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

We apply quantum-mechanical sum rules to pairs of one-dimensional systems defined by potential energy functions related by parity. Specifically, we consider symmetric potentials, V(x) = V(- x), and their parity-restricted partners, ones with V(x) but defined only on the positive half-line. We extend recent discussions of sum rules for the quantum bouncer by considering the parity-extended version of this problem, defined by the symmetric linear potential, V(z) = F|z| and find new classes of constraints on the zeros of the Airy function, Ai(ζ), and its derivative, Ai′(ζ). We also consider the parity-restricted version of the harmonic oscillator and find completely new classes of mathematical relations, unrelated to those of the ordinary oscillator problem. These two soluble quantum-mechanical systems defined by power-law potentials provide examples of how the form of the potential (both parity and continuity properties) affects the convergence of quantum-mechanical sum rules. We also discuss semi-classical predictions for expectation values and the Stark effect for these systems.