Spectral properties of a finite system of Sturm-Liouville differential operators (Q1431066)

Consider the finite system of Sturm-Liouville differential expressions \[ l_ny := -y'' + q_n (x)y_,\quad n=1,\dots, N,\;x\in \mathbb{R}_+, \] where \(q_n\) are complex-valued functions. Let \( L\) denote the nonselfadjoint operator generated in \(L^2(\mathbb{R}_+, \mathbb{C}^N)\) by \[ Ly:= (l_1(y_1), \dots, l_N (y_N))^T, \] with the boundary conditions \(y_n(0)=0\). The authors investigate the eigenvalues and the spectral singularities of the operator \(L\). In particular, they prove that \(L\) has a finite number of eigenvalues and spectral singularities with finite multiplicities under the condition \[ \sup_{x\in \mathbb{R}_+} \{ \exp(\varepsilon \sqrt{x}) | q_n(x)| \} <\infty, \;n=1,\dots,N,\;\varepsilon >0. \]