On Landis' conjecture for positive Schrödinger operators on graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913856300777472 |
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| author | Das, Ujjal Keller, Matthias Pinchover, Yehuda |
| author_facet | Das, Ujjal Keller, Matthias Pinchover, Yehuda |
| contents | In this note we study the Landis conjecture for positive Schrödin\-ger operators on graphs. More precisely, we prove a Landis-type result in the form of a decay criterion that ensures when $\mathcal{H}$-harmonic functions for a positive Schrödinger operator $\mathcal{H}$ with potentials bounded from above by $ 1 $ are trivial. The positivity assumption on the operator allows us to impose slow decay across the entire graph, while requiring fast decay in only one direction, rather than throughout the whole graph. We then specifically look at the special cases of $ \mathbb{Z}^{d} $ and regular trees for which we get a explicit decay criterion. Moreover, we consider the fractional analogue of the Landis conjecture on $ \mathbb{Z}^{d} $. Our approach relies on the discrete version of Liouville comparison principle which is also proved in this article. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02149 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Landis' conjecture for positive Schrödinger operators on graphs Das, Ujjal Keller, Matthias Pinchover, Yehuda Analysis of PDEs Mathematical Physics Functional Analysis Spectral Theory 35J10, 35B53, 35R02, 39A12 In this note we study the Landis conjecture for positive Schrödin\-ger operators on graphs. More precisely, we prove a Landis-type result in the form of a decay criterion that ensures when $\mathcal{H}$-harmonic functions for a positive Schrödinger operator $\mathcal{H}$ with potentials bounded from above by $ 1 $ are trivial. The positivity assumption on the operator allows us to impose slow decay across the entire graph, while requiring fast decay in only one direction, rather than throughout the whole graph. We then specifically look at the special cases of $ \mathbb{Z}^{d} $ and regular trees for which we get a explicit decay criterion. Moreover, we consider the fractional analogue of the Landis conjecture on $ \mathbb{Z}^{d} $. Our approach relies on the discrete version of Liouville comparison principle which is also proved in this article. |
| title | On Landis' conjecture for positive Schrödinger operators on graphs |
| topic | Analysis of PDEs Mathematical Physics Functional Analysis Spectral Theory 35J10, 35B53, 35R02, 39A12 |
| url | https://arxiv.org/abs/2408.02149 |