论文标题
半经典分解边界，用于弱腐烂的电势
Semiclassical resolvent bounds for weakly decaying potentials
论文作者
论文摘要
在本说明中，我们证明了半经典的schrödinger操作员的加权分解估计值$ -h^2δ+ v（x）：l^2（\ mathbb {r}^n）\ to l^2（\ mathbb {r}^n）$，$ n \ neq 2 $。潜在的$ v $是真实价值的，并假定在无穷大腐烂或服从径向$α$-Hölder连续性条件，$ 0 \ leqleqα\ leq 1 $，并充分衰减了局部径向$ c^α$ norm for Infinity。但是，请注意，在Hölder情况下，潜力需要\ emph {not}衰减。如果尺寸$ n \ ge 3 $，则分解键是$ \ exp \ weft（c h^{ - 1 - \ frac {1-α} {3 +α}} {（1-α）\ log（h^{ - 1}） + c] + crirp）$ n = 1 $ n = 1 $。一种新型的重量和相位功能构建使我们即使在纯$ l^\ infty $ case中也可以减少必要的衰减。
In this note, we prove weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 Δ+ V(x) : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, $n \neq 2$. The potential $V$ is real-valued, and assumed to either decay at infinity or to obey a radial $α$-Hölder continuity condition, $0\leq α\leq 1$, with sufficient decay of the local radial $C^α$ norm toward infinity. Note, however, that in the Hölder case, the potential need \emph{not} decay. If the dimension $n \ge 3$, the resolvent bound is of the form $\exp \left(C h^{-1 - \frac{1 - α}{3 + α}} [(1-α) \log(h^{-1})+c]\right)$, while for $n = 1$ it is of the form $\exp(Ch^{-1})$. A new type of weight and phase function construction allows us to reduce the necessary decay even in the pure $L^\infty$ case.
