  We consider the existence and spectral stability of multi-breather structures
in the discrete Klein-Gordon equation, both for soft and hard symmetric
potentials. To obtain analytical results, we project the system onto a
finite-dimensional Hilbert space consisting of the first $M$ Fourier modes, for
arbitrary $M$. On this approximate system, we then take a spatial dynamics
approach and use Lin's method to construct multi-breathers from a sequence of
well-separated copies of the primary, single-site breather. We then locate the
eigenmodes in the Floquet spectrum associated with the interaction between the
individual breathers of such multi-breather states by reducing the spectral
problem to a matrix equation. Expressions for these eigenmodes for the
approximate, finite-dimensional system are obtained in terms of the primary
breather and its kernel eigenfunctions, and these are found to be in very good
agreement with the numerical Floquet spectrum results. This is supplemented
with results from numerical timestepping experiments, which are interpreted
using the spectral computations.
