Constraints on Airy function zeros from quantum-mechanical sum rules

TL;DR

This paper derives new constraints on the zeros of Airy functions by applying quantum-mechanical sum rules and perturbation theory within the quantum bouncer model, providing systematic evaluation methods for related sums.

Contribution

It introduces a systematic approach to evaluate sums over Airy function zeros using quantum sum rules and perturbation theory, advancing understanding of their properties.

Findings

Derived new constraints on Airy zeros

Developed methods to evaluate sums over zeros

Connected quantum sum rules with Airy function properties

Abstract

We derive new constraints on the zeros of Airy functions by using the so-called quantum bouncer system to evaluate quantum-mechanical sum rules and perform perturbation theory calculations for the Stark effect. Using commutation and completeness relations, we show how to systematically evaluate sums of the form $S_{p}(n) = \sum_{k \neq n} 1/(\zeta_k - zeta_n)^p$, for natural $p > 1$, where $\zeta_n$ is the $n$-th zero of $Ai(\zeta)$.