Uniqueness of Generalized Schrِdinger Operators, Part II

We prove that for φ ∈ H1, 2loc(Rd; dx), φ ≠ 0 dx-a.e., the generalized Schrödinger operator S = Δ + 2φ−1∇φ · ∇, Dom(S) = C∞0(Rd), has exactly one self-adjoint extension on L2(Rd; φ2 · dx) which generates a (sub-)Markovian semigroup on L2(Rd; φ2 · dx). This is based on our previous work where a necessary and sufficient condition on φ for this to hold was proved, but which was only verified to always hold if d = 1. We also prove a corresponding result where Rd is replaced by an infinite dimensional space and the Lebesgue measure by some Gaussian measure whose covariance operator has a discrete spectrum.