Real eigenvalues in symmetrised multi-well potentials with gain and loss

Balanced gain and loss leads to stationary dynamics in open systems. This occurs naturally in $\mathcal{PT}$-symmetric systems, where the imaginary part of the potential describing gain and loss is perfectly antisymmetric. While this case seems intuitive, stationary dynamics is also possible in asymmetric open systems. We will show that open multi-well quantum systems can possess completely (or partly) real spectra if their Hamiltonian is (semi-)symmetrised. A simple matrix model for the description of two and three-mode systems is used as an example, for which analytical symmetrised solutions are derived. We will further explicitly show how symmetrisation can be used to systematically find two-mode systems with a stable, stationary ground state. It will also become clear why only $\mathcal{PT}$-symmetric two-mode systems can also have stationary excited states.