= Marginal fragmentation mode of an isothermal slab
{title2=$|k|H=1,\quad\phi\propto\operatorname{sech}(z/H)$}
For <mass density> $\rho=\rho_0\operatorname{sech}^2(z/H)$ with $H^2=c_s^2/(2\pi G\rho_0)$, a zero-frequency gravitational mode with $k\ne0$ obeys $W=0$. Its potential satisfies $\phi_{zz}+[2\operatorname{sech}^2(z/H)/H^2-k^2]\phi=0$. The substitution $\tau=\tanh(z/H)$ gives the <associated Legendre differential equation> of degree one and order $|k|H$. Solutions decaying at both infinities exist only at $|k|H=1$, when $\phi\propto\operatorname{sech}(z/H)$. Longer horizontal wavelengths are unstable. Neutral mass-preserving relabeling displacements and the uniform vertical translation at $k=0$ are distinct from this fragmentation threshold.
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