# A note on eigenvalues estimates for one-dimensional diffusion operators

## TL;DR

This paper derives sharp variational bounds for eigenvalues of one-dimensional diffusion operators, extending previous results and providing estimates for spectral gaps using intertwinings and eigenfunction properties.

## Contribution

It generalizes Chen and Wang's spectral gap result by establishing variational formulas for eigenvalues and offers methods to estimate gaps between the first two positive eigenvalues.

## Key findings

- Sharp variational bounds for eigenvalues in discrete spectrum
- Extension of spectral gap estimates using intertwinings
- Bounds coincide with actual eigenvalues when in discrete spectrum

## Abstract

Dealing with one-dimensional diffusion operators, we obtain upper and lower variational formulae on the eigenvalues given by the max-min principle, generalizing the celebrated result of Chen and Wang on the spectral gap. Our inequalities reveal to be sharp at least when the eigenvalues considered belong to the discrete spectrum of the operator, since in this case both lower and upper bounds coincide and involve the associated eigenfunctions. Based on the intertwinings between diffusion operators and some convenient gradients with weights, our approach also allows to estimate the gap between the two first positive eigenvalues when the spectral gap belongs to the discrete spectrum.