Schrödinger equation with Gaussian potential

This paper studies the Schrödinger equation with fractional Gaussian noise potential of the form $\Delta u(x)= u(x)\diamond \dot W(x)$, $x\in D$, $u(x)= \phi (x)$, $x\in \partial D$, where $\Delta$ is the Laplacian on the $d$-dimensional Euclidean space $\mathbb {R}^d$, $D\subseteq \mathbb {R}^d$ is a given domain with smooth boundary $\partial D$, $\phi$ is a given nice function on the boundary $\partial D$, and $\dot W$ is the fractional Gaussian noise of Hurst parameters $(H_1, \ldots , H_d)$ and $\diamond$ denotes the Wick product. We find a family of distribution spaces $(\mathbb {W}_{\lambda }, {\lambda }>0)$, with the property $\mathbb {W}_{{\lambda }}\subseteq \mathbb {W}_\mu$ when ${\lambda }\le \mu$, such that under the condition $\sum _{i=1}^d H_i>d-2$, the solution exists uniquely in $\mathbb {W}_{{\lambda }_0}$ when ${\lambda }_0$ is sufficiently large and the solution is not in $\mathbb {W}_{{\lambda }_1}$ when ${\lambda }_1$ is sufficiently small.