Spectral asymptotics for fourth order differential operator with two turning points

The paper is devoted to studying the asymptotics of the spectrum of a self-adjoint operator $T$ generated by a fourth-order differential expression in the space $L^{2} (0,+\infty)$ under the assumption that the coefficients of the expression have a power growth at infinity such that: a) the deficiency index of the corresponding minimal operator is $(2,2)$, b) for sufficiently large positive values of a spectral parameter, the differential equation $ Ty = \lambda y $ has two turning points: a finite one and $+\infty$, c) the roots of the characteristic equation grow not with the same rate. The latter assumption leads one to significant difficulties in studying the asymptotics of the counting function for the spectrum by the traditional Carleman–Kostyuchenko method based on estimates of the resolvent far from the spectrum and Tauberian theorems. Curiously enough, the method of reference equations used to solve the more subtle problem of finding asymptotic expansions of the eigenvalues themselves, and therefore more sensitive (compared to the Carleman–Kostyuchenko method) to the behavior of the coefficients in the differential expression is more effective in the considered situation: imposing some constraints on coefficients such as smoothness and regular growth at infinity, we obtain an asymptotic equation for the spectrum of the operator $T$. This equation allows one to write out the first few terms of the asymptotic expansion for the eigenvalues of the operator $T$ in the case when the coefficients have a power growth. We also note that so far the method of reference equations has been used only in the case of the presence of the only turning point.