Conditions of quasi-regularity of a system of differential equations of second order with oscillating coefficients (Q1909814)

The author studies a singular system of second-order differential equations \[ \begin{aligned} (p_0(x) y_2')'& + (p(x)- \lambda) y_1+ q(x) y_2= 0,\\ (p_0(x) y_1')' & + q(x) y_1+ (r(x)- \lambda) y_2= 0,\end{aligned} \] where \(p_0(x)< 0\), \(p(x)\), \(q(x)\), \(r(x)\) are real-valued functions, \((p_0(x))^{- 1}\), \(p(x)\), \(q(x)\) and \(r(x)\) are Lebesgue integrable on \([0, b]\), \(0< b< \infty\), and \(\lambda\) is a complex parameter. Conditions are given on the coefficients \(p_0\), \(p\), \(q\), \(r\), providing that the operator \(\mathcal L\) has defect number equal to 4, where \(\mathcal L\) stands for the minimal closed symmetric operator generated by the expression \(l[y]= (P_0(x) y')'+ P_1(x)y\) with \[ P_0(x)= \begin{pmatrix} 0 & p_0(x)\\ p_0(x) & 0\end{pmatrix},\quad P_1(x)= \begin{pmatrix} p(x) & q(x)\\ q(x) & r(x)\end{pmatrix} \] in \(L_2[0, \infty)\).