I will try one more time being more detailed and careful. Consider the transport equation of the form $$u_t+au_x=0, t\in[0,T],x\in \mathbb{R}, a>0$$ and initial condition $u(0,x)=u_0(x)$. I would like to establish stability of Crank-Nicolson scheme for that equation by computing the amplification factor. Consider the following iteration of CN: $$\frac{u^{n+1}_j-u^{n}_j}{k}+\frac{1}{2}a\frac{u^{n+1}_{j+1}-u^{n+1}_{j-1}}{2h}+\frac{1}{2}a\frac{u^{n}_{j+1}-u^{n}_{j-1}}{2h}=0$$ then I substitute the form $u^n_j=Q^ne^{ijhw}$ in the above equation: $$\frac{(Q-1)e^{ijhw}Q^n}{k}+\frac{1}{2}ae^{ijhw}Q^{n+1}\frac{e^{ihw}-e^{-ihw}}{2h}+\frac{1}{2}ae^{ijhw}Q^{n}\frac{e^{ihw}-e^{-ihw}}{2h}=0$$ from where with we get: $$Q=\frac{1-i\frac{ak}{2h}\sin(hw)}{1+i\frac{ak}{2h}\sin(hw)}$$ this is a complex number of the form $\frac{1-iC}{1+iC}$, thus I can calculate the absolute value as follows: $$|Q|=|\frac{(1-iC)(1-iC)}{(1+iC)(1-iC)}|=|\frac{1-C^2-2iC}{1+C^2}|=\frac{(1-C^2)^2+4C^2}{1+C^2}=\frac{(1+C^2)^2}{(1+C^2)^2}=1$$
from where I deduce that for Crank Nicolson method the amplification factor for that equation has the magnitude of: $$|Q|=1$$ If there is a mistake in the computations please let me know and I proceed from there. But if this is the amplification factor indeed I would like to know if I can deduce stability/conditional stability/instability from it? Is there a possibility of oscillations of any form? It does satisfy the condition less or equal than one, therefore I can use it and will not have problems?