= Solution
For stable density jumps, $J>0$ and $q=J/(2\alpha)>0$. Let $y=C^2$. The two roots of the <dispersion relation for two density interfaces in uniform shear> are
$$
y_\pm=1+q\pm\sqrt{4q+q^2E^2},\qquad E=e^{-2\alpha}.
$$
They are real, and $y_+>0$. An exponentially growing <normal mode> exists precisely when $y_-<0$, giving a pair of imaginary phase speeds; the member with positive imaginary part grows in $e^{ik(x-ct)}$ for $k>0$.
Since $y_+y_-=(1-q)^2-q^2E^2$, this condition is $|1-q|<qE$. Solving both inequalities yields
$$
\frac1{1+E}<q<\frac1{1-E},\qquad
\boxed{\frac{2\alpha}{1+e^{-2\alpha}}<J<\frac{2\alpha}{1-e^{-2\alpha}}.}
$$
At either endpoint $y_-=0$, so the exponential growth rate vanishes. Outside the open band both squared phase speeds are nonnegative and the interfacial modes have real frequencies. This is a modal instability test for the stably stratified configuration; unstable density inversions would require a separate analysis. Within the band the dimensional growth rate is $k\Delta U\sqrt{-y_-}/2$.
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