Lattice dynamics with second neighbor interactions. II. Green's formula

Abstract

AbstractA satisfactory definition of spectral density for the normal modes of lattice dynamics problems requires the study of singular recurrence relations which is carried out in detail for one‐dimensional chains with pth neighbor interactions. The relationship of transfer matrices to the dynamical matrix is explored in order to obtain Green's formula. By using Green's formula, a mapping is defined between Vn, whose basis is formed from the normal modes of vibration of an n‐particle chain, and V2p, which is the space of boundary conditions for the recurrsion equations. Most of the properties of this mapping may be deduced from a symplectic bilinear form in V2p which is associated with the Hermitean inner product in Vn. This symplectic form defines a geometry which is invariant under the recursion relation, as well as canonical initial and boundary conditions, and a maximal isotropic subspace which may be used to determine square summability of the normal modes and the spectral density in the limit as the number of particles becomes infinite.