A definition of self-adjoint operators derived from the Schrödinger operator with the white noise potential on the plane

For the white noise $\xi$ on $R^2$, an operator corresponding to a limit of $-\Delta +{\xi }_{\varepsilon }+c_{\varepsilon }$ as $\varepsilon \to 0$ is realized as a self-adjoint operator, where, for each $\varepsilon >0$, $c_{\varepsilon }$ is a constant, ${\xi }_{\varepsilon }$ is a smooth approximation of $\xi$ defined by $exp\left({\varepsilon }^2\Delta \right)\xi$, and $\Delta$ is the Laplacian. This result is a variant of results obtained by Allez and Chouk, Mouzard, and Ugurcan. The proof in this paper is based on the heat semigroup approach of the paracontrolled calculus, referring the proof by Mouzard. For the obtained operator, the spectral set is shown to be $R$.