We consider, for h,E>0, resolvent estimates for the semiclassical Schrödinger operator −h2Δ+V−E. Near infinity, the potential takes the form V=VL+VS, where VL is a long range potential which is Lipschitz with respect to the radial variable, while VS=O(|x|−1(log|x|)−ρ) for some ρ>1. Near the origin,|V| may behave like |x|−β, provided 0⩽β<2(3−1). We find that, for any ρ˜>1, there are C,h0>0 such that we have a resolvent bound of the form exp(Ch−2(log(h−1))1+ρ˜) for all h∈(0,h0]. The h-dependence of the bound improves if VS decays at a faster rate toward infinity.

中文翻译：

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原点处具有奇点的短程 L∞ 势的半经典解析界

我们考虑，对于 h,E>0，半经典薛定谔算子 −h2Δ+V−E 的解析估计。接近无穷大时，势的形式为 V=VL+VS，其中 VL 是长程势，相对于径向变量为 Lipschitz，而 VS=O(|x|−1(log|x|)−ρ)对于某些 ρ>1。接近原点，|V| 其行为可能类似于 |x|−β，前提是 0⩽β<2(3−1)。我们发现，对于任何 ρ~>1，都存在 C,h0>0，因此对于所有 hε，我们有一个形式为 exp(Ch−2(log(h−1))1+ρ~) 的解析界限(0,h0]。如果 VS 以更快的速率向无穷大衰减，则束缚的 h 依赖性会提高。