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We consider a linear symmetric operator in a Hilbert space that is neitherbounded from above nor from below, admits a block decomposition correspondingto an orthogonal splitting of the Hilbert space and has a variational gapproperty associated with the block decomposition. A typical example is theDirac-Coulomb operator defined on $C^\infty_c(\mathbb R^3\setminus\{0\},\mathbb C^4)$. In this paper we define a distinguished self-adjoint extensionwith a spectral gap and characterize its eigenvalues in that gap by a min-maxprinciple. This has been done in the past under technical conditions. Here weuse a different, geometric strategy, to achieve that goal by making onlyminimal assumptions. Our result applied to the Dirac-Coulomb-like Hamitonianscovers sign-changing potentials as well as molecules with an arbitrary numberof nuclei having atomic numbers less than or equal to 137.

Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac-Coulomb operators