  Properties of the wave function equivalent potentials introduced by HAL QCD
collaboration are studied in a non-relativistic coupled-channel model. The
derivative expansion is generalized, and then applied to the energy-independent
and non-local potentials. The expansion coefficients are determined from
analytic solutions to the Nambu-Bethe-Salpeter wave functions. The scattering
phase shifts computed from these potentials are compared with the exact values
to examine the convergence of the expansion. It is confirmed that the
generalized derivative expansion converges in terms of the scattering phase
shift rather than the functional structure of the non-local potentials. It is
also found that the convergence can be improved by tuning either the choice of
interpolating fields or expansion scale in the generalized derivative
expansion.
