Extension theorems for logarithmic Schrödinger and discrete Laplacian operators
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arXiv
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| Formato: | Preprint |
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2026
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| _version_ | 1866914445682278400 |
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| author | Betancor, Jorge J. de León-Contreras, Marta Rodríguez-Mesa, Lourdes |
| author_facet | Betancor, Jorge J. de León-Contreras, Marta Rodríguez-Mesa, Lourdes |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03638 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extension theorems for logarithmic Schrödinger and discrete Laplacian operators Betancor, Jorge J. de León-Contreras, Marta Rodríguez-Mesa, Lourdes Classical Analysis and ODEs 35J10, 42B37, 47D06 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. |
| title | Extension theorems for logarithmic Schrödinger and discrete Laplacian operators |
| topic | Classical Analysis and ODEs 35J10, 42B37, 47D06 |
| url | https://arxiv.org/abs/2604.03638 |