带乘性高斯噪声的一维薛定谔算子半群
Semigroups for One-Dimensional Schrödinger Operators with Multiplicative Gaussian Noise
摘要：
设$ H:=-\tfrac12Δ+V$是一维连续薛定谔算子。考虑${\hat H}:= H+ξ$，其中$ξ$是平移不变高斯噪声。在对$ξ$的一些假设下，证明了若$V$是局部可积的，下有界的，且在无穷远处比$\log$增长得快，则半群$\mathrm e^{-t {\hat H}}$是迹类，且允许通过Feynman-Kac公式的概率表示.我们的结果适用于运营商的全线$\mathbb R$，半线$(0,\infty)$，或有界区间$(0,b)$，各种边界条件。我们的证明方法包括一个全面的推广技术最近开发的随机矩阵理论文献中解决这个问题的特殊情况下，${\hat H}$是随机艾里算子。
英文摘要：
Let $ H:=-\tfrac12Δ+V$ be a one-dimensional continuum Schrödinger operator. Consider ${\hat H}:= H+ξ$, where $ξ$ is a translation invariant Gaussian noise. Under some assumptions on $ξ$, we prove that if $V$ is locally integrable, bounded below, and grows faster than $\log$ at infinity, then the semigroup $\mathrm e^{-t {\hat H}}$ is trace class and admits a probabilistic representation via a Feynman-Kac formula. Our result applies to operators acting on the whole line $\mathbb R$, the half line $(0,\infty)$, or a bounded interval $(0,b)$, with a variety of boundary conditions. Our method of proof consists of a comprehensive generalization of techniques recently developed in the random matrix theory literature to tackle this problem in the special case where ${\hat H}$ is the stochastic Airy operator.
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