Polaron states in cylindrical and spherical quantum dots with parabolic confinement potentials are investigated applying the Feynman variational principle. It is observed that for both kinds of quantum dots the polaron energy and mass increase with the increase of Fröhlich electron-phonon coupling constant and confinement frequency. In the case of a spherical quantum dot, the polaron energy for the strong coupling is found to be greater than that of a cylindrical quantum dot. The energy and mass are found to be monotonically increasing functions of the coupling constant and the confinement frequency.

Polaron states in a quasi 1D cylindrical quantum wire with a parabolic confinement potential are investigated applying the Feynman variational principle. The effect of the wire radius on the polaron ground state energy level, the mass and the Fröhlich electron-phonon-coupling constant are obtained for the case of a quasi 1D cylindrical quantum wire. The effect of anisotropy of the structure on the polaron ground state energy level and the mass are also investigated.It is observed that as the wire radius tends to zero, the polaron mass and energy diverge logarithmically. The polaron mass and energy differ from the canonical strong-coupling behavior by the Fröhlich electron-phonon coupling constant and the radius of the quasi 1D cylindrical quantum wire that are expressed through a logarithmic function. Moreover, it is observed that the polaron energy and mass for strong coupling for the case of the quasi 1D cylindrical quantum wire are greater than those for bulk crystals. It is also observed that the anisotropy of the structure considerably affects both the polaron ground state energy level and the mass. It is found that as the radius of the cylindrical wire reduces, the regimes of the weak and intermediate coupling polaron shorten while the region of the strong coupling polaron broadens and extends into those of the weak and intermediate ones.Analytic expressions for the polaron ground state energy level and mass are derived for the case of strong coupling polarons.