Logarithmic Schr\"odinger Equations in Infinite Dimensions

TL;DR

This paper investigates the logarithmic Schrödinger equation in infinite-dimensional lattice spaces, establishing a Gibbs measure, proving a logarithmic Sobolev inequality, and demonstrating the existence of weak solutions.

Contribution

It introduces a novel approach to infinite-dimensional logarithmic Schrödinger equations by constructing a Gibbs measure and analyzing solution properties.

Findings

Constructed a global Gibbs measure for the equation.

Proved the Gibbs measure satisfies a logarithmic Sobolev inequality.

Established existence of weak solutions in infinite dimensions.

Abstract

We study the logarithmic Schr\"odinger equation with finite range potential on $\mathbb{R}^{\mathbb{Z}^d}$. Through a ground-state representation, we associate and construct a global Gibbs measure and show that it satisfies a logarithmic Sobolev inequality. We find estimates on the solutions in arbitrary dimension and prove the existence of weak solutions to the infinite-dimensional Cauchy problem.