Balanced gain and loss in spatially extended non-$\mathcal{P}\mathcal{T}$-symmetric multi-well potentials

The experimental realization of balanced gain and loss in a quantum system has been a long standing goal in quantum mechanics since the introduction of the concept of $\mathcal{P}\mathcal{T}$ symmetry and has only recently been achieved. In this paper we analyze balanced gain and loss in Gaussian multi-well potentials with either only gain or loss in each well. By means of symmetrization via matrix models we can construct asymmetric extended potentials with partially real or complex conjugate spectra. This will be demonstrated explicitly for double-well and triple-well systems. Such systems can be realized with Bose-Einstein condensates in optical trapping potentials in the presence of localized particle gain and loss. The usage of asymmetric potentials in the process is more versatile and is considered beneficial in real experimental implementations.