Global modes of a quadratically confined Ginzburg-Landau equation ($s_j={\mu }_0-U^2/(4\gamma )-ϵ(2j+1)\sqrt{\gamma \nu }$)

For the whole-line linear complex Ginzburg-Landau equation with real positive diffusion $\gamma$ and $\mu (x)={\mu }_0-\nu {ϵ}^2{x}^2$, removing drift and scaling $\xi =\sqrt{ϵ}(\nu /\gamma {)}^{1/4}x$ gives the Hermite differential equation. Decay at both infinities quantizes the spectral parameter to $2j+1$. The strongest mode is $j=0$, giving global instability iff ${\mu }_0>U^2/(4\gamma )+ϵ\sqrt{\gamma \nu }$. The confinement correction vanishes in the slow-profile limit, recovering the homogeneous absolute wave-packet instability threshold.