 Backward Euler diffusion stability on a finite Dirichlet interval

On $J$ interior Dirichlet grid points, backward Euler diffusion is stable for $\mu \ge 0$ and also for the nonphysical branch $\mu \le -[2{\sin }^2(\pi /(2(J+1))){]}^{-1}$. The latter makes every amplification denominator at most $-1$.