Hybrid QLunch: Blazej Ruba

It is a classic fact that the Dirac Hamiltonian with Coulomb potential is essentially self-adjoint only for atomic numbers Z up to 118. In the range 119<Z<137 essential self-adjointness is lost, but physically distinguished self-adjoint extensions exist. I will discuss how these issues can be understood by constructing a holomorphic family of Dirac-Coulomb operators. These operators depend on charge and angular momentum parameters (both allowed to be complex) and a boundary condition at the origin. Their spectral properties, including diagonalization, will be described. Another perspective on the problem of self-adjoint extensions is offered by the analysis of scaling action. With mass parameter put to zero, one obtains formally homogeneous differential operators, whose domains of self-adjointness are acted upon by the scaling group. Distinguished extensions are the infrared attractive fixed points, while for Z above 137 the scaling action becomes periodic. If time permits, I will also review separation of variables for Dirac Hamiltonian in any dimension.