Asymptotic formulas for $n$-diameters of certain compacta in $L_2[0,1]$

In the current article the order of the Kolmogorov $n$-diameters of compacta, determined by the operators
$$Ly=p(x)\frac{dy}{dx}+q(x)y,\quad Ly=\Bigl[-\frac{d^2}{dx^2}+q(x)\frac d{dx}\Bigr]^ry$$
in $L_2[0,1]$ with a bound on the order of the error is studied and asymptotic formulas for $d_n$ as a function of $p(x)$, $g(x)$ and $r$ are derived.