Algebraic and scattering aspects of a 𝒫𝒯-symmetric solvable potential

We study a particular solvable potential and analyse the effect of symmetry on its bound state as well as scattering solutions. We determine the transmission and reflection coefficients for the -symmetric case and also formulate the problem in terms of an SU(1,1) potential group, which allows unified treatment of the discrete and the continuous spectra in a natural way. We find that (bound and scattering) states of the -symmetric problem supply a basis for the unitary irreducible representations of the SU(1,1) potential group, and this gives a straightforward group theoretical interpretation of the fact that the (complex) -invariant potential has a real energy spectrum.