# Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei

## TL;DR

This paper classifies all self-adjoint extensions of the Dirac-Coulomb operator for heavy nuclei using advanced extension theories, providing clear boundary conditions and spectral properties relevant for quantum physics.

## Contribution

It introduces a novel classification method based on Kre-Vi-Birman and Grubb's theories, differing from traditional von Neumann approaches.

## Key findings

- Boundary conditions emerge as multiplicative constraints.
- Explicit characterization of spectral gap and resolvent.
- Unified framework for self-adjoint extensions in critical Coulomb regime.

## Abstract

We derive a classification of the self-adjoint extensions of the three-dimensional Dirac-Coulomb operator in the critical regime of the Coulomb coupling. Our approach is solely based upon the Kre{\u\i}n-Vi\v{s}ik-Birman extension scheme, or also on Grubb's universal classification theory, as opposite to previous works within the standard von Neumann framework. This let the boundary condition of self-adjointness emerge, neatly and intrinsically, as a multiplicative constraint between regular and singular part of the functions in the domain of the extension, the multiplicative constant giving also immediate information on the invertibility property and on the resolvent and spectral gap of the extension.