For the white noise $\xi$ on ${\mathbb R}^2$, 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 $\xi$ such that $\xi _{\varepsilon}\to \xi$ as $\varepsilon \to 0$, and $\Delta$ is the Laplacian. This is an extension of the known results for the operator on compact spaces by Allez and Chouk, and Mouzard. For the obtained operator, the spectral set is showen to be ${\mathbb R}$. The main tools are the paracontrolled calculus and the partition of unity.