= Solution
For a <Fourier mode>, the amplification polynomial is
$$
\xi^2-(1-2\mu)(1-e^{i\theta})\xi-e^{i\theta}=0.
$$
Put $a=1-2\mu$ and $\xi=e^{i\theta/2}\eta$. The roots become
$$
\eta_\pm=-ia\sin(\theta/2)\pm\sqrt{1-a^2\sin^2(\theta/2)}.
$$
If $|a|<1$, the square root is real, both roots have <modulus> one, and their separation is at least $2\sqrt{1-a^2}>0$. The companion <matrix> has <eigenvectors> $(\xi_\pm,1)^T$, so their uniform separation gives a mesh-independent bound on its powers. By <power boundedness of a two-level Fourier scheme> and the discrete <Fourier transform>, this proves stability for $0<\mu<1$.
If $|a|>1$, choose $\theta=\pi$. The roots have product of <modulus> one and unequal <moduli>, so one has <modulus> greater than one. The scheme is unstable.
The boundary cases require more than a root-modulus check. At $\theta=\pi$, $\mu=0$ gives $(\xi-1)^2=0$, while $\mu=1$ gives $(\xi+1)^2=0$. The companion <matrices> have nontrivial <Jordan blocks> and solutions of the form $(A+Bn)(\pm1)^n$. Their linear growth rules out a uniform stability bound. This mode occurs on even periodic grids, which are enough to disprove mesh-uniform stability at the endpoints.
Therefore the stable range is exactly
$$
\boxed{0<\mu<1.}
$$
Within this range, second-order convergence additionally requires two appropriately accurate starting time levels. The unit-modulus parasitic root near $\theta=0$ does not decay, so poor initialization cannot be repaired by the scheme.
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