$L^{\infty }$-uniqueness of Schrödinger operators restricted in an open domain

Abstract

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