limits on allowed in and out states for Beliaev-Landau scattering. Only excitations with a wavevector $k$ for which $k_\text{min} < k < k_\text{max}$ can scatter resonantly to excitations with wavevectors $q$ and $k-q$ through Beliaev decay. Likewise, only excitations with a wavevector $q$ for which $q_\text{min} < q < q_\text{max}$ can combine with an excitation at $k - q$ to form one at $k$ through Landau scattering. However, the driven-dissipative nature of our setup allows energy not to be strictly conserved, so that scattering processes are possible within a finite linewidth $\gamma$ around the energy-conservation point.

In Fig. \[fig:bel\_decay\]c) we show how the extremal input and output momenta shift as a function of the interaction-renormalized detuning, defined in $\Delta$ (\[eq:delta\]), a parameter that can be tuned in experiment by changing the laser frequency $\omega_L$. When the drive is too far below resonance, i.e. when $\Delta < \Delta_0$, with $\Delta_0<0$ a critical value that can be derived from the dispersion relation, the spectral gap is too large as compared to the bandwidth and no resonant Beliaev-Landau scattering channels exist. In the limit of $\Delta \rightarrow 0$, for which the dispersion relation is linear, we find the contour of an equilibrium condensate 