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February 2022 A note on eigenvalues estimates for one-dimensional diffusion operators
Michel Bonnefont, Aldéric Joulin
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Bernoulli 28(1): 64-86 (February 2022). DOI: 10.3150/21-BEJ1333

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.

Funding Statement

MB is partially supported by the French ANR-18-CE40-0012 RAGE project.
AJ is partially supported by the French ANR-18-CE40-006 MESA project.

Citation

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Michel Bonnefont. Aldéric Joulin. "A note on eigenvalues estimates for one-dimensional diffusion operators." Bernoulli 28 (1) 64 - 86, February 2022. https://doi.org/10.3150/21-BEJ1333

Information

Received: 1 July 2020; Revised: 1 February 2021; Published: February 2022
First available in Project Euclid: 10 November 2021

Digital Object Identifier: 10.3150/21-BEJ1333

Keywords: diffusion operator , Eigenvalues , intertwining , max–min principle , ‎Schrödinger operator‎ , spectral gap

Rights: Copyright © 2022 ISI/BS

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Vol.28 • No. 1 • February 2022
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