Discreteness spectrum and asymptotic distribution of eigenvalues of operator differential equations of higher order on semi-axis

In the present paper asymptotic distribution of eigenvalues of operator-differential equations of higher order on a semi-axis is studied. The uniform asymptotics of the Green function for large values of spectral parameter is obtained. Using the asymptotics of the Green function, the discretness of the spectrum is proved. Using the Keldysh – Tauberian theorem, an asymptotic formula is obtained for the distribution function of eigenvalues of the considered operator