# Uniform local Lipschitz continuity of eigenvalues with respect to the potential in $L^1[a,b]$

## TL;DR

This paper proves that the eigenvalues of a Sturm-Liouville problem vary in a uniformly Lipschitz continuous manner with respect to the potential function in the $L^1$ space, under certain conditions.

## Contribution

It establishes the uniform Lipschitz continuity of eigenvalues with respect to the potential in $L^1$, extending previous results to a broader class of problems.

## Key findings

- Eigenvalues are uniformly Lipschitz continuous in $L^1$ on bounded sets.
- The result applies to Sturm-Liouville problems with certain monotonic weights.
- Continuity holds uniformly across the eigenvalue sequence.

## Abstract

The present paper shows that the eigenvalue sequence $\{\lambda_n(q)\}_{n\geqslant 1}$ of regular Sturm-Liouville eigenvalue problem with certain monotonic weights is uniformly Lipschitz continuous with respect to the potential $q$ on any bounded subset of $L^1([a,b],\mathbb{R})$.