Renormalization in non-relativistic quantum mechanics I read many articles about renormalization in the Internet, but as I currently don't know much of QFT (currently just studying classical field theory and QM), and as all this looks quite interesting, I'd like to still get some bit of understanding and feeling of it — in the context of non-relativistic quantum mechanics.
So, my question is: are there any (hopefully simple) examples of quantum mechanical problems, applying perturbation theory to which will give divergent series, which can then be regularized by renormalization procedure? What are they, what does this process of renormalizing them look like?
 A: Normally renormalization is necessary when the phenomenological constants acquire unnecessary perturbative corrections, not obligatorily divergent. But there may be other cases. Read, for example, http://www.physics.umd.edu/courses/Phys851/Luty/notes/renorm.pdf for QM.
A: To understand the essence of perturbative renormalization you don't need any quantum field theory nor any quantum mechanics. A simple toy model suffices.
Suppose your theory makes a prediction for two distinct observables $F$ and $G$ in terms of a perturbative parameter $g$:
$$F = g + g^2 (S+1) + g^3 (S+1)^2 + g^4 (S+1)^3 + ...$$
$$G = g + g^2 (S-1) + g^3 (S-1)^2 + g^4 (S-1)^3 + ...$$
Here, $S$ represents some divergent mathematical sum. For instance, $S=1+2+3+4+...$. As in both expressions each term beyond first order in $g$ diverges, your perturbative theory seems pretty useless. What to do?
Under perturbative renormalization, one attempts to eliminate the non-observable parameter $g$ from the theory, and rewrite the perturbative series in terms of the observable $G$. So, we write:
$$g= \alpha_1 G + \alpha_2 G^2 + \alpha_3 G^3 + ...$$
and substitute this expression in the perturbation expansion for $G$. This gives us (up to order $G^3$):
$$G =  \alpha_1 G + \alpha_2 G^2 + ... + (\alpha_1 G + ...)^2 (S-1)+ ...$$
It follows that $\alpha_1 =1$ and $\alpha_2 = -(S-1)$:
$$g= G - G^2 (S-1) + ...$$
Substituting this series in the expression for the observable $F$ yields:
$$F= G - G^2 (S-1) + ... + (G-...)^2 (S+1)+ ...$$
Which gives us the end result:
$$F = G + 2G^2 + ...$$
Note that this perturbative expression for $F$ is written solely in terms of the observable $G$, and no longer contains the divergent quantity $S$. You conclude that your perturbative theory for the observables $F$ and $G$ is renormalizable.

Note: when performing a perturbative renormalization and all infinities exactly cancel each other, it feels a bit like magic. In this toy example the magic is a direct consequence of the hidden non-perturbative relationship $F-G = 2FG$. 
A: http://www.roma1.infn.it/~amelino/appunti1.pdf
Here, page 16: "Aside on perturbative renormalizability". 
You can find a quite simple but enlightening example of renormalization applied to a non-relativistic theory where you also have the exact energy spectrum and eigenfunctions to be compared to.
