It is now common to use Feshbach resonance to tune the s-wave scattering length of a Bose-Einstein condensate.
Apparently as $a\rightarrow \infty$, the GPE would break down. The reason is that it predicts that the size of the condensate will diverge.
But at which point exactly does GPE break down? What would the bose gas look like at $a= \infty$?
TL;DR: The Gross-Pitaevskii equation is only applicable for very weakly-interacting bosons. At $a=\infty$ the gas displays universal physics.
Strictly speaking, the Gross-Pitaevskii equation (GPE) is only valid for $$na^3 \ll 1,$$ where $n$ is the density of particles and $a$ is the $s$-wave scattering length. As it is a mean-field theory, one has to look for beyond-mean-field effects in order to see where the GPE approach breaks down. One such effect is the celebrated Lee-Huang-Yang correction to the ground-state energy. It has been measured by the Solomon group in 2011.
The strongly-interacting regime ($a=\infty$), also known as the unitarity regime, has been extensively studied theoretically (see, e.g., this article and papers citing it). It has been hard to achieve experimentally (even believed to be non-existent by some) due to technical problems (in particular, three-body losses) until recently. In 2014, it has been realized in JILA by quenching the gas from the weakly-interacting regime and observing that the thermalization is faster than three-body losses.
This regime is particularly interesting, as the two scales in the gas are given by the density and the thermal de Broglie wavelength. Therefore, at very low temperatures only a single parameter remains, given by the density of the gas, and the physics is therefore "universal". That is, all thermodynamic quantities in appropriate units can then be expressed as a set of universal numbers. The latter are independent of any details of the system, such as the type of atoms used to make the condensate.
