Spacing gain and absorption in a simple $\mathcal{PT}$-symmetric model: spectral singularities and ladders of eigenvalues and resonances

Abstract: We consider a parity-time ($\mathcal{PT}$-) symmetric waveguide consisting of a localized gain and loss elements separated by a variable distance. The situation is modelled by a Schr\"odiner operator with localized complex $\mathcal{PT}$-symmetric potential. Properties of the latter Hamiltonian are considered subject to the change of the gain-to-loss distance. Resonances, spectral singularities and eigenvalues are analyzed in detail and discussed in the context of the associated laser-absorber modes and $\mathcal{PT}$-symmetry breaking phase transition. Increasing gain-to-loss distance creates new resonances and spectral singularities which do not exist in the waveguide with adjacent gain and loss. In the limit of large gain-to-loss distance, the waveguide features a ladder of resonances which can be transformed to a ladder of complex eigenvalues by means of the change of the gain-and-loss amplitude.