Determination of singularities of a function from its perturbation expansion

Abstract

The method for the determination of the position of the pair of complex conjugate branch points suggested in previous studies is generalized here. The method is modified in order to consider cases where the value of the function at the singularity is not real. A method is proposed for the determination of single isolated singularities located either on the real axis or in the complex plane. These methods are applied to three eigenvalue problems, namely the bounded delta-potential atom, the Mathieu equation and the hydrogen atom in a spherically symmetric cavity. We show that the position of the singularities can be obtained very accurately with minimal number of perturbation coefficients. If we take the characteristic polynomial for variational energy levels as an approximate implicit equation, the method can be used for the investigation of the analytic structure of the energy considered as a function of complex coupling constant. In particular, we show that the first singularity appears at the point of intersection of the ground and the first excited states. The second singularity, when the first and second excited states intersect, can be determined either from the expansion at the first singularity or from the expansion of the second excited state at the origin.