On a numerical method for trace formulas for binomial ordinary operators of high order (Q2772114)

The boundary value problem for high-order ordinary differential equations of the form NEWLINE\[NEWLINE (-1)^my^{(2m)}+q(x)y= \lambda y,\quad 0<x<\pi, NEWLINE\]NEWLINE with \(m\in\mathbb{N}\) and \(q\in C^2[0,\pi]\) with the following boundary value conditions \(y(0)=\cdots=y^{(2m-2)}(0)=0\) and \(y(\pi)=\cdots= y^{(2m-2)}(\pi)=0\) is considered. It is assumed that \(\int_0^\pi q(x) dx=0\). The main result based on Lax's method [\textit{P. D. Lax}, Commun. Pure Appl. Math. 47, No. 4, 503-512 (1994; Zbl 0802.34089)] is: Let \(\{\lambda_n\}_{n=1}^\infty\) be eigenvalues to the above problem then \(\sum_{n=1}^\infty (\lambda_n-n^{2m})=-1/4 (q(0)+q(\pi))\).