Spectral problems about many-body Dirac operators mentioned by Derezi\'{n}ski

TL;DR

This paper investigates the spectral properties of many-body Dirac operators, deriving a matrix representation for helium-like ions, proving essential self-adjointness, and analyzing the spectrum to show it covers the entire real line with no eigenvalues.

Contribution

It provides a new matrix representation of the Dirac Coulomb operator for helium-like ions and establishes key spectral properties including self-adjointness and spectrum characterization.

Findings

The Dirac Coulomb operator can be represented as a 16x16 matrix operator.

The operator is essentially self-adjoint under certain conditions.

Its spectrum covers the entire real line with no eigenvalues.

Abstract

We consider spectral problems for many-body Dirac operators mentioned by Derezi\'{n}ski in the IAMP News Bulletin of January 2012. In particular, we derive a representation of the Dirac Coulomb operator for a helium-like ion as a matrix operator of order sixteen. We show that it is essentially self-adjoint (under natural restrictions on the coupling constants), that the essential spectrum of its closure is the whole real line and that it has no eigenvalues.