of motion are $$\label{G2}
D_{2}({\mathbf{k}}, t) = -i \left\langle \left[ \hat{b}^{\dag}_{-{\mathbf{k}}} (t), \hat{b}^{\dag}_{{\mathbf{k}}} (0) \right] \right\rangle \theta (t),$$ $$\label{G2_eom}
\frac{dD_{2}({\mathbf{k}}, t)}{dt} = i \Gamma_{-{\mathbf{k}}} D_{2}({\mathbf{k}}, t) + i \Delta_{{\mathbf{k}}} D_{1}({\mathbf{k}}, t).$$ Fourier transforming equations  and in time and solving for $D_{1}({\mathbf{k}}, \omega_{{\mathbf{k}}})$ gives the following expression for the Green’s function: $$\label{G(om)}
D_{1}({\mathbf{k}}, \omega_{{\mathbf{k}}}) = \frac{\omega_{{\mathbf{k}}} + \Gamma_{{\mathbf{k}}}}{\omega^{2}_{{\mathbf{k}}} - \Gamma_{{\mathbf{k}}}^{2} + \Delta_{{\mathbf{k}}}^{2}}.$$ The phonon dispersion is therefore given by $$\begin{aligned}
\omega_{{\mathbf{k}}} & = \sqrt{\Gamma_{{\mathbf{k}}}^{2} - \Delta_{{\mathbf{k}}}^{2}} \\
& = \sqrt{\Omega \left( \Omega - 2\Delta_{{\mathbf{k}}} \right)}.\end{aligned}$$

We obtain an expression for $J_{1}$ by comparing the frequency at $k_{x} = \pi/2$ and the zone centre: $$\label{J1}
J_{1} = \frac{\hbar^{2} \left( \omega^{2}_{\pi/2} - \omega^{2}_{0} \right)}{\Omega \left( \gamma_{0} - \gamma_{\pi/2} \right)},$$ where the subscripts $\pi/2$ and 0 indicate ${\mathbf{k}}=(\pi/2,0,0)$ and ${\mathbf{k}}=(0,0,0)$, respectively. Since $\Omega$ is already known from bulk susceptibility data, $J_{1}$ can be estimated solely using the material’s phonon dispersion. Using neutron scattering data from [@cowley64], we obtained a range of $J_{1}$ values between 30 and 200 meV depending on $S$ and on how the fit was made. As will be shown in , these estimates are somewhat lower than the values required to produce a 2DEG at the LAO/STO interface, which is likely 