Phase transitions in a one-dimensional multibarrier potential of finite range

We have previously studied properties of a one-dimensional potential with $N$ equally spaced identical barriers in a (fixed) finite interval for both finite and infinite $N$. It was observed that scattering and spectral properties depend sensitively on the ratio $c$ of spacing to width of the barriers (even in the limit $N \to \infty$). We compute here the specific heat of an ensemble of such systems and show that there is critical dependence on this parameter, as well as on the temperature, strongly suggestive of phase transitions.