= Solution
Use the real diffusive interpretation $\gamma>0$, $U,\mu_0\in\mathbb R$, with $\nu>0$. Without a positive real diffusion coefficient the printed ordering inequality and real oscillator coordinate would need different hypotheses. Remove the drift by $\eta=e^{Ux/(2\gamma)}v$. Direct differentiation reduces the <linear complex Ginzburg-Landau equation> to
$$
v_t=\gamma v_{xx}+\left(\mu_0-\frac{U^2}{4\gamma}-\nu\epsilon^2x^2\right)v.
$$
Set $v=b(\xi)e^{-i(\omega_0+\epsilon\omega_1)t}$ and $\xi=\sqrt\epsilon f x$. The resulting equation is
$$
\gamma\epsilon f^2b''+\left[\mu_0-\frac{U^2}{4\gamma}+i\omega_0+\epsilon i\omega_1-\frac{\nu\epsilon}{f^2}\xi^2\right]b=0.
$$
Canceling its order-one coefficient and normalizing the quadratic term gives
$$
\boxed{\omega_0=i\left(\mu_0-\frac{U^2}{4\gamma}\right),\qquad f=(\nu/\gamma)^{1/4},\qquad \lambda=\frac{i\omega_1}{\sqrt{\nu\gamma}},\qquad b''+(\lambda-\xi^2)b=0.}
$$
The supplied decay quantization has $\lambda=2j+1$, $j=0,1,2,\ldots$. Equivalently the <Hermite functions> $b_j=e^{-\xi^2/2}H_j(\xi)$ solve this harmonic-oscillator problem, where $H_j$ is the physicists' <Hermite polynomial>. Thus $\omega_1=-i(2j+1)\sqrt{\nu\gamma}$ and the exact temporal <growth rates> are
$$
\boxed{s_j=-i(\omega_0+\epsilon\omega_1)=\mu_0-\frac{U^2}{4\gamma}-\epsilon(2j+1)\sqrt{\nu\gamma}.}
$$
The <global modes of a quadratically confined Ginzburg-Landau equation> are
$$
\eta_j(x,t)=C_j e^{Ux/(2\gamma)-\epsilon f^2x^2/2}H_j(\sqrt\epsilon f x)e^{s_jt}.
$$
For every fixed positive $\epsilon$, the negative quadratic spatial exponent dominates the linear drift exponent and the <polynomial> at both infinities; all these <normal modes> are spatially bounded. The largest <growth rate> is $s_0$, so the least-node <normal mode> is temporally non-growing, and all <normal modes> are non-growing, when
$$
\boxed{\mu_0\leq\frac{U^2}{4\gamma}+\epsilon\sqrt{\nu\gamma}.}
$$
Equality is a neutral ground-state <normal mode>; strict inequality gives decay. This proves the printed sufficient condition for a bounded stable global <normal mode> and identifies its stronger meaning: stability of the leading global <normal mode>. Existence of merely some stable higher-$j$ <normal mode> is not equivalent to the inequality, since sufficiently large $j$ gives $s_j<0$ even above the leading-mode threshold. In the quadratic-coefficient model the reduction and <eigenvalues> are exact, not just a first-order approximation to a more general profile.
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