= Solution
Use a <Fourier mode> $u_m^n=z^ne^{im\theta}$. Put $a=4\mu\sin^2(\theta/2)\ge0$. Its <amplification polynomial of a multistep method> is
$$
\boxed{z^2-(1-3a/2)z-a/2=0}.
$$
The two roots are real, one nonnegative and one nonpositive. For $a>0$, evaluation at $1$ gives $a>0$ and at zero gives $-a/2<0$, so the positive root lies below one. The negative root is at least $-1$ exactly when the value at $-1$, namely $2-2a$, is nonnegative. Thus both roots have modulus at most one exactly for $0\le a\le1$. At $a=0$ they are $1,0$, and at $a=1$ they are $1/2,-1$; all unit-modulus roots are simple, satisfying the multistep <root condition for a multistep method>. Since the largest available $a$ is $4\mu$, the <Adams-Bashforth stability for centered diffusion> condition is
$$
\boxed{0\le\mu\le1/4}.
$$
For $\mu>1/4$, the highest-frequency mode has a root less than $-1$ and grows, proving necessity as well as sufficiency.
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