Landis-type results for discrete equations
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916094756782080 |
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| author | Fernández-Bertolin, Aingeru Roncal, Luz Stan, Diana |
| author_facet | Fernández-Bertolin, Aingeru Roncal, Luz Stan, Diana |
| contents | We prove Landis-type results for both the semidiscrete heat and the stationary discrete Schrödinger equations. For the semidiscrete heat equation we show that, under the assumption of two-time spatial decay conditions on the solution $u$, then necessarily $u\equiv 0$. For the stationary discrete Schrödinger equation we deduce that, under a vanishing condition at infinity on the solution $u$, then $u\equiv 0$. In order to obtain such results, we demonstrate suitable quantitative upper and lower estimates for the $L^2$-norm of the solution within a spatial lattice $(h\mathbb{Z})^d$. These estimates manifest an interpolation phenomenon between continuum and discrete scales, showing that close-to-continuum and purely discrete regimes are different in nature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09066 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Landis-type results for discrete equations Fernández-Bertolin, Aingeru Roncal, Luz Stan, Diana Analysis of PDEs 39A12, 35B05 We prove Landis-type results for both the semidiscrete heat and the stationary discrete Schrödinger equations. For the semidiscrete heat equation we show that, under the assumption of two-time spatial decay conditions on the solution $u$, then necessarily $u\equiv 0$. For the stationary discrete Schrödinger equation we deduce that, under a vanishing condition at infinity on the solution $u$, then $u\equiv 0$. In order to obtain such results, we demonstrate suitable quantitative upper and lower estimates for the $L^2$-norm of the solution within a spatial lattice $(h\mathbb{Z})^d$. These estimates manifest an interpolation phenomenon between continuum and discrete scales, showing that close-to-continuum and purely discrete regimes are different in nature. |
| title | Landis-type results for discrete equations |
| topic | Analysis of PDEs 39A12, 35B05 |
| url | https://arxiv.org/abs/2401.09066 |