Results in Mathematics

, 74:20 | Cite as

Spectral Properties of Sturm–Liouville Problems with Strongly Singular Potentials

  • Yu Liu
  • Guoliang Shi
  • Jun YanEmail author


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 prüfer 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.


Singular boundary value problem oscillation prüfer angle continuous dependence 

Mathematics Subject Classification

Primary 34B09 34C10 Secondary 34L40 34B30 



This research was supported by the National Natural Science Foundation of China under Grant No. 11601372.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of MathematicsTianjin UniversityTianjinPeople’s Republic of China

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