Unique continuation of Schrödinger-type equations for $\bar\partial$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911476768309248 |
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| author | Pan, Yifei Zhang, Yuan |
| author_facet | Pan, Yifei Zhang, Yuan |
| contents | The purpose of this paper is to study the unique continuation property for a Schrödinger-type equation $ \bar\partial u = Vu$ on a domain in $\mathbb C^n$, where the solution $u$ may be a scalar function, or a vector-valued function. While simple examples show that the unique continuation property fails in general if the potential $V\in L^{p}, p<2n$, we first prove that, in the case when $u $ is a scalar function, the unique continuation property holds when $V\in L_{loc}^{2n}$ and is $\bar\partial$-closed. For vector-valued smooth solutions, we establish the unique continuation property either when $V\in L_{loc}^p $, $ p>2n$ for $n\ge 3$, or when $V\in L_{loc}^{2n}$ for $n = 2$. Finally, we discuss the unique continuation property for some special cases where $V\notin L_{loc}^{2n}$, for instance, $V $ is a constant multiple of $ \frac{1}{|z|}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01947 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unique continuation of Schrödinger-type equations for $\bar\partial$ Pan, Yifei Zhang, Yuan Complex Variables Analysis of PDEs Primary 32W05, Secondary 35J10, 35A02 The purpose of this paper is to study the unique continuation property for a Schrödinger-type equation $ \bar\partial u = Vu$ on a domain in $\mathbb C^n$, where the solution $u$ may be a scalar function, or a vector-valued function. While simple examples show that the unique continuation property fails in general if the potential $V\in L^{p}, p<2n$, we first prove that, in the case when $u $ is a scalar function, the unique continuation property holds when $V\in L_{loc}^{2n}$ and is $\bar\partial$-closed. For vector-valued smooth solutions, we establish the unique continuation property either when $V\in L_{loc}^p $, $ p>2n$ for $n\ge 3$, or when $V\in L_{loc}^{2n}$ for $n = 2$. Finally, we discuss the unique continuation property for some special cases where $V\notin L_{loc}^{2n}$, for instance, $V $ is a constant multiple of $ \frac{1}{|z|}$. |
| title | Unique continuation of Schrödinger-type equations for $\bar\partial$ |
| topic | Complex Variables Analysis of PDEs Primary 32W05, Secondary 35J10, 35A02 |
| url | https://arxiv.org/abs/2404.01947 |