= Solution
Put $q=k_n^2$. The mode matrix has <trace> $-1/2-(1+D)q<0$, so a growing <eigenvalue> occurs exactly when its <determinant> is negative:
$$
\boxed{\Delta(q)=Dq^2+(D-1/2)q+1<0.}
$$
This upward quadratic has two positive roots only if $D<1/2$ and $D^2-5D+1/4>0$. Within $0<D<1$ this is
$$
\boxed{0<D<D_c=\frac{5-\sqrt{24}}2.}
$$
For such $D$, define
$$
\boxed{q_\pm=\frac{1/2-D\pm\sqrt{D^2-5D+1/4}}{2D}.}
$$
The unstable modes satisfy $\boxed{\sqrt{q_-}<k_n<\sqrt{q_+}}$. These are strict inequalities: at either endpoint the leading <eigenvalue> is zero rather than positive.
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