On the equation $Du+Xu=0$ and its relation to Schroedinger ground states

TL;DR

This paper explores the connection between minimizers of a specific functional involving a vector field and ground state solutions of the Schrödinger equation, providing insights into their mathematical relationship.

Contribution

It establishes simple relations linking absolute minimizers of a functional to Schrödinger ground states, extending to more general functionals.

Findings

Identifies relations between minimizers and ground states

Provides a framework for analyzing Schrödinger solutions via functional minimization

Extends results to broader classes of functionals

Abstract

We present some simple relations between the absolute minimizers of the functional $||Du+Xu||$, where $X$ is a vector field on $R^n$, and ground state solutions to the (non-relativistic) Schroedinger equation. This article is a byproduct of the study of the more general functional $||ADu+F'(u)X||$.