A method previously derived for the approximate construction of the nucleon Green's function is here applied to that of the polaron. After making an arbitrary translation of the phonon variables, a linear integral equation for the Green's function is derived by means of a symmetrical treatment and a noncorrelation assumption, in complete analogy to the nucleon problem. This equation is solved through the introduction of a spectral representation in the special case of zero total momentum. The lowest energy state of the system is calculated in terms of the arbitrary translations and then minimized with respect to them. Using the simplest nontrivial cutoff procedure to obtain the variational equation and its solution, results are obtained for values of the coupling parameter $\ensuremath{\alpha}<~3$, and are compared with those of Feynman.
It is shown that Schrodinger's equation for a translation-invariant system consisting of N particles with arbitrary masses interacting via Hooke's law pair potentials with the same coupling constant can be solved exactly; explicit solutions are found for the case N=3. Exact solutions are also found explicitly for the translation-invariant problem in which a particle with mass m 0 interacts with N identical particles of mass m 1 via a Hooke's law pair potential with coupling constant k 0 2 , and the identical particles interact with each other via Hooke's law pair potentials with coupling constant k 1 2 . The latter solution provides a basis problem for an energy lower-bound method for translation-invariant atom-like systems.