  We prove inverse Strichartz theorems at $L^2$ regularity for a family of
Schr\"{o}dinger evolutions in one space dimension. Prior results rely on
spacetime Fourier analysis and are limited to the translation-invariant
equation $i\partial_t u = -\tfrac{1}{2} \Delta u$. Motivated by applications to
the mass-critical Schr\"odinger equation with external potentials (such as the
harmonic oscillator) we use a physical space approach.
