Fermionic interaction potentials Are there any examples of fermionic particles or quasiparticles for which the interaction potential is a globally smooth function? i.e. no singularities or branch points. 
As an example, in Flügge's Practical Quantum Mechanics, problem 148 has two repulsive particles on a circle. This is supposed to model the two helium electrons in the ground state. The equation he gives is
$$ -\frac{\hbar^2}{2mr^2}\left(\frac{\partial^2 \psi}{\partial x_1^2}+\frac{\partial^2 \psi}{\partial x_2^2}\right)+V_0\cos(x_1-x_2)\psi=E\psi$$
I don't quite follow why this potential does not have a singularity when $x_2\rightarrow x_1$. Are there other such examples?
 A: Well there's no particular reason for a textbook problem to actually model a physical system... But one can certainly write something like this as a completely valid approximation. Take Flugge's example, with He3 so its fermionic[fn.2]. Say the size of the atoms is very small, much smaller than the scales on which the ground state wavefunction varies, which is reasonable enough.  
Now there should really be a term $V_{repulse}(x_1-x_2)\psi$ where $V_{repulse}$ gets really big when $|x_1-x_2|\rightarrow 0$ to capture the fact that you can't put the two atoms on top of each other [fn.2]. But this is going to be really short ranged, almost zero if $|x_1-x_2|$ is significantly bigger then the size of the atom. On the other hand we know that $\psi(x_1,x_2)\rightarrow 0$ when $x_1 \rightarrow x_2$. So in precisely the region where $V_{repulse}$ would matter, $\psi$ is basically zero. So we can basically ignore $V_{repulse}\psi$. More exactly, the term is proportional to (size of atom)/(size of circle) squared, which could be very small.
So you don't always have to include a repulsive term. It can actually be quite negligible, even though it seems like a fact you can't ignore.
[fn. 1] There are also times when you can ignore the repulsive interactions of bosons, although its not suppressed like the fermions.
[fn. 2] Its not really true that it should diverge as $x_1\rightarrow x_2$. If you really got the two atoms on top of each they would stop behaving like pointlike atoms, so your model would stop being applicable, rather than anything going to infinity. 
A: Apparently the Gaussian Effective Potential is used a fair amount. This would be something like $$V(x_1,x_2)=V_0 \exp(-\alpha(x_1-x_2)))$$. Thanks for the responses. 
A: In atomic physics effective 1D soft-core Coulomb potentials are routinely used for the interaction between particles, particularly when external fields are present. For example, for the interaction between an electron and a (space-fixed) proton at the center of coordinates:
$$V(x) = - \frac{1}{\sqrt{x^2 + \epsilon^2}}$$
where atomic units are used and $\epsilon$ is usual fitted or taken as one. It is much simpler to integrate the time-dependent Schrödinger equation for this potential.
