Inverse Strichartz estimates for 1d Schr\"odinger operators with potentials of quadratic growth

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 $i{\mathrm{\partial }}_{t}u=-\frac{1}{2}\mathrm{\Delta }u$. Motivated by applications to the mass-critical Schr"odinger equation with external potentials (such as the harmonic oscillator) we use a physical space approach.