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Extension theorems for logarithmic Schrödinger and discrete Laplacian operators

Jorge J. Betancor, Marta de León-Contreras, Lourdes Rodríguez-Mesa

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 $\mathcal L_V=-Δ+V$, where the potential $V$ is nonnegative and satisfies a reverse Hölder inequality and, as usual, $Δ$ denotes the Euclidean Laplacian, while the discrete Laplacian $Δ_d$ on $\mathbb Z$ is given by $(Δ_df)(n)=f(n+1)-2f(n)+f(n-1)$, $n\in \mathbb Z$. Both logarithmic operators $\log \mathcal L_V$ and $\log (-Δ_d)$ 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.

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 on is defined as , where the potential is nonnegative and satisfies a reverse Hölder inequality and, as usual, denotes the Euclidean Laplacian, while the discrete Laplacian on is given by , . Both logarithmic operators and 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.

Paper Structure

This paper contains 5 sections, 5 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. The Schrödinger setting
  3. Proof of Theorem \ref{['Th1.1']}
  4. The discrete setting
  5. Proof of Theorem \ref{['Th1.2']}

Key Result

Theorem A

Let $\sigma \in (0,1)$. We consider the Dirichlet problem where $f\in\mathcal{S}(\mathbb{R}^d)$, the space of Schwartz functions, and $\nabla=(\partial_{x_1}, \dots,\partial_{x_n},\partial_t)$, and the associated Dirichlet to Neumann map, given by $f\in\mathcal{S}(\mathbb{R}^d) \mapsto -\lim_{t\to 0^+}t^{1-2\sigma}\partial_t u(\cdot, t).$ Then, the last m

Theorems & Definitions (5)

  • Theorem A
  • Theorem B
  • Theorem 1.1: Extension problem for the Schrödinger logarithmic operator
  • Theorem C
  • Theorem 1.2: Extension problem for the logarithmic discrete Laplacian operator