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

A 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 V_n, whose basis is formed from the normal modes of vibration of an n-particle chain, and V_2p, 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 V_2p which is associated with the Hermitean inner product in V_n. 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.