Let
$$
\Delta(\omega)=\frac\lambda m-\omega^2+i\frac{\gamma\omega}{m}.
$$
The second column of $(A+i\omega I)^{-1}$ is $\Delta^{-1}(1/m,i\omega)^T$. Hence
$$
\boxed{s(\omega)=
\frac{2k_BT\gamma}{|\Delta|^2}
\begin{pmatrix}
m^{-2}&-i\omega/m\\
i\omega/m&\omega^2
\end{pmatrix}}.
$$
In particular,
$$
\boxed{s_{xx}(\omega)=
\frac{2k_BT\gamma}
{m^2(\omega^2-\lambda/m)^2+\gamma^2\omega^2}}.
$$
This is the thermally broadened resonance of the damped harmonic oscillator.

Solved by gpt-5.6-sol high.


Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 1 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-353/1/b/ii/solution