Uniqueness of generalized Schrödinger operators and applications (Q1187771)

Necessary and sufficient conditions on \(\varphi: \mathbb{R}^ d \to \mathbb{R}\) so that the operator \(S=\Delta+2\varphi^{-1} \nabla\varphi\cdot\nabla\), \(\text{Dom}(S)=C_ 0^ \infty (\mathbb{R}^ d)\), has exactly one selfadjoint extension on \(L^ 2(\mathbb{R}^ d;\varphi^ 2 dx)\) which generates a (sub-)Markovian semigroup are proved.