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A novel theoretical approach to the problem of the compositeness ($X$) of aresonance or bound state is developed on the basis of the expectation values ofthe number operators of the free particles in the continuum. This formalism isspecially suitable for effective field theories in which the bare elementarystates are integrated out but that give rise to resonance and bound states whenimplemented in nonperturbative calculations. We demonstrate that $X=1$ forfinite-range energy-independent potentials, either regular or singular. Anon-trivial example for an energy-dependent potential is discussed where it isshown that $X$ is independent of any type of cutoff regulator employed. Thegeneralization of these techniques to relativistic states is developed. We alsoexplain how to obtain a meaningful compositeness with respect to the openchannels for resonances, even if it is complex in a first turn, by making useof suitable phase-factor transformations. Defining elementariness as $X=0$, wederive a new universal criterion for the elementariness of a bound state. Alongthe same lines, a necessary condition for a resonance to be qualified aselementary is given. The application of the formalism here developed might beof considerable practical interest.

New results from a number operator interpretation of the compositeness of bound and resonant states