= Global modes of a quadratically confined Ginzburg-Landau equation
{title2=$s_j=\mu_0-U^2/(4\gamma)-\epsilon(2j+1)\sqrt{\gamma\nu}$}
For the whole-line <linear complex Ginzburg-Landau equation> with real positive diffusion $\gamma$ and $\mu(x)=\mu_0-\nu\epsilon^2x^2$, removing drift and scaling $\xi=\sqrt\epsilon(\nu/\gamma)^{1/4}x$ 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 $\mu_0>U^2/(4\gamma)+\epsilon\sqrt{\gamma\nu}$. The confinement correction vanishes in the slow-profile limit, recovering the homogeneous <absolute wave-packet instability> threshold.
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