A model independent way to describe a resonance is through the frequency dependent scattering operator $S(\omega)$. Causality requires that this object is analytic in the upper half of the complex $\omega$-plane. In the lower half it may have poles, say at $\omega=\Omega-i\Gamma$. This gives a resonant response when excited at a frequency within $\Gamma$ of $\Omega$.

A model independent way to describe a resonance is through the frequency dependent scattering operator $S(\omega)$. Causality requires that this object is analytic in the upper half of the complex $\omega$-plane. In the lower half it may have poles, say at $\omega=\Omega-i\Gamma$. This gives a resonant response when excited at a frequency within $\Gamma$ of $\Omega$.

Concerning the relation between resonant frequencies and eigenfrequencies: At frequencies near resonance one can expand near the pole, $$S(\omega)=(H-\omega+i\Gamma)(H-\omega-i\Gamma)^{-1},$$ with $H$ a Hermitian operator (so that $S$ is unitary). The resonant frequency $\Omega$ then corresponds to an eigenvalue (eigenfrequency) of $H$.

The operator $H$ does not need to produce a simple wave equation. For example, this description of resonant scattering applies to nuclear physics, where the resonance is a bound state of strongly interacting elementary particles.