Explicit sharp bounds for all nodes of Sturm-Liouville operators with potentials in $L^1$ balls

TL;DR

This paper derives explicit sharp bounds for all nodes of eigenfunctions of Sturm-Liouville operators with potentials in $L^1$ balls, using optimization and nonlinear functional analysis.

Contribution

It provides the first explicit sharp bounds for all nodes of Sturm-Liouville eigenfunctions with potentials in $L^1$ balls, extending previous results to all nodes.

Findings

Explicit formulas for sharp bounds of eigenfunction nodes

Bounds depend on the $L^1$ norm of the potential

Results applicable to all nodes of eigenfunctions

Abstract

For the classical Sturm-Liouville operators, we prove the sharp bounds for all nodes of eigenfunctions by regarding these nodes as nonlinear functionals of potential $q\in L^1[0,1]$. By studying the optimization problems to minimize or to maximize the nodes $\{ T_{i,m}\}$ subject to the constraint $\|q\|_{1}=r$ with $r>0$ and using the strong continuity of the nodes in potentials, we obtain the explicit expressions for the sharp bounds, which are given as elementary functions.