Inverse Strichartz estimates for 1d Schrödinger operators with potentials of quadratic growth

Abstract

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

Improved exposition; added figures; added R. Killip and M. Visan as co-authors; 22 pages