Renormalization and regularization operators for ultracold atoms When dealing with s-wave scattering in ultracold atoms physics people usually work with the pseudopotential $U = g_0 \delta^{(3)}(r)\frac{\partial}{\partial r}(r\cdot)$.
On the other hand one (sometimes) uses a "bare" contact potential $U = g \delta^{(3)}(r)$ and then goes on and renormalises the coupling constant as $\frac{1}{g} = \frac{1}{g_0}  - \frac{1}{V}\sum_{k<\Lambda}\frac{1}{2\epsilon_k}$, where $\Lambda$ is the momentum cut-off and $\epsilon_k$ the free particle energy. This is done, for example, for the derivation of cooper pairs.
How are both methods related? Are they, after all, or do they tackle different divergencies / problems (the latter method seems to appear in the "many-body context")?
 A: These are two different regulators. The first is designed to act on coordinate space N-body wave functions, and the second one can be used in N-body perturbation theory (or on the lattice). However, they are constructed to do the same thing, to express the coefficient of a delta function potential in terms of the physical scattering length.
We are considering a short range force, modeled as a delta function interaction. In one dimension the Schroedinger equation in the delta potential has a well-defined solution, but in $d\geq 2$ the delta function is too singular and has to be regularized.
We write $V(x)=g_0\delta(x)$. In the Born approximation to the Schroedinger equation, or at tree level in diagrammatic perturbation theory, we can fix $g_0$ by matching the $k\to 0$ $T$-matrix to the scattering length. We get $g_0=4\pi a/m$.
If we iterate the Born series, or sum 2-particle bubbles, we encounter divergences. We wish to introduce a regulator to remove divergences, and renormalize to
express the bare potential in terms of $a$.
In the Schroedinger equation we can use the pseudo-potential method explained, for example, in Kerson Huang's book. The trick is that the low energy scattered wave is $\psi\sim (1-a/r)$.
Then the operator $\partial_r(r\psi)$ removes the scattered wave, and the Born approximation in the 2-body channel is exact. Huang explains how to apply this to $N$-body wave functions.
In diagrammatic perturbation theory we can impose a momentum space cutoff (the regulator), and compute the 2-particle bubble. The solution to the Schroedinger equation corresponds to summing the bubbles, which is a geometric series. Now we set $k=0$, and demand agreement with the tree level result. This gives your relation between $g$ and $g_0$ (renormalization). In a two component Fermi gas there are no further divergences, and this step fully renormalizes $N$-body perturbation theory. By the way: If you want a regulator that behaves exactly like the pseudo-potential, and removes the zero external momentum bubble, you can use dimensional regularization.
For bosons, fermions with more than 2 components, or finite range potentials, additional divergences appear.
Historically, the ground state energy of the weakly interacting Fermi gas was first computed using the pseudo-potential (by Lee and Yang), but modern presentations typically use the renormalized cutoff theory. The result is the same.
A: Both methods are closely related to each other, in the sense that they describe the same low-energy (long-distance) physics. (As long as the s-wave scattering length $a$ is the same, of course.)
For dilute gases at low enough temperature, the physics is universal, in the sense that it is independent of the microscopic details of the potential: it only depends on the s-wave scattering length (however, see below). Therefore, one can use any potential or pseudopotential, as long as it gives the correct scattering length. In practice, the delta potential is much better suited for field-theoretic calculations (which are the norm in the many-body context).
A side note concerning universality. It so happens that in three dimensions, the field theory of non-relativistic interacting particles is non-renormalizable (in the field-theoretic sense), or put it in a more modern framework, it is an effective field theory. This effective field theory is controlled at low density, with small parameter $\sqrt{n a^3}$ ($n$ the density). At each order of the expansion in $\sqrt{n a^3}$, one needs to fix more and more non-universal parameters which depend on the specifics of the interaction potential (see the papers by Braaten and Hammer).
