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

TL;DR

This paper establishes inverse Strichartz estimates for 1D Schrödinger operators with quadratic growth potentials, extending prior results beyond translation-invariant cases using a physical space approach.

Contribution

It introduces inverse Strichartz theorems for Schrödinger evolutions with quadratic potentials, broadening applicability beyond translation-invariant equations.

Findings

Proves inverse Strichartz estimates at L^2 regularity for 1D Schrödinger operators with quadratic potentials.

Develops a physical space method applicable to non-translation-invariant Schrödinger equations.

Extends inverse estimates to Schrödinger equations relevant for external potentials like the harmonic oscillator.

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.