  Eigenvalues are defined for any element of an algebra of observables and do
not require a representation in terms of wave functions or density matrices. A
systematic algebraic derivation based on moments is presented here for the
harmonic oscillator, together with a perturbative treatment of anharmonic
systems. In this process, a collection of inequalities is uncovered which
amount to uncertainty relations for higher-order moments saturated by the
harmonic-oscillator excited states. Similar saturation properties hold for
anharmonic systems order by order in perturbation theory. The new method, based
on recurrence relations for moments of a state combined with positivity
conditions, is therefore able to show new physical features.
