  In many-body theory it is often useful to renormalize short-distance,
high-momentum components of an interaction via unitary transformations. Such
transformations preserve the on-shell physical observables of the two-body
system (mostly phase-shifts, hence unitarily-connected effective interactions
are often called phase-equivalent), while modifying off-shell T-matrix elements
influential in many-body systems. In this paper I lay out a general and
systematic approach for controlling the off-shell behavior of an effective
interaction, which can be adjusted to many-body properties, and present an
application to trapped fermions at the unitary
