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

Abstract

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

Comment on ‘A solution of the bosonic and algebraic Hamiltonians by using an AIM’

“…They assumed 'that m is large enough and the states reach their asymptotic values'. Although the meaning of this statement is not clear because the states |n are the eigenvectors of the occupation-number operator n = â † â, they concluded that the eigenvalues are given by the asymptotic condition [1] q m p m+1 − q m+1 p m = 0.…”

“…Koç et al [1] developed a variant of the AIM for the calculation of the eigenvalues of bosonic Hamiltonians Ĥ = H (â † , â), where â † and â are the boson creation and annihilation operators, respectively, that satisfy [ â, â † ] = 1. They explicitly considered the occupation-number basis set {|n , n = 0, 1, .…”

“…The authors assumed that the action of Ĥ on the state |n produces the following three-term recurrence relation [1]:…”

“…The purpose of this comment is to analyze the new AIM proposed by those authors. In section 2, we criticize the validity of the main equations derived by Koç et al [1]. In section 3, we put forward a conjecture about the reasons why the authors obtained the actual eigenvalues and eigenfunctions of the bosonic operators.…”

“…In a recent paper, Koç et al [1] proposed a variant of the asymptotic iteration method (AIM) for the calculation of eigenvalues and eigenfunctions of bosonic operators. They obtained results for several models that agree with earlier calculations.…”