# Difference in spectrum between the damped harmonic oscillator and a HO in thermal equilibrium with a bath?

I'm considering a damped, NOT driven harmonic oscillator, more specific an exponentially decaying oscillation. I would like to know the power spectral density of the signal. What I did so far was taking the signal x(t)= $\exp(-\Gamma^{-1}|t|)\cos \omega t$ and taking the fourier transform of this signal. This gives me the following spectrum (i.e. the signal x(t) displayed in frequency space): two lorentzian peaks of width $\Gamma$, one centered at $\omega$ and one at $-\omega$.

Two problems with this:

1. Maybe it is helpful to know what spectrum I expected to get: a lorentzian ONLY around $-\omega$.

2. If I would do this thought experiment the other way, i.e. see a spectrum conisting of two lorentzian peaks of width $\Gamma$, one centered at $\omega$ and one at $-\omega$, I would interpretate this as a harmonic oscillator that is in thermal equilibrium and is certainly NOT damped (i.e. having an amplitude decaying to zero on a timescale relating to $\Gamma^{-1}$). I would interprete the lorentzian around $-\omega$ as dissipation of energy, BUT I would interpret the lorentzian around $\omega$ as the reverse process, namely (re)absorption of energy: since the spectrum is symmetric for $\omega \to - \omega$, I would say indeed there are fluctuations, i.e. heat or energy is exchanged between the oscillator and the bath BUT in a noisy way: the signal decays a bit (dissipates energy to the bath), then regains amplitude/energy (absorbs energy from the bath) and this goes on in a fluctuating way.

3. The signal x(t)= $\exp(-\Gamma^{-1}|t|)\cos \omega t$ is as you may have noticed not the exact description of an exponentially decaying oscillation. They only correspond for times t>0. I used this description because I do not know how to take the fourier transform of $x(t)= \exp(-\Gamma^{-1}t)\cos \omega t$. Not sure if this is relevant or not.

P.S.: I stumbled upon this problem while reading into quantum optics (or rather cavity optomechanics), where these spectra are ubiquitous but never really justified, and -as it seems to me- they appear in contradictory contexts. I have in mind specifically the review paper 'Cavity Optomechanics' by Aspelmeyer (2014).