Geodesic
Asymptotics for the eigenvalues of a~fourth order differential operator in a~``degenerate'' case
Ufa mathematical journal, Tome 8 (2016) no. 3, pp. 79-94

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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 $L$.
Keywords: differential operators, asymptotics of spectrum, turning point. 
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     title = {Asymptotics for the eigenvalues of a~fourth order differential operator in a~``degenerate'' case},
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Kh. K. Ishkin; Kh. Kh. Murtazin. Asymptotics for the eigenvalues of a~fourth order differential operator in a~``degenerate'' case. Ufa mathematical journal, Tome 8 (2016) no. 3, pp. 79-94. http://geodesic.mathdoc.fr/item/UFA_2016_8_3_a7/