Remarks on the derivation of the virial identity for nonlinear Schr\"odinger equations

TL;DR

This paper offers a new derivation of the virial identity for nonlinear Schrödinger equations, avoiding approximate solutions and regularization, by leveraging intrinsic solution properties.

Contribution

It introduces a novel derivation method for the virial identity that bypasses traditional approximation and regularization techniques.

Findings

Derives virial identity using solution properties without approximation.

Avoids regularizing weights in the derivation process.

Provides a more direct approach based on solution characteristics.

Abstract

We revisit the derivation of the virial identity for nonlinear Schr\"odinger equations. In \cite{O06, FM17}, several conservation laws, such as for the charge and the energy, were derived without constructing a sequence of approximate solutions. Their approach involves additional properties of solutions due to Strichartz' estimate. In this paper, we derive the virial identity without constructing the sequence of approximate solutions or employing a regularizing argument for weights, by exploiting the properties of solutions.