We broaden the scope of the recent work of Fernandez et al. with respect to the hypervirial method for one-dimensional quantum problems. The modulus squared of the solutions to the Schrödinger equation uniquely satisfy a third-order linear differential equation. Because of the inherent nonnegativity of the physical solutions to this third-order system, one can use the moment methods of Handy and Bessis to generate rapidly converging lower and upper bounds to the individual physical discrete spectrum states. Crucial in this regard is the application of linear programming methods to solve the “missing moment problem”. We discuss these matters in the context of some one-dimensional examples cited in the literature.