We give an elementary proof of weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 \Delta + V(x) - E$ in dimension $n \neq 2$, where $h, \, E> 0$. The potential is real valued and $V$ and $\partial _r V$ exhibit long-range decay at infinity and may grow like a sufficiently small negative power of $r$ as $r \to 0$. The resolvent norm grows exponentially in $h^{-1}$, but near infinity it grows linearly. When $V$ is compactly supported, we obtain linear growth if the resolvent is multiplied by weights supported outside a ball of radius $CE^{-1/2}$ for some $C> 0$. This $E$-dependence is sharp and answers a question of Datchev and Jin.

中文翻译：

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长程 Lipschitz 势的半经典解析界

我们给出了半经典薛定谔算子 $-h^2 \Delta + V(x) - E$ 在维度 $n \neq 2$ 的加权分解估计的基本证明，其中 $h, \, E>; 0 美元。势是实值的，$V$ 和 $\partial_r V$ 在无穷远处表现出长程衰减，并且可能会像 $r$ 的足够小的负幂一样增长，因为 $r \to 0$。解析范数在 $h^{-1}$ 中呈指数增长，但在接近无穷大时呈线性增长。当 $V$ 被紧支撑时，如果将解析器乘以在半径为 $CE^{-1/2}$ 的球外支持的权重，对于某些 $C>，我们将获得线性增长。0 美元。这种$E$-依赖是尖锐的并且回答了Datchev 和Jin 的问题。