On the existence and continuity of the pressure in quantum statistical mechanics

It is shown that in the case of all three statistics (Maxwell–Boltzmann; Bose–Einstein, and Fermi–Dirac) the pressure in the canonical ensemble is a continuous function that satisfies a Lipschitz condition provided the pair interaction potential $\Phi(r)$ for $r\eqslantgtr a$ ($a\eqslantgtr0$ is the hardcore radius) is a twice continuously differentiable function. Apart from the usual conditions needed to ensure the existence of the thermodynamic limit, this function satisfies for some $\varepsilon>0$ the further inequality
$$ \tilde U_N(x_1,x_2,\dots,x_N)=\sum_{i<j}\tilde{\Phi}(|x_i-x_j|)\eqslantgtr-\tilde BN,\quad\tilde B\eqslantgtr0, $$
where $\tilde{\Phi}(r)=\Phi(r)+\varepsilon(2r\Phi'(r)-r^2\Phi''(r)).$ Some sufficient conditions to be imposed on $\Phi(r)$ for this inequality to hold are given.