Two classes of extensions for generalized Schrödinger operators (Q1908911)

The following so-called generalized Schrödinger operator \(L_\rho\) in \(L^2 (\Omega, \rho^2 dx)\) \((\Omega \subset \mathbb{R}^d\) a domain, \(\rho\in W^{1, 2}_{\text{loc}} (\Omega, dx))\) is considered: \[ L_\rho f= \Delta f+ 2\sum^d_{i =1} {{\nabla_i \rho} \over \rho} \nabla_i f, \qquad f\in C_0^\infty (\Omega). \] Two classes of extensions of the associated energy form \((-L_\rho f, g)_{\rho^2 dx}\) are compared: the Markovian selfadjoint extensions and, on the other side, the extensions in Silverstein's sense. The result is that both classes coincide. Using this, some relations concerning the closures of \((-L_\rho \cdot, \cdot)_{dx}\) and the associated drift transformation of the canonical Brownian motion determined by the multiplicative functional \[ L_t^{[ \rho]}= \exp \Biggl( \int^t_0 {\nabla_\rho \over \rho} (B_s) dB_s- {1\over 2} \int^t_0 \biggl|{\nabla_\rho \over \rho} \biggr|^2 (B_s) ds\;\mathbf{1}_{[t< \tau]} \Biggr), \] \(\tau= \inf\{t: B_t\in \{x: 0< \widetilde {\rho} (x)< \infty\}\}\) (\(\widetilde {\rho}\) a quasi-continuous version of \(\rho\)) are derived.