Pragmatic SAE procedure in the Schrodinger equation for the inverse-square-like potentials

Abstract: The Self-Adjoint Extension in the Schrodinger equation for potentials behaved as an attractive inverse square at the origin is critically reviewed. Original results are also presented. It is shown that the additional non-regular solutions must be retained for definite interval of parameters, which requires a necessity of performing a Self-Adjoint Extension (SAE) procedure of radial Hamiltonian.The Pragmatic approach is used and some of its consequences are considered for wide class of transitive potentials. Our consideration is based on the established earlier by us a boundary condition for the radial wave function and the corresponding consequences are derived. Various relevant applications are presented as well. They are: inverse square potential in the Schrodinger equation is solved when the additional non-regular solution is retained. Valence electron model and the Klein-Gordon equation with the Coulomb potential is considered and the hydrino -like levels are discussed.