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Exciton-polaron formation and self-trapping (ST) due to short-ranged coupling with phonons depend strongly on the degree of freedom of exciton motion on a lattice. When excitons can move three-dimensionally, the self-trapped (S) state appears suddenly as a strongly contracted localized state when the coupling constant (g) exceeds a certain critical value. When exciton motion is limited only in one dimension, excitons are always self-trapped and free (F) states are unstable however small g ($NEQ 0) is in the adiabatic approximation. The S state appears as a strongly extended localized state in the limit of g yields 0, and its spatial extension decreases as g increases. For excitons mobile in two dimensions, ST takes place when g exceeds a certain critical value, as in three dimensions. In two dimensions, however, ST begins with either a strongly contracted or a strongly extended S state, depending on whether the exciton-phonon interaction is caused respectively by modulation of site energies of excitons by phonons or by that of transfer integrals of excitons. Spatially extended large-radius S states characterize ST in low dimensions, being called the states of weak self-trapping in tune with weak localization in a low-dimensional-random lattice. Absorption spectra of these states including their phonon structures were investigated, by the dynamical coherent-potential approximation extended so as to incorporate spatially extended lattice distortions. It is shown that even an S state is mobile when it is a large-radius one, forming an energy band of Bloch waves of exciton polarons. Its bandwidth is, however, smaller than the phonon-energy quantum, and decrease to a value much smaller than it as the S state changes itself into a strongly contracted immobile state with increasing g.
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