$\eta$-weak-Pseudo-Hermiticity generators and exact solvability

Exact solvability of some non-Hermitian $\eta$-weak-pseudo-Hermitian Hamiltonians is explored as a byproduct of $\eta$-weak-pseudo-Hermiticity generators. A class of V_{eff}(x)=V(x)+iW(x) potentials is considered, where the imaginary part W(x) is used as an $\eta$-weak-pseudo-Hermiticity generator to obtain exactly solvable $\eta$-weak-pseudo-Hermitian Hamiltonian models.