Pt-symmetric Non-polynomial Oscillators and Hyperbolic Potential with Two Known Real Eigenvalues in a Susy Framework

Extending the supersymmetric method proposed by Tkachuk to the complex domain, we obtain general expressions for superpotentials allowing generation of quasiexactly solvable PT-symmetric potentials with two known real eigenvalues (the ground state and first-excited state energies). We construct examples, namely those of complexified non-polynomial oscillators and of a complexified hyperbolic potential, to demonstrate how our scheme works in practice. For the former we provide a connection with the sl(2) method, illustrating the comparative advantages of the supersymmetric one. Running head: SUSY and PT-symmetric potentials PACS: 03.65.Fd, 03.65.Ge