Holomorphic Family of Dirac–Coulomb Hamiltonians in Arbitrary Dimension

Abstract

AbstractWe study massless one-dimensional Dirac–Coulomb Hamiltonians, that is, operators on the half-line of the form $$D_{\omega ,\lambda }:=\begin{bmatrix} -\frac{\lambda +\omega }{x} &{} - \partial _x \\ \partial _x &{} -\frac{\lambda -\omega }{x} \end{bmatrix}$$ D ω , λ : = - λ + ω x - ∂ x ∂ x - λ - ω x . We describe their closed realizations in the sense of the Hilbert space $$L^2(\mathbb {R}_+,\mathbb {C}^2)$$ L 2 ( R + , C 2 ) , allowing for complex values of the parameters $$\lambda ,\omega $$ λ , ω . In physical situations, $$\lambda $$ λ is proportional to the electric charge and $$\omega $$ ω is related to the angular momentum. We focus on realizations of $$D_{\omega ,\lambda }$$ D ω , λ homogeneous of degree $$-1$$ - 1 . They can be organized in a single holomorphic family of closed operators parametrized by a certain two-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. We describe the point spectrum of their nonhomogeneous realizations. It is well-known that $$D_{\omega ,\lambda }$$ 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. Furthermore, we explain the relationship of spherically symmetric Dirac operators with the Dirac operator on the sphere and its eigenproblem. Our work is mainly motivated by a large literature devoted to distinguished self-adjoint 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.