Does electron-positron pair annihilation occur when $E$ is negative? Suppose I release an electron-positron pair from rest at a distance of $r$. Then the particles attract each other and collide. The total energy $E$ is
$$E = 2m_ec^2-\frac{e^2}{4\pi\varepsilon_0r},$$
which converts into gamma rays.
If radius is small enough  to make $E$ negative, then do they collide? Or do they act like bosons without colliding each other?
 A: The do collide if we are using description of classical physics, but for this situation it does not work so well, which is well known. Consider this: if we imagine electron and positron as point particles interacting via Coulomb forces, there is no limit to energy that can be extracted from them. By putting them so close that total energy is negative, an energy of 2x511keV and more has already been extracted (perhaps radiated away). Further approach of the two particles would release even more energy which should manifest as a runaway radiation event.
We do not have convincing record of such processes, so here it is where the classical description seem to deviate from reality and is not reliable.
In non-relativistic quantum theory, the pair's expected average energy can never be so negative, because the Schroedinger equation predicts existence of stationary states of low negative energy(net energy including the rest energy is positive). But this also deviates from reality, because the pairs with low speed are observed to decay in hundreds of nanoseconds. ( More on the lifetime of positronium see https://physicsworld.com/a/positronium-puzzle-is-solved/ )
In relativistic quantum theory, there is an explanation for the instability and the lifetime can be calculated to high accuracy with measurements, but at the same time, the calculations require some tricks to handle new problems with energy - net energy comes out infinite (quantum IR and UV catastrophe) and it has to be handled by various tricks to get some reasonable results. This kind of works for some problems but less so for others (doesn't work for vacuum energy density), so there is no completely satisfactory theory for these things.
A: Electrons and positrons are not classical entities, they will be attracted to each other and will form a positronium, i.e. a bound state of electron positron for a while.

Positronium (Ps) is a system consisting of an electron and its anti-particle, a positron, bound together into an exotic atom, specifically an onium. The system is unstable: the two particles annihilate each other to produce two or three gamma-rays, depending on the relative spin states. The orbit and energy levels of the two particles are similar to that of the hydrogen atom (electron and proton). However, because of the reduced mass, the frequencies of the spectral lines are less than half of the corresponding hydrogen lines.

The quantum mechanical  probability of passing through each other (your r going to zero)  is not zero for l=0 ( l the angular momentum quantum number) but it is controlled by the wavefunction, the solution of the potential equation, squared. When they overlap they will annihilate.
