UNIQUENESS OF GENERALIZED SCHRODINGER-OPERATORS .2.

We prove that for phi is-an-element-of H(loc)1.2(R(d); dx), phi not-equal dx-a.e., the generalized Schrodinger operator S = DELTA + 2phi-1 delphi.del, Dom(S) = C0infinity(R(d)), has exactly one self-adjoint extension on L2(R(d);phi2.dx) which generates a (sub-)Markovian semigroup on L2(R(d);phi.dx). This is based on our previous work where a necessary and sufficient condition on phi for this to hold was proved, but which was only verified to always hold if d= 1. We also prove a corresponding result where R(d) is replaced by an infinite dimensional space and the Lebesgue measure by some Gaussian measure whose covariance operator has a discrete spectrum. (C) 1994 Academic Press, Inc.