# On the Second-order Frechet Derivatives of Eigenvalues of Sturm-Liouville Problems in Potentials

## TL;DR

This paper derives the second-order Frechet derivatives of eigenvalues in Sturm-Liouville problems with respect to potentials, revealing their negative definiteness in certain cases, which advances understanding of spectral sensitivity.

## Contribution

It provides the first explicit formulas for second-order Frechet derivatives of eigenvalues in Sturm-Liouville problems, extending previous first-order analyses.

## Key findings

- Second-order derivatives are explicitly calculated.
- Second-order derivatives are negative definite quadratic forms in some cases.
- Enhances understanding of eigenvalue sensitivity to potential variations.

## Abstract

The works of V. A. Vinokurov have shown that eigenvalues and normalized eigenfunctions of Sturm-Liouville problems are analytic in potentials, considered as mappings from the Lebesgue space to the space of real numbers and the Banach space of continuous functions respectively. Moreover, the first-order Frechet derivatives are known and paly an important role in many problems. In this paper, we will find the second-order Frechet derivatives of eigenvalues in potentials, which are also proved to be negative definite quadratic forms for some cases.