AI 中文总结
研究 d = 1,2 时带白噪声的随机薛定谔算子，利用费曼 - 卡茨公式和布朗桥局部时间估计，给出 Tr[e^(-sH)] 与 Tr[e^(-tH)] 协方差最优渐近界，改进 d = 1 时结果，首证 d = 2 情况，还证明了迹的相关性质。
AI 中文摘要
对于 d∈{1,2}，设 H = -1/2Δ + V + ξ 是 L²(R^d) 上的随机薛定谔算子，其中 ξ 是标准高斯白噪声，V 是在无穷远处具有幂律增长的确定性势。利用薛定谔半群迹的费曼 - 卡茨公式，通过对布朗桥局部时间的估计，给出了 Tr[e^(-sH)] 和 Tr[e^(-tH)] 协方差在 s,t→0 时的最优渐近上下界。这些估计在 d = 1 的情况下显著改进了先前的界，并在 d = 2 时为首创。作为新估计的应用，证明了迹在 s,t→0 时的定量超均匀性型性质和去相关率。
英文摘要
For $d\in\{1,2\}$, let $H=-\frac{1}{2}Δ+ V +ξ$ be the random Schrödinger operator on $L^2(\mathbb{R}^d)$ where $ξ$ 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 $\mathrm{Tr}[e^{-sH}]$ and $\mathrm{Tr}[e^{-tH}]$ as $s,t\to0$ 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 $s,t\to0$.
Comments51 pages
