. By considering the double commutator
$$
\left[\left[H, e^{i \mathbf{k} \cdot \mathbf{r}}\right], e^{-i \mathbf{k} \cdot \mathbf{r}}\right]
$$
obtain as a generalization of the Thomas-Reiche-Kuhn sum rule the formula
$$
\sum_n\left(E_n-E_s\right)\left|\left\langle n\left|e^{i \mathbf{k} \cdot \mathbf{r}}\right| s\right\rangle\right|^2=\frac{\hbar^2 k^2}{2 m}
$$
Specify the conditions on the Hamiltonian $H$ required for the validity of this sum rule.