL;DR

This paper studies a class of non-Hermitian Hamiltonians built from Lie algebra generators, exploring conditions for real spectra and constructing their Hermitian counterparts using algebraic and transformation methods.

Contribution

It introduces a systematic analysis of Lie algebraic non-Hermitian Hamiltonians, deriving constraints for real spectra and providing explicit methods to find their Hermitian equivalents.

Findings

Derived conditions for real spectra of Lie algebraic non-Hermitian Hamiltonians.

Constructed isospectral Hermitian counterparts for specific models.

Explicitly computed real energy spectra using generalized Bogoliubov transformations.

Abstract

We analyse a class of non-Hermitian Hamiltonians, which can be expressed bilinearly in terms of generators of a sl(2,R)-Lie algebra or their isomorphic su(1,1)-counterparts. The Hamlitonians are prototypes for solvable models of Lie algebraic type. Demanding a real spectrum and the existence of a well defined metric, we systematically investigate the constraints these requirements impose on the coupling constants of the model and the parameters in the metric operator. We compute isospectral Hermitian counterparts for some of the original non-Hermitian Hamiltonian. Alternatively we employ a generalized Bogoliubov transformation, which allows to compute explicitly real energy eigenvalue spectra for these type of Hamiltonians, together with their eigenstates. We compare the two approaches.