  Resonant solutions of the quantum Schr\"odinger equation occur at complex
energies where the S-matrix becomes singular. Knowledge of such resonances is
important in the study of the underlying physical system. Often the
Schr\"odinger equation is dependent on some parameter and one is interested in
following the path of the resonances in the complex energy plane as the
parameter changes. This is particularly true in coupled channel systems where
the resonant behavior is highly dependent on the strength of the channel
coupling, the energy separation of the channels and other factors. In previous
work it was shown that numerical continuation, a technique familiar in the
study of dynamical systems, can be brought to bear on the problem of following
the resonance path in one dimensional problems and multi-channel problems
without energy separation between the channels. A regularization can be defined
that eliminates coalescing poles and zeros that appear in the S-matrix at the
origin due to symmetries. Following the zeros of this regularized function then
traces the resonance path. In this work we show that this approach can be
extended to channels with energy separation, albeit limited to two channels.
The issue here is that the energy separation introduces branch cuts in the
complex energy domain that need to be eliminated with a so-called
uniformization. We demonstrate that the resulting approach is suitable for
investigating resonances in two-channel systems and provide an extensive
example.
