= Solution
In the formal static-background limit, $H=0$ and $\bar\rho$ is constant. The gravity-dominated <linear cosmological density perturbation> equation reduces to
$$
\ddot\delta_{\mathbf k}-\omega_J^2\delta_{\mathbf k}=0,\qquad
\omega_J=\sqrt{4\pi G\bar\rho}.
$$
The <Static-universe Jeans modes> are
$$
\boxed{\delta_{\mathbf k}=A_+e^{t/t_J}+A_-e^{-t/t_J},\qquad
t_J=(4\pi G\bar\rho)^{-1/2}.}
$$
The unstable perturbation grows exponentially rather than as $t^{2/3}$. In the expanding case, the $2H\dot\delta$ term resists peculiar motion and the falling background <density> also lengthens the gravitational growth timescale. A globally homogeneous static self-gravitating fluid is not obtained merely by turning off expansion; this calculation is the conventional local Jeans-background idealization, with the uniform background force removed or otherwise supported.
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