Time Dependent Problems of the Localized Lattice Vibration

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Abstract
The time dependent problems of the vibrational motion are investigated for the cases of infinitely extended one-dimensional lattice which contains one or two impurity atoms (isotopes). Starting from the equations of motion of these systems, we derive the integral equations which show various dependent properties of the lattice vibration of these perturbed one-dimensional lattices. Namely, the asymptotic solutions of these integral equations represent the localized vibration which is preserved by the impurity atom when its mass is smaller than that of base atoms. The integral equations are actually solved by means of the perturbation calculation and also by the use of Laplace transforms, and the behaviors of the lattice vibration especially the capture of the vibrational energy by the impurity atoms, are examined.

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