An example of double confluent Heun equation: Schrödinger equation with supersingular plus Coulomb potential

A recently proposed algorithm to obtain global solutions of the double confluent Heun equation is applied to solve the quantum mechanical problem of finding the energies and wave functions of a particle bound in a potential sum of a repulsive supersingular term, A r, plus an attractive Coulombian one, −Z r. The existence of exact algebraic solutions for certain values of A is discussed. Supersingular potentials, i. e., potentials presenting at the origin a singularity of the type r, with α > 2, were firstly discussed in the context of collision of particles [1– 6], the main issue being the adequate definition of the S matrix. On the other hand, concerned with bound states, a considerable interest on repulsive singular potentials added to an attractive regular one was arisen by the seminal paper of Klauder [7], where the today known as “Klauder phenomenon” was reported. As shown by Detwiler and Klauder [8], supersingular potentials cannot be treated by conventional WKB or perturbative methods. In view of this, a great diversity of approximate methods have been suggested. Extensive lists of articles applying those methods to supersingular potentials can be found in recent papers by Saad, Hall and Katatbeh [9] and by Liverts and Mandelzweig [10]. The validity of an approximate method can be inferred from comparison of its results with those obtained with an exact method. Up to now, direct numerical [email protected]