Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 344 / 2 / b / Solution

The Fourier transform sends each spatial derivative to $i{q}_i$, so at Gaussian levelFor $\kappa <0<\gamma$, minimizing over $s=q^2\ge 0$ givesThe minimum kernel is $G(q_0)=a-{\kappa }^2/(4\gamma )$. Gaussian fluctuations first diverge when this vanishes, so the nonzero-wavevector soft-mode sphere becomes unstable at