Asymptotics of the Weyl-Titchmarsh m-function (Q1112232)

We consider the Sturm-Liouville problem \(-y''+q(x)y=\lambda y\), \(0\leq x<\infty\), \(y'(0)=0\), m(z) is the Weyl-Titchmarsh function of this problem. Under the assumption that \(q(x)\in C^ n[0,\delta)\) for some \(\delta >0\) it is proved by the wave equation method that as \(| z| \to \infty\) outside of any angle \(0<\epsilon <\arg z<\pi - \epsilon\) the following asymptotic expansion holds: \[ m(z)=\frac{i}{\sqrt{z}}+\sum^{n}_{k=1}\beta_{k^ 2}^{- (k+2)/2}+O(\frac{1}{| z|^{(n+3)/2}}), \] in which the coefficients \(\beta_ k\) depend only on \(q(0),q'(0),...,q^{(n)}(0)\).