Quasi-exact l-states of Klein–Gordon equation with position-dependent effective mass for exponential scalar and vector fields

In the study, the eigenvalues of the N-dimensional Klein–Gordon equation with an exponentially position-dependent mass are obtained in an analytical form for any l-states in the presence of exponential scalar and vector potentials. A closed form of the eigenvalues for such a system has been examined in the literature. Our motivation for this study is to describe any energy-state of the system with fewer independent parameters. According to this aim, energy-eigenvalues are obtained in an analytical form by correlating the mass parameter with the potential parameters in the system. By using the Asymptotic Iteration Method as a mathematical tool, the \(l=0\) $l=0$ and \(l\ne 0\) $l\ne 0$ states are treated separately. Furthermore, it is seen from the results that the \(l=0\) $l=0$ state in our study corresponds to the \(E_{n,0}=-m_{0}\) $E_{n,0}=-m_0$ stable state of the reference studies in the literature, as expected. As for the \(l\ne 0\) $l\ne 0$ states, it is shown that the obtained analytical expression works properly, being compatible with the numerical energy eigenvalues. This compatibility is also seen for various cases of the potentials and of the position-dependent mass. Accordingly, the study gives an alternative to the literature for constructing the energy spectrum of the system held.