论文标题
远程Lipschitz电位的半经典分解边界
Semiclassical resolvent bounds for long range Lipschitz potentials
论文作者
论文摘要
我们给出了半经典schrödinger操作员加权分解估计的基本证明，$ -h^2δ+ v（x） - e $ in Dimension $ n \ neq 2 $，其中$ h，\，e> 0 $。潜力是真实价值的，$ v $和$ \ partial_r v $在Infinity处表现出远距离衰减，并且可能像$ r $ as $ r \至0 $的足够小的负功率一样生长。分解标准在$ h^{ - 1} $中呈指数增长，但在无穷大近乎生长。当$ v $紧凑地支持$ V $时，如果将分辨率乘以在半径$ ce^{ - 1/2} $外部支撑的权重乘以$ C> 0 $的情况下，我们将获得线性增长。这个$ e $依赖性很清晰，并回答了Datchev和Jin的问题。
We give an elementary proof of weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 Δ+ V(x) - E$ in dimension $n \neq 2$, where $h, \, E > 0$. The potential is real-valued, $V$ 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.
