Spectral Properties of Sturm–Liouville Problems with Strongly Singular Potentials

Abstract

This paper studies a class of singular differential equations $$\begin{aligned} -\left( \frac{\mathrm {d}}{{\mathrm {d}}x}-\frac{k}{x^{l}}-v\right) \left( \frac{{\mathrm {d}}}{{\mathrm {d}}x}+\frac{k}{x^{l}}+v\right) y=\lambda y \text { on }J=(0,1) , \end{aligned}$$ where $$k\ge \frac{1}{2},1\le l<2$$ and $$v\in L^{1}(J, {\mathbb {R}} )$$ which is bounded below. Using the prufer transformation, we get the oscillation property of the eigenfunctions. In particular, the location of eigenvalues are also described. Furthermore, we establish the continuous dependence of nth eigenvalue on the boundary condition.