New trace formula for the matrix Sturm-Liouville equation with eigenparameter dependent boundary conditions (Q2839029)

Let the function \(Q:(0,\pi)\to \mathbb{C}^{d\times d}\) be continuously differentiable on \((0,\pi)\). Consider the Sturm-Liouville spectral problem on a finite interval \((0,\pi)\) NEWLINE\[NEWLINE -Y''(x)+Q(x)Y(x)=\lambda\, Y(x),\quad x\in (0,\pi) NEWLINE\]NEWLINE subject to separated boundary conditions depending on a spectral parameter NEWLINE\[NEWLINE \lambda(Y'(0)-LY(0))=L_1Y'(0)-L_2Y(0),\quad \lambda(Y'(\pi)-HY(\pi))=H_1Y'(\pi)-H_2Y(\pi). NEWLINE\]NEWLINE The matrices \(L\), \(L_1\), \(L_2\) and \(H\), \(H_1\), \(H_2\) are assumed to satisfy the following conditions NEWLINE\[NEWLINE L_1L_2=L_2L_1,\quad L_1L=LL_1,\quad \text{and}\quad H_1H_2=H_2H_1,\quad H_1H=HH_1. NEWLINE\]NEWLINE The main result of the paper under review is the trace formula for the above mentioned spectral problem.