A Class of Exactly Solvable Real and Complex $PT$ Symmetric Reflectionless Potentials

We consider the question of the number of exactly solvable complex but PT-invariant reflectionless potentials with $N$ bound states. By carefully considering the $X_m$ rationally extended reflectionless potentials, we argue that the total number of exactly solvable complex PT-invariant reflectionless potentials are $2[(2N-1)m+N]$.