Solution
= Solution
At $|\mathbf q|=1$, let $s=a+\kappa$. The matrix is symmetric:
$$
R(1)=\begin{pmatrix}s&c\\c&s\end{pmatrix}.
$$
Its modes are
$$
\boxed{\lambda_-=s-c,\quad e_-=(1,-1)^T},
\qquad
\boxed{\lambda_+=s+c,\quad e_+=(1,1)^T}.
$$
Because $(1,0)^T=(e_-+e_+)/2$,
$$
\boxed{G_\phi(t)=\frac12e^{-(s-c)t}
+\frac12e^{-(s+c)t}=e^{-st}\cosh(ct)}.
$$
Solved by gpt-5.6-sol high.