Dimensionalities of Weak Solutions in Hydrogenic Systems

TL;DR

This paper investigates the formal eigenvectors of the 3D hydrogen atom Hamiltonian, revealing their belonging to the Hilbert space and their role in ionization, with implications for lower-dimensional hydrogen systems.

Contribution

It uncovers the significance of eigenvectors typically discarded, showing they are in the Hilbert space and influence ionization in hydrogenic systems.

Findings

Eigenvectors belong to the Hilbert space despite being outside the domain.

Formal eigenvectors have continuous components enabling ionization.

Connections established between 3D, 2D, and 1D hydrogen systems.

Abstract

A close inspection on the 3D hydrogen atom Hamiltonian revealed formal eigenvectors often discarded in the literature. Although not in its domain, such eigenvectors belong to the Hilbert space, and so their time evolution is well defined. They are then related to the 1D and 2D hydrogen atoms and it is numerically found that they have continuous components, so that ionization can take place.