Harmonic oscillator with random frequency

Exercise. Apply the result to the harmonic oscillator (1.3) with frequency co2(t) = cog l H- a (t), where t) is a stationary random process with zero mean and autocorrelation time tc. The answer is... 

Having found a relationship between the absorption lineshape associated with a periodically modulated system variable and the power spectrum of this variable, we now consider a specific example. Consider a harmonic oscillator which is randomly perturbed so that its frequency changes in time as ... 

Fig. 3 Dynamic PDF calculated for a diatomic molecule a oscillating harmonically at a frequency a>o, and b two atoms randomly hopping between two states with different bond lengths. (Dmowski W, Vakhrushev SB, Jeong I-K, Hehlen M, Trouw F, Egami T (2006) Abstracts American conference on neutron scattering, St. Charles, IL, 18-22 June 2006)...

The unperturbed harmonic vibration of an Isolated bond of length b would lead to a correlation function, Cv(t), which oscillates without decay for all times. However, shifts in either the frequency or phase of the oscillation results in an eventual decay of C (t) to zero after a time when the phase of Ab(t) is, on the average, completely random with respect to that of Ab(0). 

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