Minimax Principles, Hardy-Dirac Inequalities, and Operator Cores for Two and Three Dimensional Coulomb-Dirac Operators

For $n\in{2,3}$ we prove minimax characterisations of eigenvalues in the gap of the $n$ dimensional Dirac operator with an potential, which may have a Coulomb singularity with a coupling constant up to the critical value $1/(4-n)$. This result implies a so-called Hardy-Dirac inequality, which can be used to define a distinguished self-adjoint extension of the Coulomb-Dirac operator defined on $C₀^\infty(\{R}ⁿ\setminus{0};\{C}^2(n-1))$, as long as the coupling constant does not exceed $1/(4-n)$. We also find an explicit description of an operator core of this operator.

2010 Mathematics Subject Classification: 49R05, 49J35, 81Q10

Keywords and Phrases: Minimax Principle, Hardy-Dirac Inequality, Coulomb-Dirac Operator

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