Abstract

This article deals with fundamental solutions of the Sturm–Liouville differential equations \(\left( P\left( x\right) y^{\prime }\right) ^{\prime }+\left( R\left( x\right) -\xi ^{2}Q\left( x\right) \right) y=0\) with a complex parameter \(\xi \), \(\left\vert \xi \right\vert \gg 1\), whose coefficients \(P\left( x\right) \) and \(Q\left( x\right) \) are positive piecewise continuous functions while \(\left( P\left( x\right) Q\left( x\right) \right) ^{\prime }\) and \(R\left( x\right) \) belong to \(L^{p}\), \(p>1\). Two methods are suggested to reduce the problem to special Volterra or Volterra–Hammerstein integral equations with “small” integral operators, which can be treated by iterations that constitute asymptotic scales or by direct numerical methods. Asymptotic analysis of the iterations permits constructing uniform asymptotic approximations for fundamental solutions and their derivatives. A uniform with respect to integration limits analog of Watson’s lemma for finite range Laplace integrals is proved in this connection. When the coefficients belong to \(C^{N}\) on intervals of continuity, a new simple direct method is suggested in order to derive explicit uniform asymptotic approximations of fundamental solutions as \(\operatorname{Re} \left( \xi \right) \gg 1\) by a certain recurrent procedure. This method is expected to be more efficient than the WKB method. Theorems estimating errors of such approximations are given. Efficacy of the suggested uniform approximations is demonstrated by computing approximations for a combination of Bessel functions of large indices and arguments.

中文翻译：

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具有大复参数和非光滑系数的Sturm-Liouville微分方程解的渐近逼近

摘要

本文讨论Sturm–Liouville微分方程\（\ left（P \ left（x \ right）y ^ {\ prime} \ right）^ {\ prime} + \ left（R \ left（x \ right）-\ xi ^ {2} Q \ left（x \ right）\ right）y = 0 \）具有复杂参数\（\ xi \），\（\ left \ vert \ xi \ right \ vert \ gg 1 \），其系数\（P \ left（x \ right）\）和\（Q \ left（x \ right）\）是正分段连续函数，而\（\ left（P \ left（x \ right） Q \ left（x \ right）\ right）^ {\ prime} \）和\（R \ left（x \ right）\）属于\（L ^ {p} \），\（p> 1 \）。建议使用两种方法将问题简化为带有“小”积分算子的特殊Volterra或Volterra-Hammerstein积分方程，可以通过构成渐近标度的迭代或直接数值方法来处理。迭代的渐近分析允许构造基本解及其导数的一致渐近逼近。在这方面证明了关于有限范围拉普拉斯积分的沃森引理的积分极限模拟的统一形式。当系数在连续区间上属于\（C ^ {N} \）时，建议一种新的简单直接方法，以便导出基本解的显式统一渐近近似为\（\ operatorname {Re} \ left（\ xi \ right）\ gg 1 \）通过一定的循环程序。期望该方法比WKB方法更有效。给出了估计这种近似误差的定理。通过计算大索引和自变量的Bessel函数组合的近似值，可以证明所建议的统一近似值的有效性。