Phase shift approach in the case of a magnetic flux tube

In Refs. [3,1] two different approaches have been developed for the numerical computation of the effective energy in the presence of a magnetic flux tube, by using the phase shift method. However, the opinion has been expressed that these two variations of the phase shift method are not equivalent. In this paper we aim to solve this ambiguity by comparing these two different approaches and showing that they give identical numerical results, within the numerical accuracy of the method.