We prove a Feynman path integral formula for the unitary group $ \exp(-itL_{v,\theta})$, $t\geq 0$, associated with a discrete magnetic Schr\"odinger operator $L_{v,\theta}$ on a large class of weighted infinite graphs. As a consequence, we get a new Kato-Simon estimate $$ |\exp(-itL_{v,\theta})(x,y)|\leq \exp(-tL_{-\mathrm{deg},0})(x,y), $$ which controls the unitary group uniformly in the potentials in terms of a Schr\"odinger semigroup, where the potential $\mathrm{deg}$ is the weighted degree function of the graph.

中文翻译：

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无限加权图上磁性薛定谔算子的费曼路径积分

我们证明了幺正群$\exp(-itL_{v,\theta})$, $t\geq 0$ 的费曼路径积分公式，与离散磁Schr\"odinger 算子$L_{v,\theta }$ 在一大类加权无限图上。因此，我们得到一个新的 Kato-Simon 估计 $$ |\exp(-itL_{v,\theta})(x,y)|\leq \exp(- tL_{-\mathrm{deg},0})(x,y), $$ 以 Schr\"odinger 半群的形式均匀地控制酉群，其中势 $\mathrm{deg}$ 是图的加权度函数。