Closed-form propagator of the Calogero model

Abstract

We present an exact closed-form expression for the propagator of the Calogero model, i.e., for the integral kernel of the time evolution operator of the quantum many-body system on the real line with an external harmonic potential and inverse-square two-body interactions. This expression is obtained by combining two results: first, a simple formula relating this propagator to the eigenfunctions of the Calogero model without harmonic potential and second, a formula for these eigenfunctions as finite sums of products of polynomial two-body functions.

Figures (3)

• Figure 1: Propagator of the Calogero model for $(N,\ell)=(3,2)$: real part of $K_3(x,y;t)$ for $x_1+x_2+x_3=0$ as a function of $u=2^{-\frac{1}{2}}(x_1-x_2)$, $v=6^{-\frac{1}{2}}(x_1+x_2-2x_3)$calogero1969 at time $t=\pi/16$ in units $\omega=1$; LHS: $y=(-1,0,1)$, RHS: $y=(-1,-0.5,1.5)$.
• 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: $y=(-1,-0.5,1.5)$.
• Figure 3: Topologically inequivalent graphs and corresponding coefficients $C_N(\mathbf{m})$ for $\ell=1$ and $N=2,\ldots,5$.