Closed-Form Propagator of the Calogero Model

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.

Figure 1

Topologically inequivalent graphs and corresponding coefficients $C_N(\mathbf{m})$ for $\ell =1$ and $N=2,\dots ,5$.

Figure 2

Propagator of the Calogero model for $(N,\ell )=(3,2)$: Absolute value of $K_3(x,y;t)$ for $x_1+x_2+x_3=0$ as a function of $u=2^{-\frac{1}{2}}(x_1-x_2)$, $v=6^{-\frac{1}{2}}(x_1+x_2-2x_3)$ at time $t=\pi /16$ in units $\omega =1$; lhs, $y=(-1,0,1)$; rhs, $y=(-1,-0.5,1.5)$.