Asymptotic Approximations for Solutions of Sturm–Liouville Differential Equations with a Large Complex Parameter and Nonsmooth Coefficients

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.