Spectral properties of some differential operators of Sturm-Liouville type with homogeneous delay

In this paper we observe the operator $ D^{2}=D^{2}insert ignore into journalissuearticles values(h,H,q,\alpha);$, $h, H\in \overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,\infty\}$, $qinsert ignore into journalissuearticles values(x);\in L_{2}\left[0,\pi \right]$, $\alpha\in insert ignore into journalissuearticles values(0,1); $ and construct and partially transform its characteristic function. Those transformations enable more complete asymptotic decomposition of the zeroes and eigenvalues of the operator. The goal of this paper is to contribute to the development of the spectral theory of differential operators with homogeneous delay.