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

Abstract

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.