Uniqueness of Schrِdinger Operators Restricted in a Domain

LetS :=−Δ/2+Vbe the Schrödingerʹs operator defined onC∞0(D) whereDis a (open) domain inRd. By means of the asymptotic behavior ofVnear the boundary ∂D, we give the necessary and sufficient conditions to the essential Markovian self-adjointness ofSfor the nonnegative potentialV, and to the uniqueness ofSinL1(D) for generalV.