= Solution
Retain the factor $\epsilon$ in the last temporal phase, as printed in the original PDF. Let $a=U/(2\gamma)$ and put $\eta=e^{ax}b(\xi)e^{st}$ with $s=-i\omega_0-i\epsilon\omega_1$. The <drift-removing substitution for a Dirichlet Sturm-Liouville problem> cancels the first derivative:
$$
sb=\gamma b_{xx}+\left(\mu_0-\frac{U^2}{4\gamma}-\nu\epsilon^2x^2\right)b.
$$
With $\xi=\sqrt\epsilon f x$, this becomes
$$
b_{\xi\xi}+\left[\frac{\mu_0-U^2/(4\gamma)+i\omega_0+i\epsilon\omega_1}{\gamma\epsilon f^2}-\frac\nu{\gamma f^4}\xi^2\right]b=0.
$$
Choose the positive scale $f=(\nu/\gamma)^{1/4}$. To leave an order-one spectral parameter, cancel the numerator's order-one term. Then
$$
\boxed{\omega_0=i\left(\mu_0-\frac{U^2}{4\gamma}\right),\qquad f=(\nu/\gamma)^{1/4},\qquad\lambda=\frac{i\omega_1}{\sqrt{\gamma\nu}}.}
$$
The assumed decaying solutions of the <Hermite differential equation> require $\lambda=2j+1$, $j=0,1,\ldots$. They are proportional to $e^{-\xi^2/2}H_j(\xi)$, with physicists' <Hermite polynomials>. Therefore the <global modes of a quadratically confined Ginzburg-Landau equation> have
$$
\boxed{\omega_1=-i(2j+1)\sqrt{\gamma\nu},\qquad s_j=\mu_0-\frac{U^2}{4\gamma}-\epsilon(2j+1)\sqrt{\gamma\nu}.}
$$
For every fixed positive $\epsilon$, multiplying the Hermite-Gaussian by $e^{Ux/(2\gamma)}$ still gives decay at both infinities. The strongest mode is $j=0$, so the whole-line system is \b[globally unstable precisely when]
$$
\boxed{\mu_0>\frac{U^2}{4\gamma}+\epsilon\sqrt{\gamma\nu}.}
$$
Equality is a neutral ground mode. This spectrum is exact for the stated quadratic profile, even though the notation organizes it as an expansion in $\epsilon$.
The homogeneous absolute-instability threshold in part (a) is $\mu=U^2/(4\gamma)$. The global threshold approaches that value as the profile varies increasingly slowly, but at finite $\epsilon$ the quadratic confinement adds the loss $\epsilon\sqrt{\gamma\nu}$. A locally absolutely unstable patch at the profile maximum is necessary but must be strong and broad enough to overcome that loss. Merely having positive local temporal growth $\mu_0>0$, which can be only <convective wave-packet instability>, is insufficient for a growing decaying global <normal mode>. This connects the local impulse classification to the discrete global spectrum.
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