Eigenvalues of Sturm-Liouville Operators with Distributional Potentials

• DOI: 10.48550/arxiv.1711.07032
• Publication Date: 2017-01-01
• Authors: Yan, Jun; Shi, Guoliang; Zhao, Jia
• Keywords: Mathematics - Spectral Theory; FOS: Mathematics; Primary 34B24, Secondary 34L15, 34L05, 34L10, 34C10; 0101 mathematics; 01 natural sciences; Spectral Theory (math.SP)

We introduce a novel approach for dealing with eigenvalue problems of Sturm-Liouville operators generated by the differential expression \begin{equation*} Ly=\frac{1}{r}\left( -(p\left[ y^{\prime }+sy\right] )^{\prime }+sp\left[ y^{\prime }+sy\right] +qy\right) \end{equation*} which is based on norm resolvent convergence of classical Sturm-Liouville operators. This enables us to describe the continuous dependence of the $n$-th eigenvalue on the space of self-adjoint boundary conditions and the coefficients of the differential equation after giving the inequalities among the eigenvalues. Moreover, oscillation properties of the eigenfunctions are also characterized. In particular, our main results can be applied to solve a class of Sturm-Liouville problems with transmission conditions.