Asymptotic Properties of Solvable $\mathcal{PT}$-Symmetric Potentials

The asymptotic region of potentials has strong impact on their general properties. This problem is especially interesting for PT\mathcal{PT}-symmetric potentials, the real and imaginary components of which allow for a wider variety of asymptotic properties than in the case of purely real potentials. We consider exactly solvable potentials defined on an infinite domain and investigate their scattering and bound states with special attention to the boundary conditions determined by the asymptotic regions. The examples include potentials with asymptotically vanishing and non-vanishing real and imaginary potential components (Scarf II, Rosen-Morse II, Coulomb). We also compare the results with the asymptotic properties of some exactly non-solvable PT\mathcal{PT}-symmetric potentials. These studies might be relevant to the experimental realization of PT\mathcal{PT}-symmetric systems.