  We consider a certain class of exactly solvable models, describing spectral
properties an electron moving in random in time external field with different
statistical characteristics. This electron can be band - like or belong to a
quantum well. The known dynamical Keldysh model is generalized for the case of
fields with finite correlation time of fluctuations and for finite transfer
frequencies of these fluctuations. In all cases we are able to perform the
complete summation of all Feynman diagrams of corresponding perturbation series
for the Green's function. This can be done either by the reduction of this
series to some continuous fraction or by the use of the generalized Ward
identity from which we can derive recurrence relations for the Green's
function. In the case of a random field with finite transferred frequency there
appear the interesting effects of modulation of spectral density and density of
states.
  Dedicated to 130-th anniversary of Pyotr Leonidovich Kapitza.
