A Liouville-type theorem for Schrödinger equations with nonnegative potential
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908677049417728 |
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| author | Ueberschaer, Henrik |
| author_facet | Ueberschaer, Henrik |
| contents | Let $u$ be a solution of $Δu=Vu$ on $\mathbb{R}^d$, where $V$ be continuous, nonnegative and bounded. We prove that the condition $$\int_{r_j\leq|x|\leq r_j+1}|u(x)|^2dx\to 0,$$ along any sequence $(r_j)$, $r_j\nearrow+\infty$, implies $u\equiv 0$ on $\mathbb{R}^d$. In particular, this implies the Landis conjecture for solutions satisfying a sufficiently fast algebraic decay. These results are generalized to exterior domains as well as for a class of nonlinear Schrödinger equations under suitable conditions on the zero set of the potential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21275 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Liouville-type theorem for Schrödinger equations with nonnegative potential Ueberschaer, Henrik Analysis of PDEs Mathematical Physics 35J10 Let $u$ be a solution of $Δu=Vu$ on $\mathbb{R}^d$, where $V$ be continuous, nonnegative and bounded. We prove that the condition $$\int_{r_j\leq|x|\leq r_j+1}|u(x)|^2dx\to 0,$$ along any sequence $(r_j)$, $r_j\nearrow+\infty$, implies $u\equiv 0$ on $\mathbb{R}^d$. In particular, this implies the Landis conjecture for solutions satisfying a sufficiently fast algebraic decay. These results are generalized to exterior domains as well as for a class of nonlinear Schrödinger equations under suitable conditions on the zero set of the potential. |
| title | A Liouville-type theorem for Schrödinger equations with nonnegative potential |
| topic | Analysis of PDEs Mathematical Physics 35J10 |
| url | https://arxiv.org/abs/2511.21275 |