$L^\infty$-uniqueness of Schr\"odinger operators restricted in an open domain

Consider the Schr\"odinger operator ${\cal A}=-\frac{\Delta}{2}+V$ acting on space $C_0^\infty(D)$, where $D$ is an open domain in $\R^d$. The main purpose of this paper is to present the $L^\infty(D,dx)$-uniqueness for Schr\"odinger operators which is equivalent to the $L^1(D,dx)$-uniqueness of weak solutions of the heat diffusion equation associated to the operator $\cal A$.