On one of the methods for calculating the regularized trace of the Dirac operator with peculiarity in the potential (Q2772126)

The boundary value problem for the system of ordinary differential equation of the form NEWLINE\[NEWLINE\begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}\begin{pmatrix} y'_1\\ y'_2\end{pmatrix}+ \begin{pmatrix} p(x)&-\frac 1x+q(x)\\ \frac 1x+q(x)&-p(x)\end{pmatrix} \begin{pmatrix} y_1\\ y_2\end{pmatrix}=\lambda\begin{pmatrix} y_1\\ y_2 \end{pmatrix}, \qquad 0<x<\pi, NEWLINE\]NEWLINE with \(p,q\in C^2[0,\pi]\) and the following boundary value conditions \(y_2(0)= y_2(\pi)=0\) is considered. The main result is: Let \(\{\lambda_n\}_{n=-\infty}^\infty\) be eigenvalues to the above problem. Then \(\lambda_0+\sum_{n=1}^\infty (\lambda_n+\lambda_{-n})= 1/2(p(0)+p(\pi))\).