Bessel quotients and Robin eigenvalues

Pedro Freitas

Vol. 315 (2021), No. 1, 75–87

DOI: 10.2140/pjm.2021.315.75

Abstract

We obtain sharp bounds for the quotient $x{J}_{\nu +1}∕J_{\nu }\left(x\right)$, where $J_{\nu }$ are Bessel functions of the first kind and $\nu >-1$. These bounds are asymptotically correct close to the zeros of $J_{\nu }$, allowing us to derive sharp estimates for the zeros of the function $x{J}_{\nu +1}\left(x\right)-\beta J_{\nu }\left(x\right)$, with applications to eigenvalue problems associated with the Laplace and Dirac operators.

Keywords

Bessel functions, eigenvalues, Mittag–Leffler expansion

Mathematical Subject Classification

Primary: 33C10, 34B30

Secondary: 35P15

Milestones

Received: 4 September 2020

Accepted: 11 October 2021

Published: 13 December 2021