Extension theorems for logarithmic Schr\"odinger and discrete Laplacian operators

TL;DR

This paper develops extension theorems for logarithmic Schr"odinger and discrete Laplacian operators, defining them via boundary value problems inspired by Caffarelli and Silvestre's fractional Laplacian extension.

Contribution

It introduces new extension problems for logarithmic operators in both continuous Schr"odinger and discrete settings, expanding the theoretical framework for nonlocal operators.

Findings

Defined logarithmic operators through boundary value problems

Extended the Caffarelli-Silvestre approach to logarithmic operators

Provided a unified framework for continuous and discrete logarithmic operators

Abstract

In this paper we consider logarithmic operators in two different contexts: the adapted to (continuous) Schr\"odinger operators and the classical discrete setting. The Schr\"odinger operator $\mathcal L_V$ on $\mathbb R^d$ is defined as $\mathcal L_V=-\Delta+V$, where the potential $V$ is nonnegative and satisfies a reverse H\"older inequality and, as usual, $\Delta$ denotes the Euclidean Laplacian, while the discrete Laplacian $\Delta_d$ on $\mathbb Z$ is given by $(\Delta_df)(n)=f(n+1)-2f(n)+f(n-1)$, $n\in \mathbb Z$. Both logarithmic operators $\log \mathcal L_V$ and $\log (-\Delta_d)$ are nonlocal operators and we will define them through suitable extension problems. The extension problems for logarithmic operators are inspired by the one introduced by Caffarelli and Silvestre for the fractional Laplacian but, in this case, the logarithmic operators are obtained as the boundary values of the extension in a more involved way.