Spectral asymptotics of solutions of a \(2\times 2\) system of first-order ordinary differential equations (Q2063691)

The authors study the completeness property of the root functions of the spectral problem \begin{align*} & {{\left( \begin{matrix} {{y}_{1}} \\ {{y}_{2}} \\ \end{matrix} \right)}^{\prime }}-\left( \begin{matrix} p & q \\ r & s \\ \end{matrix} \right)\left( \begin{matrix} {{y}_{1}} \\ {{y}_{2}} \\ \end{matrix} \right)=\lambda \left( \begin{matrix} g & 0 \\ 0 & -h \\ \end{matrix} \right)\left( \begin{matrix} {{y}_{1}} \\ {{y}_{2}} \\ \end{matrix} \right),\,\,\,\,\,\,\,\,\,x\in [0,1], \\ & {{y}_{1}}\left( 0 \right)=0,\,\,\,\,\,\,\,{{y}_{2}}\left( 1 \right)=0. \\ \end{align*} Here $\lambda$ is a spectral parameter, both functions g and h are positive, and the functions $p, q, r$, and s are complex-valued. It is only assumed that all these functions are integrable on the segment $[0, 1]$.