Spectral problems about many-body Dirac operators mentioned by Dereziński

Abstract

We consider spectral problems for many-body Dirac operators mentioned by Dereziński 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.

Distinguished Self-Adjoint Extension of the Two-Body Dirac Operator with Coulomb Interaction

“…The Appendix A contains a comment on [20], the only publication of which we are aware that also treats self-adjointness of H DC .…”

“…Hence, from the essential self-adjointness of H + on D a that was proven in [20], it does not follow that H DC is essentially self-adjoint on D a . On the contrary, as the article at hand shows, H 0 exhibits a non-trivial nullspace structure in the relative coordinate, i.e., the coordinate of the interaction, whereas one always has Ker(H + ) = {0}.…”

“…We briefly comment on the article [20] by Okaji et al which was written in response to [5]. As already mentioned in the introduction, the proof of their claim, i.e., essential self-adjointness of H DC (even including external Coulomb potentials) as operator in the underlying Hilbert space…”

“…To our best knowledge, there is only one publication so far that touches upon this topic [20]. In their work, the two-body Dirac operator for Coulomb interaction potential and Coulomb external potential is studied.…”