Abstract. The properties of translationally invariant polaron functional have been investigated in the region of strong and extremely strong coupling. It has been shown that the Gross–Tulub polaron functional obtained earlier using the methods of field theory was derived only for the region , where is the Fröhlich constant of the electron-phonon coupling. Various representations of exact and approximate polaron functionals have been considered. Asymptotic dependences of the polaron energy have been obtained using a functional extending the Gross–Tulub functional to the region of extremely strong coupling. The asymptotic dependence of polaron energies for an extremely strong coupling are (for the one-parameter variational function fk), and (for a two-parameter function ). It has been shown that the virial theorem 1:3:4 holds for the two-parameter function . Minimization of the approximate functional obtained by expanding the exact Gross–Tulub functional in a series on leads to a quadratic dependence of the polaron energy. This approximation is justified for . For a two-parameter function , the corresponding dependence has the form . However, the use of approximate functionals, in contrast to the strict variational procedure, when the exact polaron functional varies, does not guarantee obtaining the upper limit for the polaron energy.

Keywords: polaron, Fröhlich Hamiltonian, electron-phonon interaction.

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