Holomorphic family of Dirac-Coulomb Hamiltonians in arbitrary dimension

Abstract

We study massless 1-dimensional Dirac-Coulomb Hamiltonians, that is, operators on the half-line of the form D ω,λ :=. We describe their closed realizations in the sense of the Hilbert space L 2 (R + , C 2 ), allowing for complex values of the parameters λ, ω. In physical situations, λ is proportional to the electric charge and ω is related to the angular momentum. We focus on realizations of D ω,λ homogeneous of degree −1. They can be organized in a single holomorphic family of closed operators parametrized by a certain 2-dimensional complex manifold. We describe the spectrum and the numerical range of these realizations. We give an explicit formula for the integral kernel of their resolvent in terms of Whittaker functions. We also describe their stationary scattering theory, providing formulas for a natural pair of diagonalizing operators and for the scattering operator.It is well-known that D ω,λ arise after separation of variables of the Dirac-Coulomb operator in dimension 3. We give a simple argument why this is still true in any dimension.Our work is mainly motivated by a large literature devoted to distinguished selfadjoint realizations of Dirac-Coulomb Hamiltonians. We show that these realizations arise naturally if the holomorphy is taken as the guiding principle. Furthermore, they are infrared attractive fixed points of the scaling action. Beside applications in relativistic quantum mechanics, Dirac-Coulomb Hamiltonians are argued to provide a natural setting for the study of Whittaker (or, equivalently, confluent hypergeometric) functions.

Dirac-Coulomb Operators with Infinite Mass Boundary Conditions in Sectors

“…Indeed, the operator h ν,k is precisely in the form of the one defined in [14, Equation (2. 19)] with m = λ = µ = 0 and k j = −λ k .…”

“…This result was achieved correctly in [26] for [30] for |ν| > 1 and in [14,15] for any ν ∈ R with different techniques: the adaptation of Kreȋn-Višik-Birman-Grubb extension scheme, von Neumann extension theory and the restriction of the domain of the adjoint and boundary triplets respectively. More recently, Dereziński and Ruba [19] classified and carefully analyzed closed extensions with complex-valued potentials. Leaving the realm of electrostatic fields generated by one point charge, we mention [22], where the authors prove the existence of a distinguished self-adjoint extension for a generic (in a certain sense "subcritical") charge distribution.…”