Asymptotics of fundamental solutions to Sturm--Liouville problem with respect to spectral parameter

V. E. Vladykina

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2019), pp. 57-61

Résumé

We consider the Sturm–Liouville equation $$-(r^2y')'+py'+qy=\lambda^2\rho^2 y,\qquad x\in[a,b]\subset\mathbb{R},$$ where $\lambda^2$ is a spectral parameter, $r$ and $\rho$ are positive functions while $p$ and $q$ are complex-valued ones. An asymptotic representation for the fundamental system of solutions with respect to the spectral parameter $\lambda\to\infty$ is obtained in the half-planes $\operatorname{Im}\lambda\geqslant\operatorname{const}$ and $\operatorname{Im}\lambda\leqslant\operatorname{const}$ under the following conditions on the coefficients: $$p\in L_1[a,b],\quad q\in W_2^{-1}[a,b],\quad\rho,r\in W_1^1[a,b],\quad\rho'u,r'u,pu\in L_1[a,b], \quad\text{where}\quad u=\int q~dx,$$ and the antiderivative is understood in the sense of distributions.