New eigenvalue estimates involving Bessel functions

TL;DR

This paper derives new eigenvalue estimates for Laplacians and Dirac operators on Riemannian manifolds using Bessel functions, extending classical inequalities and spectral bounds with novel analytical techniques.

Contribution

It introduces Bessel function-based estimates for eigenvalues of Laplacians and Dirac operators, and extends Robin Laplacian to differential forms with spectral analysis.

Findings

Proves Faber-Krahn inequalities using Bessel functions.

Establishes a lower bound for Dirac operator eigenvalues involving Bessel functions.

Extends Robin Laplacian to differential forms with spectral characterization.

Abstract

Given a compact Riemannian manifold (M n , g) with boundary $\partial$M , we give an estimate for the quotient $\partial$M f d$\mu$ g M f d$\mu$ g , where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [37], we provide a differential inequality for f which, under some curvature assumptions, can be interpreted in terms of Bessel functions. As an application of our main result, a direct proof is given of the Faber-Krahn inequalities for Dirichlet and Robin Laplacian. Also, a new estimate is established for the eigenvalues of the Dirac operator that involves a positive root of Bessel function besides the scalar curvature. Independently, we extend the Robin Laplacian on functions to differential forms. We prove that this natural extension defines a self-adjoint and elliptic operator whose spectrum is discrete and consists of positive real eigenvalues. In particular, we characterize its first eigenvalue and provide a lower bound of it in terms of Bessel functions.