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 <$Ealpha~symbol Г~10>, where alpha is the Frohlich constant of the electron-phonon coupling. Various representations of exact and approximate polaron functionals have been considered. Asymptotic dependences of the polaron energy have been obtained using a functional extending the Gross - Tulub functional to the region of extremely strong coupling. The asymptotic dependence of polaron energies for an extremely strong coupling are <$EE sub p ~symbol Ы~-0,31683~alpha sup {4 "/" 3 }> (for the one-parameter variational function fk), and <$EE sub p ~symbol Ы~-0,31767~alpha sup {4 "/" 3 }> (for a two-parameter function <$Ef sub k prime>). It has been shown that the virial theorem 1:3:4 holds for the two-parameter function <$Ef sub k prime>. Minimization of the approximate functional obtained by expanding the exact Gross - Tulub functional in a series on <$E1 "/" alpha> leads to a quadratic dependence of the polaron energy. This approximation is justified for <$Ealpha~symbol Ы~7,5~-~8>. For a two-parameter function <$Ef sub k prime>, the corresponding dependence has the form <$EE sub p ~symbol Ы~-0,125~alpha sup 2>. 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.

• Каширіна Наталія Іванівна (фізико-математичні науки)

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