Solve the time evolution for the ladder model approximately
$$
H=\left(\begin{array}{cccc}
0 & V & \cdots & V \\
V & E_1 & & \\
\vdots & & \ddots & \\
V & & & E_n
\end{array}\right), \quad E_j=\alpha+(j-1) \Delta \epsilon.
$$
First derive an integral equation for $C_0(t)$ only by substitution. Then replace the sum by integration over $\omega=j(\Delta \epsilon / \hbar)$ and extend the integration over the whole real axis. Replace the integral by a $\delta$-function and show that the initial state decays exponentially with a rate
$$
k=\frac{2 \pi V^2}{\hbar \Delta \epsilon}.
$$