A solution of the bosonic and algebraic Hamiltonians by using an AIM

We apply the notion of an asymptotic iteration method (AIM) to determine the eigenvalues of the bosonic Hamiltonians that include a wide class of quantum optical models. We consider the solutions of the Hamiltonians, which are even the polynomials of the fourth order with respect to the Boson operators. We also demonstrate the applicability of the method for obtaining the eigenvalues of the simple Lie algebraic structures. The eigenvalues of the multi-boson Hamiltonians have been obtained by transforming them into the form of the single boson Hamiltonian in the framework of the AIM.