Lower bounds for self-adjoint Sturm–Liouville operators

Abstract

In this note we provide estimates for the lower bound of the self-adjoint operator associated with the three-coefficient Sturm–Liouville differential expression 1 r ( − d d x p d d x + q ) \begin{equation*} \frac {1}{r} \left (-\frac {\mathrm d}{\mathrm dx} p \frac {\mathrm d}{\mathrm dx} + q\right ) \end{equation*} in the weighted L 2 L^2 -Hilbert space L 2 ( R ; r d x ) L^2(\mathbb R; rdx) .