The description of quasiparticles seems to come in two flavors: Completely qualitatively, where it is simply said that different (quasi-)particles interact to "form" a quasiparticle, or quantitatively, but indirectly via characterization of, e.g., the effective mass of interacting electrons, or via association with peaks in spectral functions.
This makes my current understanding of the mathematical definition of a quasiparticle rather unsatisfactory. However, the characterization via peaks in the spectral functions makes me wonder: Is a quasiparticle simply an eigenstate of a (complicated, many-body) Hamiltonian?
I mean this in the following sense: If $|\psi_m\rangle$ and $|\psi_n\rangle$ are eigenstates of $H$, then is a quasiparticle simply the excitation created by the operator $a^{\dagger} = |\psi_m\rangle \langle \psi_n|$ (for appropriately chosen $m,n$)? If not, then what is the relationship between the two?