A damped oscillator has the time evolution:
$$ y(t) = e^{-\Gamma t}\cos^2(\tilde{\omega}_0 t)$$
where $\Gamma$ is the damping rate, $\tilde{\omega}_0^2=\omega_0^2-\Gamma^2$ and $\omega_0$ is the undamped frequency. I understand that damping causes: (1) the oscillations to no longer occur at a single frequency but cover a Lorentzian distribution of values with (2) a peak at the new frequency $2\tilde{\omega}_0$. However, in the Fourier Spectrum two peaks are observed - one at $2\tilde{\omega}_0$ and another at $\omega=0$. For large damping these 2 peaks can overlap and can shift the fundamental peak to a new (lower frequency). Is the broad peak at $\omega=0$ and the additional shift on the other peak physical, if so what is its meaning?
Note: The Fourier Transform of undamped oscillations (i.e. $y(t) = \cos^2(\omega_0 t)$) produces a Dirac Delta peak at $\omega=0$ due to a non-zero mean, which is then presumably broadened by the damping. I just do not understand if/why this can produce a physical shift in the frequency?
