There are at least two possible ways to go about computing the amplitude for $2\to2$ scattering of indistinguishable particles in non-relativistic quantum mechanics. The first is the method we all learn in courses: first move to the center of mass frame and then solve the single-particle Schrödinger equation with prescribed boundary conditions.
The second approach is based on perturbative expansion of the $S$-operator using the many-body Hamiltonian. This produces a Feynman diagram expansion similar to relativistic QFT (except without antiparticles). Remarkably, the infinite resummation of Feynman diagrams can be shown to agree with the result of solving the single-particle Schrödinger equation.
I am interested in learning more about the latter approach. The only reference I am aware of (which discusses the $\delta$-function potential) is the following lectures notes (page 10 specifically). Can anyone recommend a canonical text?