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

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