Moments and saturation properties of eigenstates: Oscillator systems

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.