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Properties of the polaron functional obtained as a result of averaging the Fröhlich Hamiltonian on the translation-invariant function have been investigated. The polaron functional can be represented in two different forms. It has been shown that the functional of translationally invariant Gross–Tulub polaron cannot be applied in the strong coupling region, where the real part ) ( Re s D of the complex quantity ( ) 2 s D takes negative values. The function ( ) 2 s D coincides in its structure with the dynamic susceptibility of degenerate electron gas. The necessary condition for obtaining correct results is investigation of the region of admissible values of the Gross–Tulub functional depending on properties of the function ( ) 2 s D , variational parameters, and the electronphonon interaction parameter α (Fröhlich coupling constant). A simple and exact formula for the recoil energy of the translationally invariant polaron has been derived, which makes it possible to extend the range of admissible values of the parameters of the electron-phonon interaction to the region of extremely strong coupling (α > 10), where 0 ) ( Re ≤ s D . Numerical investigation of different forms of polaron functionals obtained using the field theory methods has been carried out.

Gross–Tulub polaron functional in the region of intermediate and strong coupling