某些转向点为复值时特征值和的渐近性
Asymptotics of eigenvalue sums when some turning points are complex
摘要：
最近的工作表明，在密度泛函理论中的半局部近似和本征值的WKB半经典展开之和的渐近性之间有着深刻的联系。然而，迄今为止研究的所有例子都有潜力，只有真正的经典转折点。但是，具有复杂转折点的系统产生的次优势项超出了WKB级数。最简单的情况是纯四次振子。我们展示了如何推广的渐近特征值和，包括次显性贡献的总和，如果他们是已知的特征值。对WKB的这些修正极大地提高了特征值和的精度，特别是对于许多水平。我们通过超渐近性得到了进一步的改进。对于最低水平，我们的求和方法具有低于$2 \times 10^{-4}$的误差。对于最低10个级别的总和，我们的误差小于$10^{-22}$。我们报告的所有结果，以许多数字，包括丰富的细节的渐近展开及其推导。
英文摘要：
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.
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