The SO(3,1) x U(1)-gauge invariant approach to charged bosons in relativistic magnetars

Using a perturbative method, we investigate solutions of the Klein-Gordon equations for a charged massive field in the background of a magnetar, both in the interior solution and outside the star. A special attention is given to cases where the variables can be separated and the wave function is expressed in terms of the Heun's general or confluent functions. By imposing various conditions on the parameters, one gets the energy quantization law and simple polynomial forms of the Heun's functions, which can be used in computing first-order transition amplitudes.