Which nuclear shape generates the strongest attraction on a relativistic electron? An open problem in relativistic quantum mechanics

TL;DR

This paper investigates whether a relativistic quantum electron experiences the strongest attraction when the nuclear charge is concentrated at a single point, addressing a longstanding open problem in relativistic quantum mechanics.

Contribution

It formulates conjectures about the lowest eigenvalue of a Dirac operator with electrostatic potential, exploring the impact of nuclear shape on electron attraction in the relativistic regime.

Findings

Proposes conjectures on eigenvalue minimization

Highlights the open problem in relativistic quantum mechanics

Connects nuclear shape to electron attraction strength

Abstract

In this article we formulate several conjectures concerning the lowest eigenvalue of a Dirac operator with an external electrostatic potential. The latter describes a relativistic quantum electron moving in the field of some (pointwise or extended) nuclei. The main question we ask is whether the eigenvalue is minimal when the nuclear charge is concentrated at one single point. This well-known property in nonrelativistic quantum mechanics has escaped all attempts of proof in the relativistic case.