Balancing gain and loss in symmetrised multi-well potentials

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 symmetrised or semi-symmetrised, respectively. In contrast to similar concepts, symmetrisation allows for the description of physical multi-well potentials with gain and loss. 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 have stationary excited states.