We give an elementary proof of a weighted resolvent estimate for semiclassical Schrödinger operators in dimension $n \ge 1$. We require the potential belong to $L^\infty(\mathbb{R}^n)$ and have compact support, but do not require that it have distributional derivatives in $L^\infty(\mathbb{R}^n)$. The weighted resolvent norm is bounded by $e^{Ch^{-4/3}\log(h^{-1})}$, where $h$ is the semiclassical parameter.

中文翻译：

---

半经典分解物必将紧紧支撑$ L ^ \ infty $势

我们给出维度为$ n \ ge 1 $的半经典Schrödinger算子的加权分解估计的初等证明。我们要求势能属于$ L ^ \ infty（\ mathbb {R} ^ n）$并具有紧凑支持，但不要求其在$ L ^ \ infty（\ mathbb {R} ^ n）中具有分布导数$。加权分解范数以$ e ^ {Ch ^ {-4/3} \ log（h ^ {-1}）} $为界，其中$ h $是半经典参数。