the expression for $P_{\Phi,k}$ according to Eq. . To do that one would first diagonalize the matrix $P_{\Phi,k}$ by finding its eigenvalues $\lambda_i(p)$ and eigenfunctions $\varphi_i(p,\vec q)$. This is a problem of similar complexity as solving the Schrödinger equation for the two-particle problem. It is clear that for $p=(E,0,0,0)$ and $k=0$ the spectrum of eigenvalues of $P_\Phi$ contains as many vanishing eigenvalues as bound states exist with energy $E$.

Note, that the on-shell information obtained from solving Schrödingers equation (i.e. the binding energies and corresponding eigenfunctions) is not sufficient to determine the bound state propagator $G_k$ uniquely. In particular one can add to $G_k$ a term that is regular as a function of the center of mass momentum $p$ without changing the poles and therefore the on-shell information. It is especially useful to consider the combination $$\tilde G_k(p) = G_k(p)-G_\Lambda(p) = G_k(p)-V^{-1}.$$ This is just the part of $G_k$ that is generated from the flow equation with the “classical” part subtracted. Using Eq. this can be written as $$\tilde G_k(p) = (A(p)-V)^{-1}.$$ In this representation it is particularly clear that the poles of $\tilde G_k$ correspond to the solutions of Schrödingers equation for two particles.

Moreover, it is clear how 