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Continuing a line of investigation initiated in [11] exploring theconnections between Jost and Evans functions and (modified) Fredholmdeterminants of Birman-Schwinger type integral operators, we here examine thestability index, or sign of the first nonvanishing derivative at frequency zeroof the characteristic determinant, an object that has found considerable use inthe study by Evans function techniques of stability of standing and travelingwave solutions of partial differential equations (PDE) in one dimension. Thisleads us to the derivation of general perturbation expansions foranalytically-varying modified Fredholm determinants of abstract operators. Ourmain conclusion, similarly in the analysis of the determinant itself, is thatthe derivative of the characteristic Fredholm determinant may be efficientlycomputed from first principles for integral operators with semi-separableintegral kernels, which include in particular the general one-dimensional case,and for sums thereof, which latter possibility appears to offer applications inthe multi-dimensional case.  A second main result is to show that the multi-dimensional characteristicFredholm determinant is the renormalized limit of a sequence of Evans functionsdefined in [23] on successive Galerkin subspaces, giving a natural extension ofthe one-dimensional results of [11] and answering a question of [27] whetherthis sequence might possibly converge (in general, no, but withrenormalization, yes). Convergence is useful in practice for numerical errorcontrol and acceleration.

Derivatives of (Modified) Fredholm Determinants and Stability of Standing and Traveling Waves