We study the uniform weighted resolvent estimates of Schrödinger operator with scaling-critical electromagnetic potentials which, in particular, include the Aharonov-Bohm magnetic potential and inverse-square potential. The potentials are critical due to the scaling invariance of the model and the singularities of the potentials, which are not locally integrable. In contrast to the Laplacian −Δ on ${\mathbb{R}}^2$, we prove some new uniform weighted resolvent estimates for this 2D Schrödinger operator and, as applications, we show local smoothing estimates for the Schrödinger equation in this setting.

中文翻译：

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在Aharonov-Bohm磁场中对Schrödinger算符的统一分辨估计

我们研究具有定标临界电磁势的Schrödinger算子的统一加权分解溶剂估计，其中特别是包括Aharonov-Bohm磁势和平方反比电势。由于模型的比例不变性和电位的奇异性（局部不可积分），因此电位至关重要。与拉普拉斯算子上的-Δ相反${\mathbb{[R}}^{\mathrm{2个}}$，我们证明了该二维Schrödinger算子的一些新的统一加权分解量估计，并且在应用中，我们显示了此设置下Schrödinger方程的局部平滑估计。