  Alpha($^{4}$He)-cluster models have often been used to describe light nuclei.
Towards the application to multi-cluster systems involving heavy clusters, we
study the relative wave functions of the $\alpha+^{16}$O and $\alpha+^{40}$Ca
systems generated from phase-shift-equivalent potentials. In general, a
potential between clusters is deep accommodating several redundant bound states
which should be removed in an appropriate way. To avoid such a complicated
computation, we generate a shallow-singular potential by using supersymmetric
transformations from the original deep potential. Changes in the relative wave
functions by the transformations are quantified with electric-multipole
transitions which give a different radial sensitivity to the wave function
depending on their multipolarity. Despite the fact that the original and
transformed potentials give exactly the same phase shift, some observables are
unfavorably modified. A possible way to obtain a desired supersymmetric
potential is proposed.
