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

TL;DR

This paper investigates the singular behavior of the Laplace operator in spherical coordinates, revealing an overlooked Dirac delta term in the radial Schrödinger equation and discussing conditions to avoid it.

Contribution

It uncovers the hidden Dirac delta contribution in the reduced radial wave function and establishes conditions for its elimination, independent of potential regularity.

Findings

Identifies an overlooked Dirac delta term in the radial Schrödinger equation.

Shows that the wave function must decay rapidly at the origin to avoid this term.

Demonstrates the universality of this behavior regardless of potential type.

Abstract

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.