Higher-order asymptotic formula for the eigenvalues of Sturm–Liouville problem with n turning points

In this paper, we investigate the asymptotic behavior of the differential equationy″+(λr(x)−q(x))y=0,0⩽x⩽1, where [0,1][0,1] contains a finite number of zeros of r(x)r(x), the so-called turning points, λ   is a real parameter and the function q(x)q(x) is bounded and integrable in [0,1][0,1]. Using a technique used previously in [B.J. Harris, S.T. Talarico, On the eigenvalues of second-order linear differential equations with fractional transition points, Math. Proc. R. Ir. Acad. Ser. A 99 (1) (1999) 29–38], we derive the higher-order asymptotic distribution of the positive eigenvalues associated with this equation for the Dirichlet problem (i.e., y(0)=y(1)=0y(0)=y(1)=0). This method is different from Eberhard's method [W. Eberhard, G. Freiling, The distribution of the eigenvalues for second-order eigenvalues problems in the presence of an arbitrary number of turning points, Results Math. 21 (1992) 24–41]. They have used asymptotic solution in order to obtain asymptotic distribution of the eigenvalues. In most differential equations with variable coefficient it is impossible to obtain an exact solution, so we want to obtain asymptotic distribution of the eigenvalues without solving equation. Note that in similar case, the leading term of the asymptotic distribution of positive and negative eigenvalues was derived previously by Atkinson and Mingarelli [F.V. Atkinson, A. Mingarelli, Asymptotics of the number of zeros and the eigenvalues of general weighted Sturm–Liouville problems, J. Reine Angew. Math. 395 (1986) 380–93].