Conjecture on the thermodynamic-limit excitation spectrum of an interacting Fermi gas

Let $H^L$ be the finite-volume Hamiltonian of an interacting Fermi gas in a periodic box, with ground-state energy $E^L$. For momentum ${\bf k}$, let $\epsilon^\pm({\bf k})$ denote the thermodynamic-limit infimum of the excitation spectrum in the even and odd particle-number parity sectors, respectively. Thermodynamic-limit excitation-spectrum conjecture. For a large class of potentials with attractive interactions: (1) the functions $\epsilon^\pm:{\mathbb R}^d\to{\mathbb R}$ are continuous; (2) if ${\bf k}_s\to{\bf k}$, $L_s\to\infty$, and ${\bf k}_s\in\frac{2\pi}{L_s}{\mathbb Z}^d$, then $\epsilon^{L_s,\pm}({\bf k}_s)\to\epsilon^\pm({\bf k})$; (3) for $d\geq2$, $\inf_{\bf k}\min(\epsilon^-({\bf k}),\epsilon^+({\bf k}))=:\varepsilon>0$; (4) for $d\geq2$, $\inf_{{\bf k}\neq0}\frac{\min(\epsilon^-({\bf k}),\epsilon^+({\bf k}))}{|{\bf k}|}=:c_{\operatorname{cr}}>0$; and (5) the three stated subadditivity inequalities hold:

$$\epsilon^-({\bf k}_1+{\bf k}_2)\leq\epsilon^-({\bf k}_1)+\epsilon^+({\bf k}_2),$$ $$\epsilon^+({\bf k}_1+{\bf k}_2)\leq\epsilon^-({\bf k}_1)+\epsilon^-({\bf k}_2),$$ $$\epsilon^+({\bf k}_1+{\bf k}_2)\leq\epsilon^+({\bf k}_1)+\epsilon^+({\bf k}_2).$$

The conjectures are presented as unproved. The positivity assertions are expected only in dimensions $d\geq2$, because of a Galilean-covariance argument in one dimension.