Unique continuation of Schr\"odinger-type equations for ∂

The purpose of this paper is to study the unique continuation property for a Schr\"odinger-type equation ∂ u = Vu on a domain in Cⁿ, 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∈ Lᵖ, p<2n, we first prove that, in the case when u is a scalar function, the unique continuation property holds when V∈ Lloc²ⁿ and is ∂-closed. For vector-valued smooth solutions, we establish the unique continuation property either when V∈ Llocᵖ, $ p>2n$ for n≥ 3, or when V∈ Lloc²ⁿ for $n = 2$. Finally, we discuss the unique continuation property for some special cases where V∉ Lloc²ⁿ, for instance, V is a constant multiple of 1/|z|.