  The energy spectrum of two short-range interacting particles in a harmonic
potential trap has previously been related to free-space scattering phase
shifts. But the existing formula for this purpose is exact only in the limit of
an infinitely shallow trap. Here we provide a systematically improved
formula---describing the low-energy dynamics---that enables the use of finite
traps. This paves the way for extracting nuclear scattering phase shifts from
{\it ab initio} nuclear many-body structure calculations, a long-sought goal in
nuclear physics. The derivation establishes effective field theory as a
powerful framework for studying the connection between structure information of
a trapped system (with two or more sub-clusters) and continuum physics in the
fields of both nuclear and condensed-matter physics.
