Uniqueness of Schrodinger operators restricted in a domain

Let S := Delta/2 + V be the Schrodinger's operator defined on C-0(x)(D) where D is a (open) domain in R-d. By means of the asymptotic behavior of V near the boundary partial derivative D, we give the necessary and sufficient conditions to the essential Markovian self-adjointness of S for the nonnegative potential V, and to the uniqueness of S in L-1(D) for general V. (C) 1998 Academic Press.