P T$-symmetric potentials and the so(2, 2) algebra

Starting from a differential realization of the generators of the so(2, 2) algebra we connect the eigenvalue equation of the Casimir invariant either with the hypergeometric equation, or the Schrödinger equation. In the latter case we consider problems for which so(2, 2) appears as a potential algebra, connecting states with the same energy in different potentials. We analyse the role of the two so(2, 1) subalgebras and point out their importance for -symmetric problems, where the doubling of bound states is known to occur. We present two mechanisms for this and illustrate them with the example of the Scarf and the Pöschl–Teller II potentials. We also analyse scattering states, transmission and reflection coefficients for these potentials.