Dirac-Coulomb operators with general charge distribution: results and open problems

Abstract: This talk is based on joint works with M.J. Esteban and M. Lewin. Consider an electron moving in the attractive Coulomb potential generated by a non-negative finite measure representing an external charge density. If the total charge is fixed, it is well known that the lowest eigenvalue of the corresponding Schrodinger operator is minimized when the measure is a delta. We investigate the conjecture that the same holds for the relativistic Dirac-Coulomb operator. First we give conditions ensuring that this operator has a natural self-adjoint realisation and that its eigenvalues are given by min-max formulas. Then we define a critical charge such that, if the total charge is fixed below it, then there exists a measure minimising the first eigenvalue of the Dirac-Coulomb operator. Moreover this optimal measure concentrates on a compact set of Lebesgue measure zero. The last property is proved using a new unique continuation principle for Dirac operators.

mathematical physicsanalysis of PDEsquantum physics

Audience: researchers in the topic

Mathematical Challenges in Quantum Mechanics 2021 Workshop