Final CN equations
We can rewrite the Crank-Nicolson scheme as follows
$$
\begin{equation*}
V_{j}=
\left(2\hat{I}+\alpha\hat{B}\right)^{-1}\left(2\hat{I}-\alpha\hat{B}\right)V_{j-1}.
\end{equation*}
$$
We have already obtained the eigenvalues for the two matrices
\( \left(2\hat{I}+\alpha\hat{B}\right) \) and \( \left(2\hat{I}-\alpha\hat{B}\right) \).
This means that the spectral function has to satisfy
$$
\begin{equation*}
\rho(\left(2\hat{I}+\alpha\hat{B}\right)^{-1}\left(2\hat{I}-\alpha\hat{B}\right)) < 1,
\end{equation*}
$$
meaning that
$$
\begin{equation*}
\left|(\left(2+\alpha\mu_i\right)^{-1}\left(2-\alpha\mu_i\right)\right| < 1,
\end{equation*}
$$
and since \( \mu_i = 2-2cos(\theta) \) we have \( 0 < \mu_i < 4 \). A little algebra shows that
the algorithm is stable for all possible values of \( \Delta t \) and \( \Delta x \).