We prove semiclassical resolvent estimates for the Schrödinger operator in $${\mathbb {R}}^d$$ R d , $$d\ge 3$$ d ≥ 3 , with real-valued radial potentials $$V\in L^\infty ({\mathbb {R}}^d)$$ V ∈ L ∞ ( R d ) . In particular, we show that if $$V(x)={{\mathcal {O}}}\left( \langle x\rangle ^{-\delta }\right) $$ V ( x ) = O ⟨ x ⟩ - δ with $$\delta >2$$ δ > 2 , then the resolvent bound is of the form $$\exp \left( Ch^{-4/3}\right) $$ exp C h - 4 / 3 with some constant $$C>0$$ C > 0 . We also get resolvent bounds when $$1<\delta \le 2$$ 1 < δ ≤ 2 . For slowly decaying $$\alpha $$ α —Hölder potentials, we get better resolvent bounds of the form $$\exp \left( Ch^{-4/(\alpha +3)}\right) $$ exp C h - 4 / ( α + 3 ) .

中文翻译：

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改进了径向电位的解析度界限

我们证明了 $${\mathbb {R}}^d$$ R d , $$d\ge 3$$ d ≥ 3 中 Schrödinger 算子的半经典解析解估计，具有实值径向势 $$V\in L ^\infty ({\mathbb {R}}^d)$$ V ∈ L ∞ ( R d ) 。特别地，我们证明如果 $$V(x)={{\mathcal {O}}}\left( \langle x\rangle ^{-\delta }\right) $$ V ( x ) = O ⟨ x ⟩ - δ 且 $$\delta >2$$ δ > 2 ，则解析器边界的形式为 $$\exp \left( Ch^{-4/3}\right) $$ exp Ch - 4 / 3 有一些常数 $$C>0$$ C > 0 。当 $$1<\delta \le 2$$ 1 < δ ≤ 2 时，我们也得到解析器边界。对于缓慢衰减的 $$\alpha $$ α -Hölder 势，我们得到更好的解析边界，形式为 $$\exp \left( Ch^{-4/(\alpha +3)}\right) $$ exp C h - 4 / (α + 3)。