We want to use the Fourier discrete transform to analyze the stability of leapfrog scheme for 1D diffusion eqn $v_t = \nu v_{xx}$
Thoughts
The leapfrog numerical discretization is given by
$$ \frac{ u_k^{n+1} - u_{k}^{n-1} }{2 \Delta t } = \nu \frac{ u_{k+1}^n - 2 u_k^n + u_{k-1}^n }{\Delta x } $$
With $r = 2 \nu \Delta t / \Delta x$, we can write our numerical scheme
$$ u_{k}^{n+1} = u_k^{n-1} + 2r ( u_{k+1}^n - 2u_k^n + u_{k-1}^n ) $$
(Def:) the discrete Fourier discrete transform of $u \in l_2$ is the function defined in $L_2[- \pi, \pi] $ by
$$ \hat{u}(\xi) = \frac{1}{ \sqrt{2 \pi } } \sum_{k=-\infty}^{\infty} e^{-i k \xi }u_k$$
for $\xi \in [-\pi, \pi]$.
If we apply the discrete Fourier transform to our scheme and then shifting the index $k$ appropriately, one obtains
$$ \hat{u}^{n+1} ( \xi) = \hat{u}^{n-1} ( \xi) + (2 r e^{i \xi} - 4r + 2r e^{- i \xi} ) \hat{u}^n(\xi) $$
And here is where I'm looking for some guidance. I know how to perform Von Neumann analysis for 1-level schemes, but how do we handle these situations when we have 2-level scheme?
The characteristic equation for the recursion in $\hat u^n(ξ)$ is $$ q^2+4r(1-\cosξ)q-1=0\iff q_\pm(ξ)=q_\pm=-2r(1-\cosξ)\pm\sqrt{1+4r^2(1-\cosξ)^2}. $$ As both characteristic roots are real and their product is $-1$ with negative sum, the absolute value of the negative one will be larger than $1$. Which means that the resulting sequence $$\hat u^n(ξ)=c_1(ξ)q_+(ξ)^n+c_2(ξ)(-q_+(ξ))^{-n}$$ is almost always unstable. Even if initially $c_2(ξ)$ is very small, the accumulation of numerical errors will add to it in every step, and the alternating exponential term $$(-q_+(ξ))^{-n}\approx(-1)^n\exp(2nr(1-\cosξ))$$ will quickly grow, as the exponent $2nr(1-\cosξ)=(nΔt)\left(\frac{2\sinξ/2}{Δx}\right)^2$ is is positive and grows linearly in time $t_n=t_0+nΔt$.