  We study the spectrum of the Dirac hamiltonian in one space dimension for a
single electron in the electrostatic potential of a point nucleus, in the
Born-Oppenheimer approximation where the nucleus is assumed fixed at the
origin. The potential is screened at large distances so that it goes to zero
exponentially at spatial infinity. We show that the hamiltonian is essentially
self-adjoint, the essential spectrum has the usual gap $(-mc^2,mc^2)$ in it,
and that there are only finitely many eigenvalues in that gap, corresponding to
ground and excited states for the system. We find a one-to-one correspondence
between the eigenfunctions of this hamiltonian and the heteroclinic
saddle-saddle connectors of a certain dynamical system on a finite cylinder. We
use this correspondence to study how the number of bound states changes with
the nuclear charge.
