Different representations for solving one-dimensional harmonic model of a crystal

One-dimensional harmonic model of an ideal crystalline system composed of particles is considered. For the potential corresponding to a pair interaction of particles with the nearest neighbors, we constructed the fundamental solution in the own basis matrix of this potential. It is shown how to write the solution using Chebyshev polynomials and Bessel functions, as well as to obtain an integral representation on the complex plane and using the Laplace transformation. The application of the results are presented for the potential matrix perturbed with respect to the diagonal elements.