Solution
= Solution
The nontotally symmetric longitudinal modes are the two $\Sigma_u^+$ coordinates $t$ and $a$. Both have $q_3-q_2=0$, so the middle spring is not extended. Hence its <spring constant> cannot alter their energies:
$$
Dt=0,\qquad Da=\frac{2k}{m}a,
\qquad\boxed{\omega_t=0,\quad\omega_a=\sqrt{2k/m}.}
$$
These results are exact for every positive $\gamma$, not just perturbative. The zero mode expresses overall <translation invariance>.