Consider the two Hamiltonians

$$\begin{aligned} &H_{1}=\frac{\mathbf{p}^{2}}{2 m}+V(|\mathbf{r}|), \\ &H_{2}=\frac{\mathbf{p}^{2}}{2 m}+\sum_{n_{i} \in \mathbb{Z}} V\left(\left|\mathbf{r}-n_{1} \mathbf{a}_{1}-n_{2} \mathbf{a}_{2}-n_{3} \mathbf{a}_{3}\right|\right), \end{aligned}$$

where $\mathbf{a}_{i}$ are three linearly independent vectors. For each of the Hamiltonians $H=H_{1}$ and $H=H_{2}$, what are the symmetries of $H$ and what unitary operators $U$ are there such that $U H U^{-1}=H$ ?

For $\mathrm{H}_{2}$ derive Bloch's theorem. Suppose that $H_{1}$ has energy eigenfunction $\psi_{0}(\mathbf{r})$ with energy $E_{0}$ where $\psi_{0}(\mathbf{r}) \sim N e^{-K r}$ for large $r=|\mathbf{r}|$. Assume that $K\left|\mathbf{a}_{i}\right| \gg 1$ for each $i$. In a suitable approximation derive the energy eigenvalues for $H_{2}$ when $E \approx E_{0}$. Verify that the energy eigenfunctions and energy eigenvalues satisfy Bloch's theorem. What happens if $K\left|\mathbf{a}_{i}\right| \rightarrow \infty$ ?