We investigate a two-parametric family of one-dimensional non-Hermitian complex potentials with parity-time ($\mathcal{PT}$) symmetry. We find that there exist two distinct types of phase transitions, from an unbroken phase (characterized by a real spectrum) to a broken phase (where the spectrum becomes complex). The first type involves the emergence of a pair of complex eigenvalues bifurcating from the continuous spectrum. The second type is associated with the collision of such pairs at the bottom of the continuous spectrum. The first transition type is closely related to spectral singularities (SSs), the transmission of the spectral singularities and reflection coefficients are divergent simultaneously. The second transition type is associated with the emergence of bound states. In particular, under specific parameter conditions, we construct an exact bound-state solution. Additionally, signatures of the SSs in a scattering process with wave packets are elucidated. These findings provide a fundamental theoretical framework for manipulating wave scattering in non-Hermitian systems, offering promising implications for designing advanced optical and quantum devices.

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• Scattering theory