  The properties of a dilute electron gas, coupled to the lattice degrees of
freedom, are studied and compared with the properties of an electron gas at
half-filling, where spinless fermions with two orbitals per lattice site are
considered. The simplest model which includes both the local electron-lattice
interaction of the Jahn-Teller type and the electronic correlations is the
$E\otimes\beta$-Jahn-Teller-Hubbard model. We analyze the formation and
stability of Jahn-Teller polarons and bipolarons, respectively. Our approach is
based on a hopping expansion in the strong-coupling regime. The results are
compared with recently published findings for the Hubbard-Holstein model [1,2].
The special case of the Jahn-Teller-Hubbard model at half-filling is mapped on
a spin-1/2 Heisenberg model with phonon-dependent coupling constants. This has
been derived within a projection formalism that provides a continued-fraction
representation of the Green's function. We study the exact solution for two and
three particles and compare it with the effective theory on the infinite
lattice with one particle per site.
