The inverse problem for a class of ordinary differential operators with periodic coefficients

The direct and inverse problem of spectral analyses of a class of ordinary differential equations of order $2m$ with coefficients polynomially depending on the spectral parameter are investigated. It is shown that, the spectrum of the operator pencil is continuous, fill in the rays $\{k\omega_j/\, 0\le k<\infty,\ j=\overline{0,2m-1}\}$, $\omega_j=\exp\left(\frac{ij\pi}{m}\right)$, and there exist spectral singularities on the continues spectrum which coincide with the numbers $\frac{n\omega_j}2$, $j=\overline{0,2m-1}$, $n=1,2,\dots$ The inverse problem of reconstructing of the coefficients by generalized normalizing numbers is solved.