Two exactly-solvable problems in one-dimensional quantum mechanics on circle

In this note we establish a relation between two exactly-solvable problems on circle, namely singular Coulomb and singular oscillator systems. In a recent paper [1] have constructed a series of complex mappings S 2 C → S 2 , S 4 C → S 3 and S 8 C → S 5 , which extend to spherical geometry the Levi-Civita, Kustaanheimo-Stiefel and Hurwitz transformations, well known for Euclidean space. We have shown that these transformations establish a correspondence between Coulomb and oscillator problems in classical and quantum mechanics for dimensions (2,2), (3,4) and (5,8) on the spheres. A detailed analysis of the real mapping on the curved space has been done in [2]. It was remarked that in the stereographic projection the relation between Coulomb and oscillator problems functionally coincide with the flat space Levi-Civita and Kustaanheimo-Stiefel transformations. The relation between the quasiradial Schr¨odinger equations for Coulomb and oscillator problems on the n -dimensional spheres and one- and two-sheeted hyperboloids for n ≥ 2 was found in the article [3]. The present note are devoted to two one-dimensional exactly-solvable potentials on circle