Inverse Sturm-Liouville problems with homogeneous delay

We study the inverse Sturm-Liouville problem with a delay \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- y''(x) + q(x)y(\alpha x), q \in L_2 [0,\pi ], \alpha \in (0,1],$$\end{document} and the boundary conditions y(0) = y(π) = 0.