Asymptotics for the eigenvalues of a fourth order differential operator in a “degenerate” case

In the paper we consider operator $L$ in $L^2[0,+\infty)$ generated by the differential expression $\mathcal L(y)=y^{(4)}-2(p(x)y')'+q(x)y$ and boundary conditions $y(0)=y''(0)=0$ in the “degenerate” case, when the roots of associated characteristic equation has different growth rate at the infinity. Assuming a power growth for functions $p$ and $q$ under some additional conditions of smoothness and regularity kind, we obtain an asymptotic equation for the spectrum allowing us to write out several first terms in the asymptotic expansion for the eigenvalues of operator $L$.