Unique continuation of Schrödinger-type equations for $\bar{\partial }$

Abstract

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 $\frac{1}{|z|}$.

27 pages