Is a quasiparticle just an eigenstate of the Hamiltonian? 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?
 A: In some contexts yes and in others no.
When we talk about quasiparticles as corresponding to peaks in the one-particle spectral function, the width of the peak tells us the quasiparticle lifetime $\tau$. The very fact we are talking about a finite lifetime means it is not an energy eigenstate (which would of course be a stationary state).

However, sometimes we use the term "quasiparticle" to describe exact excitation states of a mean field many-body Hamiltonian. Such a Hamiltonian might look like
$$ H_{MF} = \sum_k \psi_k^\dagger \mathcal{H}_k \psi_k + \text{const.}$$
where $\psi_k$ is a vector of creation/annihilation operators $c_k^\dagger/c_k$ for real electrons and $\mathcal{H}_k$ is a matrix. Through a canonical (unitary or Bogoliubov) transformation of our operators, we can obtain the diagonal form
$$ H_{MF} = \sum_kE_k\gamma_k^\dagger \gamma_k +E_0 $$
where the new operators $\gamma_k^\dagger/\gamma$ are linear combinations of the real electron operators $c_k^\dagger/c_k$. Since the transformation is canonical, these new operators behave just like the original $c_k^\dagger/c_k$ and so they can be interpreted as creating/destroying something. We call this thing the quasiparticle. And indeed, from the above diagonal Hamiltonian, we see that these quasiparticles behave like free particles with dispersion $E_k$ and ground state energy $E_0$.
