Extension theorems for logarithmic Schrödinger and discrete Laplacian operators

Abstract

In this paper we consider logarithmic operators in two different contexts: the adapted to (continuous) Schrödinger operators and the classical discrete setting. The Schrödinger operator ${\mathcal{L}}_V$ on ${\mathbb{R}}^d$ is defined as ${\mathcal{L}}_V=-\hat{I}”+V$, where the potential $V$ is nonnegative and satisfies a reverse Hölder inequality and, as usual, $\hat{I}”$ denotes the Euclidean Laplacian, while the discrete Laplacian $\hat{I}{”}_d$ on $\mathbb{Z}$ is given by $(\hat{I}{”}_{d}f)(n)=f(n+1)-2f(n)+f(n-1)$, $n\in \mathbb{Z}$. Both logarithmic operators $\log {\mathcal{L}}_V$ and $\log (-\hat{I}{”}_d)$ 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.