Sharp convergence rate on Schr\"{o}dinger type operators

TL;DR

This paper investigates the convergence rates of Schrödinger type operators in one dimension, establishing sharp estimates related to initial data regularity and extending results to optimal wave operator ranges across dimensions.

Contribution

It provides sharp convergence rate results and optimal wave operator ranges for Schrödinger type operators, including endpoint cases, in one and multiple dimensions.

Findings

Sharp convergence rates established for one-dimensional Schrödinger operators.

Optimal range for the wave operator obtained in all dimensions.

Frequency-localized maximal estimates proved up to endpoint cases.

Abstract

For Schr\"{o}dinger type operators in one dimension, we consider the relationship between the convergence rate and the regularity for initial data. By establishing the associated frequency-localized maximal estimates, we prove sharp results up to the endpoints. The optimal range for the wave operator in all dimensions is also obtained.