Higher order asymptotic distribution of the eigenvalues of nondefinite Sturm–Liouville problems with one turning point

In this paper we derive the higher order asymptotic distribution of the positive eigenvalues associated with a linear real second order equationy″+(λxα−q(x))y=0,of Sturm–Liouville type on [a,b] with Dirichlet boundary condition (i.e., y(a)=y(b)=0), where −∞<a<0<b<∞, q is a real-valued sign-indefinite member of C1[a,b], λ is a real parameter and α>−1 is chosen so that the boundary problem is non-definite.