Unique continuation properties for one dimensional higher order Schr\'{o}dinger equations
| Autor: | Huang, Tianxiao, Huang, Shanlin, Zheng, Quan |
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| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s12220-022-00906-2 |
| Popis: | We study two types of unique continuation properties for the higher order Schr\"{o}dinger equation with potential $$ i\partial_tu=(-\Delta_x)^mu+V(t,x)u,\quad(t,x)\in\mathbb{R}^{1+n},\,2\leq m\in\mathbb{N}_+. $$ The first one says if $u$ has certain exponential decay at two times, then $u\equiv0$, and this result is sharp by constructing critical non-trivial solutions. The second one says if $u\equiv0$ in an arbitrary half-space of $\mathbb{R}^{1+n}$, then $u\equiv0$ identically. The uniqueness theorems are given when $n=1$, but we also prove partial results when $n\in\mathbb{N}_+$ for their own interests. Possibility or obstacles to proving these unique continuation properties in higher spatial dimensions are also discussed. Comment: We have refined the whole paper to be more comprehensive, mainly the introduction. We also correct some mistakes concerning Examples 1.2 |
| Databáze: | arXiv |
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