  We study massless 1-dimensional Dirac-Coulomb Hamiltonians, that is,
operators on the half-line of the form
$D_{\omega,\lambda}:=\begin{bmatrix}-\frac{\lambda+\omega}{x}&-\partial_x \\
\partial_x & -\frac{\lambda-\omega}{x}\end{bmatrix}$. We describe their closed
realizations in the sense of the Hilbert space $L^2(\mathbb R_+,\mathbb C^2)$,
allowing for complex values of the parameters $\lambda,\omega$. In physical
situations, $\lambda$ is proportional to the electric charge and $\omega$ is
related to the angular momentum. We focus on realizations of
$D_{\omega,\lambda}$ 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
$D_{\omega,\lambda}$ 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.
