  Continuing a line of investigation initiated in [11] exploring the
connections between Jost and Evans functions and (modified) Fredholm
determinants of Birman-Schwinger type integral operators, we here examine the
stability index, or sign of the first nonvanishing derivative at frequency zero
of the characteristic determinant, an object that has found considerable use in
the study by Evans function techniques of stability of standing and traveling
wave solutions of partial differential equations (PDE) in one dimension. This
leads us to the derivation of general perturbation expansions for
analytically-varying modified Fredholm determinants of abstract operators. Our
main conclusion, similarly in the analysis of the determinant itself, is that
the derivative of the characteristic Fredholm determinant may be efficiently
computed from first principles for integral operators with semi-separable
integral kernels, which include in particular the general one-dimensional case,
and for sums thereof, which latter possibility appears to offer applications in
the multi-dimensional case.
  A second main result is to show that the multi-dimensional characteristic
Fredholm determinant is the renormalized limit of a sequence of Evans functions
defined in [23] on successive Galerkin subspaces, giving a natural extension of
the one-dimensional results of [11] and answering a question of [27] whether
this sequence might possibly converge (in general, no, but with
renormalization, yes). Convergence is useful in practice for numerical error
control and acceleration.
