Consider the Hamiltonian


\begin{displaymath}
H = {\sum\limits_{i = 1}^{N} {}} {\frac{{p_{i}^{2}}} {{2m}}}...
..._{j} ) -
\rho _{0} {\sum\limits_{i} {}}\int dr' v(r_{i}-r' ) ,
\end{displaymath}

with $v(r_{i} - r_{j} ) = {\frac{{e^{2}}}{{\vert r_{i} - r_{j} \vert}} }$ and $\rho _{0} = {\frac{{N}}{{V}}}$ a background neutralizing charge.

Assume the system crystallizes in a simple cubic lattice.

Write an expression for the matrix that determines the phonon frequencies.

Estimate the behavior of the phonon modes at long wavelength.