Constraints on Airy function zeros from quantum-mechanical sum rules

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 $S_p(n)={\sum }_{k\mathrm{\ne }n}1\mathrm{/}({\zeta }_k-zet{a}_n{)}^p$, for natural $p>1$, where ${\zeta }_n$ is the $n$-th zero of $Ai(\zeta )$.