This is the third paper in a series analyzing the asymptotic distribution of the phase shifts in the semiclassical limit. We analyze the distribution of phase shifts, or equivalently, eigenvalues of the scattering matrix, $S_h(E)$, for semiclassical Schr\"odinger operators on $\mathbb{R}^d$ which are perturbations of the free Hamiltonian by a potential $V$ with polynomial decay. Our assumption is that $V(x) \sim |x|^{-\alpha} v(\hat x)$ as $x \to \infty$, for some $\alpha > d$, with corresponding derivative estimates. In the semiclassical limit $h \to 0$, we show that the atomic measure on the unit circle defined by these eigenvalues, after suitable scaling in $h$, tends to a measure $\mu$ on $\mathbb{S}^1$. Moreover, $\mu$ is the pushforward from $\mathbb{R}$ to $\mathbb{R} / 2 \pi \mathbb{Z} = \mathbb{S}^1$ of a homogeneous distribution $\nu$ of order $\beta$ depending on the dimension $d$ and the rate of decay $\alpha$ of the potential function. As a corollary we obtain an asymptotic formula for the accumulation of phase shifts in a sector of $\mathbb{S}^1$. The proof relies on an extension of results of the second author and Wunsch on the classical Hamiltonian dynamics and semiclassical Poisson operator to the class of potentials under consideration here.

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具有多项式衰减的半经典势的相移分布

这是分析半经典极限中相移渐近分布的系列论文中的第三篇。我们分析了相移的分布，或等效地，散射矩阵 $S_h(E)$ 的特征值，对于 $\mathbb{R}^d$ 上的半经典 Schr\"odinger 算子，它们是自由哈密顿量的扰动具有多项式衰减的潜在 $V$。我们的假设是 $V(x) \sim |x|^{-\alpha} v(\hat x)$ 作为 $x \to \infty$，对于某些 $\alpha > d$，以及相应的导数估计。在半经典极限 $h \to 0$ 中，我们表明由这些特征值定义的单位圆上的原子测度，在 $h$ 中适当缩放后，趋于测度 $\mu$在 $\mathbb{S}^1$ 上。此外，$\mu$ 是从 $\mathbb{R}$ 到 $\mathbb{R} / 2 \pi \mathbb{Z} = \mathbb{S}^1$ 的有序分布 $\nu$ 的推进$\beta$ 取决于维度 $d$ 和势函数的衰减率 $\alpha$。作为推论，我们获得了 $\mathbb{S}^1$ 扇区中相移累积的渐近公式。证明依赖于将第二作者和 Wunsch 关于经典哈密顿动力学和半经典泊松算子的结果扩展到这里考虑的势类。