Comments about the boundary condition for reduced radial wave function in multi-dimensional equation

TL;DR

This paper critically examines the boundary conditions at the origin for the D-dimensional Schrödinger equation in hyper spherical coordinates, challenging the common assumption of Dirichlet conditions and highlighting unresolved issues for singular potentials.

Contribution

It clarifies the mathematical justification for boundary conditions in multi-dimensional quantum systems, questioning the naturalness of Dirichlet conditions used in prior studies.

Findings

Dirichlet boundary condition is not mathematically justified in general D-dimensional cases.

Time independence of wave function norm supports Dirichlet boundary condition.

Open problem remains for singular potentials at the origin.

Abstract

The problem of boundary behaviour at the origin of coordinates is discussed for D-dimensional Schrodinger equation in the framework of hyper spherical formalism, which have been often considered last time. We show that the Dirichlet condition, which seems as natural, is not mathematically well justified, on the contrary to the 3-dimensional case. The stronger argument in favour of Dirichlet boundary condition is the requirement of time independence of wave functions norm. The problem remains open for singular potentials.