We present a generalization of the Mehler kernel, providing the exact analytic time evolution of the Calogero model that describes bosons or fermions propagating on the real line with inverse-square two-body interactions and harmonic confinement. The key to our result is a simple relation between the exact propagator of the Calogero model and a Baker-Akhiezer function, together with explicit combinatorial formulas for the latter function.

The two-dimensional Gross-Neveu model is anticipated to undergo a crystalline phase transition at high baryon charge densities. This conclusion is drawn from the mean-field approximation, which closely resembles models of Peierls instability. We demonstrate that this transition indeed occurs when both the rank of the symmetry group and the dimension of the particle representation contributing to the baryon density are large (the large N limit). We derive this result through the exact solution of the model, developing the large N limit of the Bethe ansatz. Our analytical construction of the large-N solution of the Bethe ansatz equations aligns perfectly with the periodic (finite-gap) solution of the Korteweg-de Vries (KdV) of the mean-field analysis.