Constraints on Airy function zeros from quantum-mechanical sum rules

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) = k =n 1/(ζ k − ζ n ) p , for natural p > 1, where −ζ n is the n th zero of Ai(ζ).

Less than perfect quantum wavefunctions in momentum-space: How ϕ(p) senses disturbances in the force

“…The properties of the Airy function solutions have been re-examined in the light of this renewed interest with new analytic results, [11][12][13][14][15] using identities which appeared some time ago. 16 For example, the normalized wavefunctions are,…”

New identities from quantum-mechanical sum rules of parity-related potentials

“…In this note, we extend the results of Ref. [4] to discuss mathematical constraints which arise from the study of quantum-mechanical sum rules in a closely related system, namely the symmetric linear potential, defined by…”

“…The normalizations required in Eqns. (4) and (6) are obtained by using integrals first derived by Gordon [13] and Albright [14] collected in the Appendix in Sec. A, specifically…”

“…Ref. [4], Eqns. (19) and (20) could not be applied in the quantum bouncer system due to the discontinuous nature of the potential in Eqn.…”

“…For completeness, we recall from Ref. [4] that for the quantum bouncer system constraints on sums were of the form…”