 Solution

For weak anchoring, $k_2\ll k_1$, use the normalized free-molecule eigenvectors $u_T=(1,1)/\sqrt{2}$ and $u_S=(1,-1)/\sqrt{2}$. The perturbation isIts diagonal matrix elements in this basis are both $k_2/(2m)$. First-order time-independent perturbation theory for the eigenvalues therefore givesHence ${\omega }_T\simeq \sqrt{k_2/(2m)}$ and ${\omega }_S\simeq \sqrt{2k_1/m}[1+k_2/(8k_1)]$. The zero free translation requires taking a square root of its new eigenvalue, rather than Taylor-expanding the frequency about zero. It is not a degenerate eigenvalue in the stipulated one-dimensional two-coordinate model.

As a check, the exact squared frequencies areExpansion gives the same first-order results; the next lower-mode correction is $-k_2^2/(8k_{1}m)$. The perturbed eigenvectors contain small translation/stretch admixtures of order $k_2/(4k_1)$, consistent with the failure of uniform translation to remain an exact mode in orientation (1).