  The exact formulae for spectra of equilibrium diffusion in a fixed bistable
piecewise linear potential and in a randomly flipping monostable potential are
derived. Our results are valid for arbitrary intensity of driving white
Gaussian noise and arbitrary parameters of potential profiles. We find: (i) an
exponentially rapid narrowing of the spectrum with increasing height of the
potential barrier, for fixed bistable potential; (ii) a nonlinear phenomenon,
which manifests in the narrowing of the spectrum with increasing mean rate of
flippings, and (iii) a nonmonotonic behaviour of the spectrum at zero
frequency, as a function of the mean rate of switchings, for randomly switching
potential. The last feature is a new characterization of resonant activation
phenomenon.
