Unique continuation of Schr\"odinger-type equations for $\bar\partial$

Abstract

The purpose of this paper is to study the unique continuation property for a Schr\"odinger-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|}$.