A remark on the Schr\"odinger smoothing effect

TL;DR

This paper establishes a connection between the smoothing effect of Schr"odinger operators and the decay properties of spectral projectors, providing new insights and applications in one-dimensional cases.

Contribution

It proves the equivalence between smoothing effects and spectral projector decay, offering novel theoretical results and applications for Schr"odinger operators in one dimension.

Findings

Smoothing effect is equivalent to spectral projector decay.

Applications demonstrated for one-dimensional Schr"odinger operators.

Abstract

We prove the equivalence between the smoothing effect for a Schr\"odinger operator and the decay of the associate spectral projectors. We give two applications to the Schr\"odinger operator in dimension one.