Spectral parameter asymptotics of the Weil solutions of Sturm-Liouville equations

V. A. Marchenko

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 17, Tome 170 (1989), pp. 184-206

Résumé

In the article the dependence with respect to $\lambda$ of the Weil solution $\psi(\lambda,x)=c(\lambda,x)+n(\lambda)s(\lambda,x)$ of the Sturm–Liouville equation $-y''+q(x)y=\lambda^2y$ is investigated. For a semi-bounded $q$ such that $q(x)\leqslant\exp(c_0+c_1|x|)$ it is proved that $\lim\limits_{\substack{|\lambda|\to\infty\\ |\mathop{\mathrm{Im}}\lambda|\geqslant\varepsilon}}(\sup\limits_{|x|\leqslant A}|e^{-i\lambda x}\psi(\lambda,x)-1|)=0$ for any positive $\varepsilon$ and $A$.