非齐次弦的精确和定则
Exact sum rules for inhomogeneous strings
摘要：
我们导出了具有任意密度和不同边界条件的非齐次弦的本征值求和规则的显式表达式。我们表明，为了$N$的总和规则可以得到的一个图形的扩展，与$(N-1)!/2$独立的图表。这些总和规则是用来推导出上界和下界的能量的基本模式的非均匀字符串，我们还表明，它是可能的，以改善这些近似考虑到渐近行为的频谱和应用的Shanks变换序列的近似获得不同的订单。我们讨论这些结果的三个应用。
英文摘要：
We derive explicit expressions for the sum rules of the eigenvalues of inhomogeneous strings with arbitrary density and with different boundary conditions. We show that the sum rule of order $N$ may be obtained in terms of a diagrammatic expansion, with $(N-1)!/2$ independent diagrams. These sum rules are used to derive upper and lower bounds to the energy of the fundamental mode of an inhomogeneous string; we also show that it is possible to improve these approximations taking into account the asymptotic behaviour of the spectrum and applying the Shanks transformation to the sequence of approximations obtained to the different orders. We discuss three applications of these results.
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