D ec 1 99 9 New set of exactly solvable complex potentials giving the real energies

We deform the real potential V (x) of Pöschl and Teller by a shift ε ∈ (0, π/2) of x in imaginary direction. We show that the new model V (x) = F/ sinh(x − i ε) + G/ cosh(x− i ε) remains exactly solvable. Its bound states are constructed in closed form. Wave functions are complex and proportional to Jacobi polynomials. Some of them diverge in the limit ε → 0 or ε → π/2. In contrast, all their energies prove real and ε−independent. In this sense the loss of Hermiticity of our family of Hamiltonians seems well counter-balanced by their accidental PT symmetry. PACS 03.65.Ge, 03.65.Fd Among all the exactly solvable models in quantum mechanics the one-dimensional Schrödinger equation