Self-adjointness via partial Hardy-like inequalities

TL;DR

This paper establishes a method to determine distinguished selfadjoint extensions of non-semibounded operators using Hardy-like inequalities, with applications to Dirac-Coulomb operators and optimal coupling constants.

Contribution

It introduces a new approach linking the positivity of the Schur Complement to Hardy-like inequalities for identifying selfadjoint extensions.

Findings

Derived criteria for selfadjoint extensions using Hardy-like inequalities

Applied the method to Dirac-Coulomb operators for optimal coupling constants

Provided a practical framework for operators not semibounded

Abstract

Distinguished selfadjoint extensions of operators which are not semibounded can be deduced from the positivity of the Schur Complement (as a quadratic form). In practical applications this amounts to proving a Hardy-like inequality. Particular cases are Dirac-Coulomb operators where distinguished selfadjoint extensions are obtained for the optimal range of coupling constants.