Optimal Covariance Estimates for Schrödinger Semigroups with White Noise in $d=1,2$

For $d\in \{1,2\}$, let $H=-\frac{1}{2}\mathrm{\Delta }+V+\xi$ be the random Schrödinger operator on $L^2({\mathbb{R}}^d)$ where $\xi$ is a standard Gaussian white noise and $V$ is a deterministic potential with power-law growth at infinity. Using a Feynman-Kac formula for the trace of the Schrödinger semigroup, we give optimal asymptotic upper and lower bounds on the covariance of $\mathrm{T}\mathrm{r}[e^{-sH}]$ and $\mathrm{T}\mathrm{r}[e^{-tH}]$ as $s,t\to 0$ through estimates on Brownian bridge local times. These estimates are a significant improvement on previous bounds in the case $d=1$ and are the first of their kind for $d=2$. As an application of these new estimates, we prove a quantitative hyperuniformity-type property and decorrelation rate for the trace as $s,t\to 0$.