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, $Ai(z)$, and its derivative $Ai'(z)$. We also consider the parity-restricted version of the harmonic oscillator and find completely new classes of mathematical relations, unrleated 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.