Sharp weights in the Cauchy problem for nonlinear Schrodinger equations with potential

TL;DR

This paper reviews the conditions under which the nonlinear Schrödinger equation with potential is well-posed, focusing on decay properties and potential growth rates for different nonlinearities.

Contribution

It provides a detailed analysis of decay requirements for well-posedness in nonlinear Schrödinger equations with various potential types.

Findings

Decay conditions depend on the potential's growth rate.

Energy-subcritical nonlinearities require specific decay for well-posedness.

Super-quadratic potentials alter the minimal decay characterization.

Abstract

We review different properties related to the Cauchy problem for the (nonlinear) Schrodinger equation with a smooth potential. For energy-subcritical nonlinearities and at most quadratic potentials, we investigate the necessary decay in space in order for the Cauchy problem to be locally (and globally) well-posed. The characterization of the minimal decay is different in the case of super-quadratic potentials.