Singular Behavior of the Laplace Operator in Polar Spherical Coordinates and Some of Its Consequences for the Radial Wave Function at the Origin of Coordinates

摘要

Singular behavior of the Laplace operator in spherical coordinates is investigated. It is shown that in course of transition to the reduced radial wave function in the Schrodinger equation there appears addi- tional term consisting the Dirac delta function, which was unnoted during the full history of physics and mathematics. The possibility of avoiding this contribution from the reduced radial equation is discussed. It is demonstrated that for this aim the necessary and sufficient condition is requirement the fast enough falling of the wave function at the origin. The result does not depend on character of potential-is it regular or sin- gular. The various manifestations and consequences of this observation are considered as well. The corner- stone in our approach is the natural requirement that the solution of the radial equation at the same time must obey to the full equation.