Spectrum of an oscillator with jumping frequency and the interference of partial susceptibilities

We study an underdamped oscillator with shot-noise frequency fluctuations. The oscillator spectrum is determined by the interference of the susceptibilities for different eigenfrequencies. Depending on the parameters, it has a fine structure or displays a single asymmetric peak. For nano-mechanical resonators with a fluctuating number of attached molecules, the spectrum is found in a simple analytical form. The results bear on various types of systems where the reciprocal correlation time of frequency fluctuations can be comparable to the typical frequency jumps.