By using the Lee, Low, and Pines variational method, we have studied the electron-confined phonon, interaction within a rectangular quantum wire under an additional parabolic potential. Formulas for the polaron self-energy, the electron effective mass along the wire, and the ground-state energy are derived. Numerical calculations are performed for a typical GaAs quantum wire within the mesoscopic size using the idea of Fourier decomposition of the wave function. In comparison with previous calculations, our results show that the effect of phonon confinement always reduces the magnitude of the electron-phonon interaction and the associated physical quantities, whereas the additional parabolic potential tends to enhance not only this interaction but also the ground-state energy.
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