Casimir squared correction to the standard rotator Hamiltonian for the O($n$) sigma-model in the delta-regime

In a previous paper we found that the isospin susceptibility of the O($n$) sigma-model calculated in the standard rotator approximation differs from the next-to-next to leading order chiral perturbation theory result in terms vanishing like $1\mathrm{/}\mathrm{\ell },$ for $\mathrm{\ell }=L_t\mathrm{/}L\to \mathrm{\infty }$ and further showed that this deviation could be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant. Here we confront this expectation with analytic nonperturbative results on the spectrum in 2 dimensions, by Balog and Heged"us for $n=3,4$ and by Gromov, Kazakov and Vieira for $n=4$. We also consider the case of 3 dimensions.