  Using Feynman path integral technique estimations of the ground state energy
have been found for a conduction electron interacting with order parameter
fluctuations near quantum critical points. In some cases only \textit{singular}
perturbation theory in the coupling constant emerges for the electron ground
state energy. It is shown that an autolocalized state (quantum fluctuon) can be
formed and its characteristics have been calculated depending on critical
exponents for both weak and strong coupling regimes. The concept of fluctuon is
considered also for the classical critical point (at finite temperatures) and
the difference between quantum and classical cases has been investigated. It is
shown that, whereas the quantum fluctuon energy is connected with a true
boundary of the energy spectrum, for classical fluctuon it is just a
saddle-point solution for the chemical potential in the exponential density of
states fluctuation tail.
