Eigenvalues of Sturm-Liouville Operators with Distributional Potentials

TL;DR

This paper develops a new method for analyzing eigenvalues of Sturm-Liouville operators with distributional potentials, focusing on their dependence on boundary conditions and coefficients, and characterizes eigenfunction oscillations.

Contribution

Introduces a novel approach using norm resolvent convergence to study eigenvalues of Sturm-Liouville operators with distributional potentials, including dependence and oscillation properties.

Findings

Eigenvalues depend continuously on boundary conditions and coefficients.

Inequalities among eigenvalues are established.

Oscillation properties of eigenfunctions are characterized.

Abstract

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.