论文标题
当某些转折点复杂时，特征值总和的渐近学总和
Asymptotics of eigenvalue sums when some turning points are complex
论文作者
论文摘要
最近的工作表明，密度功能理论中的半近似值与特征值WKB半经典膨胀之和的渐近学之间有着深厚的联系。但是，迄今为止研究的所有示例都有潜力，只有实际的经典转折点。但是，具有复杂转折点的系统产生的次要术语超出了WKB系列的术语。最简单的情况是纯正的四分之一振荡器。我们展示了如何概括特征值总和的渐近学，以包括对总和的亚义贡献，如果它们以特征值而闻名。这些校正以极大地提高了特征值总和的准确性，尤其是在许多层面上。我们通过高空函数来进一步改善总和。对于最低级别，我们的求和方法的错误低于$ 2 \ times 10^{ - 4} $。对于最低10个级别的总和，我们的错误小于$ 10^{-22} $。我们将所有结果报告给许多数字，并包括渐近扩张及其推导的大量细节。
Recent work has shown a deep connection between semilocal approximations in density functional theory and the asymptotics of the sum of the WKB semiclassical expansion for the eigenvalues. However, all examples studied to date have potentials with only real classical turning points. But systems with complex turning points generate subdominant terms beyond those in the WKB series. The simplest case is a pure quartic oscillator. We show how to generalize the asymptotics of eigenvalue sums to include subdominant contributions to the sums, if they are known for the eigenvalues. These corrections to WKB greatly improve accuracy for eigenvalue sums, especially for many levels. We obtain further improvements to the sums through hyperasymptotics. For the lowest level, our summation method has error below $2 \times 10^{-4}$. For the sum of the lowest 10 levels, our error is less than $10^{-22}$. We report all results to many digits and include copious details of the asymptotic expansions and their derivation.
