There is much discussion in the mathematical physics literature as well as in quantum mechanics textbooks on spherically symmetric potentials. Nevertheless, there is no consensus about the behavior of the radial function at the origin, particularly for singular potentials. A careful derivation of the radial Schrödinger equation leads to the appearance of a delta function term when the Laplace operator is written in spherical coordinates. As a result, regardless of the behavior of the potential, an additional constraint is imposed on the radial wave function in the form of a vanishing boundary condition at the origin.

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 additional 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 singular. The various manifestations and consequences of this observation are considered as well. The cornerstone in our approach is the natural requirement that the solution of the radial equation at the same time must obey to the full equation.

In this article we address the singular behavior of the Laplace operator in spherical coordinates, which has been established in our earlier works. This singularity has many non-trivial consequences. In this article we consider only the simplest ones, which are connected to the solution of the Laplace equation in various classical books and lectures. We show how solutions, which have a fictitious singular behavior at the origin can be avoided. This material may be useful in physics education both for students and teachers as well.

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We show that additional solutions must be ignored (in differences of the Schrodinger and Klein-Gordon equations) in the Dirac equation, where usually passed the second order radial equation, called the reduced equation, instead of a system. Analogously to the Schrodinger equation, in this process the Dirac's delta function appears, which was unnoted during the full history of quantum mechanics. This unphysical term we remove by a boundary condition at the origin. However, the distribution theory imposes on the radial function strong restriction and by this reason practically for all potentials, even regular, use of these reduced equations is not permissible. At the end we include consideration in the framework of two-dimensional Dirac equation. We show that even here the additional solution does not survives as a result of usual physical requirements.