  Using the variable phase method, we reformulate the Dirac equation governing
the charge carriers in graphene into a nonlinear first-order differential
equation from which we can treat both confined-state problems in electron
waveguides and above-barrier scattering problems for arbitrary-shaped potential
barriers and wells, decaying at large distances. We show that this method
agrees with a known analytic result for a hyperbolic secant potential and go on
to investigate the nature of more experimentally realizable electron
waveguides, showing that, when the Fermi energy is set at the Dirac point,
truly confined states are supported in pristine graphene. In contrast to
exponentially-decaying potentials, we discover that the threshold potential
strength at which the first confined state appears is vanishingly small for
potentials decaying at large distances as a power law, but nonetheless further
confined states are formed when the strength and spread of the potential reach
a certain threshold.
