English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Let $ H:=-\tfrac12\Delta+V$ be a one-dimensional continuum Schr\"odingeroperator. Consider ${\hat H}:= H+\xi$, where $\xi$ is a translation invariantGaussian noise. Under some assumptions on $\xi$, we prove that if $V$ islocally integrable, bounded below, and grows faster than $\log$ at infinity,then the semigroup $\mathrm e^{-t {\hat H}}$ is trace class and admits aprobabilistic representation via a Feynman-Kac formula. Our result applies tooperators acting on the whole line $\mathbb R$, the half line $(0,\infty)$, ora bounded interval $(0,b)$, with a variety of boundary conditions. Our methodof proof consists of a comprehensive generalization of techniques recentlydeveloped in the random matrix theory literature to tackle this problem in thespecial case where ${\hat H}$ is the stochastic Airy operator.

Semigroups for One-Dimensional Schr\"odinger Operators with Multiplicative Gaussian Noise